A survey of the foundations of four-manifold theory in the topological category
AA SURVEY OF THE FOUNDATIONS OF FOUR-MANIFOLD THEORYIN THE TOPOLOGICAL CATEGORY
STEFAN FRIEDL, MATTHIAS NAGEL, PATRICK ORSON, AND MARK POWELL
Abstract.
The goal of this survey is to state some foundational theorems on 4-manifolds,especially in the topological category, give precise references, and provide indications ofthe strategies employed in the proofs. Where appropriate we give statements for manifoldsof all dimensions. Introduction
Here and throughout the paper “manifold” refers to what is often called a “topologicalmanifold”; see Section 2 for a precise definition. Here are some of the statements discussedin this article.(1) Existence and uniqueness of collar neighbourhoods (Theorem 2.5).(2) The Isotopy Extension theorem (Theorem 2.10).(3) Existence of CW structures (Theorem 4.5).(4) Multiplicativity of the Euler characteristic under finite covers (Corollary 4.8).(5) The Annulus theorem 5.1 and the Stable Homeomorphism Theorem 5.3.(6) Connected sum of two oriented connected 4-manifolds is well-defined (Theorem 5.11).(7) The intersection form of the connected sum of two 4-manifolds is the sum of theintersection forms of the summands (Proposition 5.15).(8) Existence and uniqueness of tubular neighbourhoods of submanifolds (Theorems 6.8and 6.9).(9) Noncompact connected 4-manifolds admit a smooth structure (Theorem 8.1).(10) When the Kirby-Siebenmann invariant of a 4-manifold vanishes, both connectedsum with copies of S × S and taking the product with R yield smoothable manifolds(Theorem 8.6).(11) Transversality for submanifolds and for maps (Theorems 10.3 and 10.8).(12) Codimension one and two homology classes can be represented by submanifolds(Theorem 10.9).(13) Classification of 4-manifolds up to homeomorphism with trivial and cyclic funda-mental groups (Section 11).(14) Compact orientable manifolds that are homeomorphic are stably diffeomorphic(Theorem 12.2 and Corollary 12.5).(15) Multiplicativity of signatures under finite covers (Theorem 13.1). a r X i v : . [ m a t h . G T ] J u l S. FRIEDL, M. NAGEL, P. ORSON, AND M. POWELL (16) The definition of Reidemeister torsion for compact manifolds and some of its keytechnical properties (Section 14.3).(17) Obstructions to concordance of knots and links (Theorem 15.2).(18) Poincar´e duality for compact manifolds with twisted coefficients (Theorem A.15 andTheorem A.16).Many of these results are essential tools for the geometric topologist. Our hope is thatwith the statements from this note the “working topologist” will be equipped to handle mostsituations. For many of the topics discussed in this paper the corresponding statementsfor 4-manifolds with a smooth atlas are basic results in differential topology. However forgeneral 4-manifolds, it can be difficult to find precise references. We aim to rectify thissituation to some extent. Throughout the paper we make absolutely no claims of originality.
Conventions. (1) Given a subset A of a topological space X we denote the interior by Int A .(2) For n ∈ N we write D n = { x ∈ R n | (cid:107) x (cid:107) ≤ } for the closed unit ball in R n . Werefer to Int D n = { x ∈ R n | (cid:107) x (cid:107) < } as the open n -ball.(3) Unless indicated otherwise I denotes the interval I = [0 , Acknowledgments.
Thanks to Steve Boyer, Anthony Conway, Jim Davis, Fabian Hebe-streit, Jonathan Hillman, Min Hoon Kim, Alexander Kupers, Markus Land, Tye Lid-man, Erik Pedersen, George Raptis, Arunima Ray, Eamonn Tweedy, and Chuck Liv-ingston for very helpful conversations. We are particularly grateful to Gerrit Herrmannfor providing us with detailed notes which form the basis of the Appendix A. SF was sup-ported by the SFB 1085 ‘Higher Invariants’ at the University of Regensburg, funded bythe Deutsche Forschungsgemeinschaft (DFG). MN gratefully acknowledges support by theSNSF Grant 181199. MP was supported by an NSERC Discovery Grant. SF wishes tothank the Universit´e du Qu´ebec `a Montr´eal and Durham University for warm hospitality.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1. Collar neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. The isotopy extension theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. Smooth 4-manifolds and their intersection forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. CW structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
HE FOUNDATIONS OF 4-MANIFOLD THEORY 3
5. The annulus theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1. The stable homeomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2. The connected sum operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3. Further results on connected sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4. The product structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196. Tubular neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.1. Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2. Tubular neighbourhoods: existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 236.3. Normal vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4. Tubular neighbourhoods: proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287. Background on bundle structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298. Smoothing 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.1. Smoothing non-compact 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.2. The Kirby-Siebenmann invariant and stable smoothing of 4-manifolds . . . . . 359. Tubing of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110. Topological transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.1. Transversality for submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.2. Transversality for maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3. Representing homology classes by submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911. Classification results for 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.1. Simply connected 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.2. Non simply connected 4-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5512. Stable smoothing of homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.1. Kreck’s modified surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2. Stable diffeomorphism of homeomorphic orientable 4-manifolds . . . . . . . . . . . 6212.3. Non-orientable 4-manifolds and stable diffeomorphism . . . . . . . . . . . . . . . . . . . . 6413. Twisted intersection forms and twisted signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6614. Reidemeister torsion in the topological category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.1. The simple homotopy type of a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.2. The cellular chain complex and Poincar´e triads . . . . . . . . . . . . . . . . . . . . . . . . . . . 7114.3. Reidemeister torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7215. Obstructions to being topologically slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.1. The Fox-Milnor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7415.2. A proof of the Fox-Milnor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Appendix A. Poincar´e Duality with twisted coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.1. Twisted homology and cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.2. Cup and cap products on twisted (co-) chain complexes . . . . . . . . . . . . . . . . . . . 83A.3. The Poincar´e duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84A.4. Preparations for the proof of the Twisted Poincar´e Duality Theorem A.15 . 86A.5. The main technical theorem regarding Poincar´e Duality . . . . . . . . . . . . . . . . . . . 88A.6. Proof of the Twisted Poincar´e Duality Theorem A.15 . . . . . . . . . . . . . . . . . . . . . 93References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
S. FRIEDL, M. NAGEL, P. ORSON, AND M. POWELL Manifolds
In the literature the notion of a “manifold” gets defined differently, depending on thepreferences of the authors. Thus we state in the following what we mean by a manifold.
Definition 2.1.
Let X be a topological space.(1) We say that X is second countable if there exists a countable basis for the topology.(2) An n -dimensional chart for X at a point x ∈ X is a homeomorphism Φ : U → V where U is an open neighbourhood of x and(i) V is an open subset of R n or(ii) V is an open subset of the half-space H n = { ( x , . . . , x n ) ∈ R n | x n ≥ } andΦ( x ) lies on E n − = { ( x , . . . , x n ) ∈ R n | x n = 0 } .In the former case we say that Φ is a chart of type (i) in the latter case we say thatΦ is a chart of type (ii).(3) We say that X is an n -dimensional manifold if X is second countable and Hausdorff,and if for every x ∈ X there exists an n -dimensional chart Φ : U → V at x .(4) We say that a point x on a manifold is a boundary point if x admits a chart oftype (ii). (A point cannot admit charts of both types [Hat02, Theorem 2B.3].) Wedenote the set of all boundary points of X by ∂X .(5) An atlas for a manifold X consists of a family of charts such that the domains coverall of X . An atlas is smooth if all transition maps are smooth. A smooth manifold is a manifold together with a smooth atlas. Usually one suppresses the choice of asmooth atlas from the notation.To avoid misunderstandings we want to stress once again that what we call a “manifold”is often referred to as a “topological manifold”. Definition 2.2. An orientation of an n -manifold M is a choice of generators α x ∈ H n ( M, M \{ x } ; Z ) for each x ∈ M \ ∂M such that for every x ∈ M \ ∂M there exists an open neigh-borhood U ⊂ M \ ∂M and a class β ∈ H n ( M, M \ U, Z ) such that β projects to α y for each y ∈ U .Using the cross product one can show that the product of two oriented manifolds admitsa natural orientation. Furthermore, the boundary of an oriented manifold also comes witha natural orientation. The proof of the latter statement is slightly delicate; we refer to[GH81, Chapter 28] or to [Fri19, Chapter 45.9] for details.2.1. Collar neighbourhoods.
We discuss the existence of a collar neighbourhood of theboundary. First we recall the definition of a neighbourhood.
Definition 2.3 (Neighbourhood) . Let X be a space. A neighbourhood of a subset A ⊆ X is a set U ⊆ X for which there is an open set V satisfying A ⊆ V ⊆ U .Next we give our definition of a collar neighbourhood. HE FOUNDATIONS OF 4-MANIFOLD THEORY 5
Definition 2.4 (Collar neighbourhood) . Let M be a manifold and let B be a compactsubmanifold of ∂M . A collar neighbourhood is a map Φ : B × [0 , r ] → M for some r > P ∈ B we have Φ( P,
0) = P ,(3) we have Φ − ( B × [0 , r ]) ∩ ∂M = B .Often, by a slight abuse of language, we identify B × [0 , r ] with its image Φ( B × [0 , r ]) andwe refer to B × [0 , r ] also as a collar neighbourhood.It is a consequence of the invariance of domain theorem that a collar neighbourhoodof ∂M is a neighbourhood of ∂M . Now we can state the collar neighbourhood theoremin the formulation of [Arm70, Theorem 1]. The existence of collars is originally due toBrown [Bro62]. Theorem 2.5 (Collar neighbourhood theorem) . Let M be an n -manifold. Let C be acompact ( n − -dimensional submanifold of ∂M . ( In most cases we take C = ∅ . ) Givenany collar neighbourhood C × [0 , , the restriction to C × [0 , can be extended to a collarneighbourhood ∂M × [0 , . To formulate a uniqueness result for collar neighbourhoods it helps to introduce thefollowing definition.
Definition 2.6.
Let f, g : X → Y be two maps between topological spaces and let Z bea subset of Y . We say f and g are ambiently isotopic rel. Z if there exists an isotopy H = { H t } t ∈ [0 , : Y × [0 , → Y such that H = Id, H t | Z = Id Z for all t and such that H ◦ f = g . Theorem 2.7.
Let M be a compact manifold. Given two collar neighbourhoods Φ : ∂M × [0 , → M and Ψ : ∂M × [0 , → M , their restrictions Φ | ∂M × [0 , and Ψ | ∂M × [0 , are ambi-ently isotopic rel. ∂M × { } .Proof. The theorem is due to [Arm70, Theorem 2], although Armstrong comments thatthe theorem is not new, and the proof he gives was told to him by Lashof. See also [KS77,Essay I, Theorem A.2]. (cid:3)
The following corollary is a straightforward consequence of the Collar neighbourhoodtheorem 2.5. The corollary often makes it possible to reduce arguments about manifoldswith boundary to the case of closed manifolds.
Corollary 2.8. (1)
Let N be an n -manifold, possibly disconnected, and let f : A → B be a homeomor-phism between disjoint collections of boundary components of N . Then the quotient N/ ∼ under the relation a ∼ f ( a ) is an n -manifold with boundary ∂ ( N/ ∼ ) = ∂N \ ( A ∪ B ) . (2) Let M be an n -manifold. Its double DM := M ∪ ∂M = ∂M M is an n -manifold withempty boundary. S. FRIEDL, M. NAGEL, P. ORSON, AND M. POWELL
The isotopy extension theorem.Definition 2.9.
Let X be a k -dimensional manifold and let M be an m -dimensionalmanifold. Let h : X × [0 , → M be a homotopy.(1) We say h is locally flat if for every ( x, t ) ∈ X × [0 ,
1] there exists a neighbourhood[ t , t ] of t and if there are level-preserving embeddings α : D k × [ t , t ] → X × [0 , β : D k × D m − k × [ t , t ] → M × [0 ,
1] to neighbourhoods of ( x, t ) and ( h t ( x ) , t )respectively, such that the following diagram commutes: D k × { } × [ t , t ] α (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) D k × D m − k × [ t , t ] β (cid:15) (cid:15) X × [0 , ( x,t ) (cid:55)→ ( h t ( x ) ,t ) (cid:47) (cid:47) M × [0 , . (2) We say h is proper if for every t ∈ [0 ,
1] we have h t ( X ) ∩ ∂M = h t ( ∂X ).This definition allows us to formulate the following useful theorem [EK71, Corollary 1.4]. Theorem 2.10. (Isotopy Extension Theorem)
Let h : X × [0 , → M be a locally flatproper isotopy of a compact manifold X into a manifold M . Then h can be covered by anambient isotopy of M , i.e. there exists an isotopy H : M × [0 , → M such that H = Id and h t = H t ◦ h for all t ∈ [0 , . As an application of the isotopy extension theorem we will prove the following theorem.We encourage the reader to find a more direct proof.
Proposition 2.11.
Let M be a connected n –manifold. Then for any two points x , y , thereexists a chart φ : U → R n with x, y ∈ U . Especially, the points x , y are connected by alocally flat embedded arc.Proof. Since M is path-connected, there exist points x = x , x , . . . , x k +1 = y such thatthere are charts ( U i , ψ i ) for i = 0 , . . . , k and both x i , x i +1 are contained in U i . Additionally,arrange that y / ∈ U i for i = 0 , . . . , k − h moving x to x k . Pick a function β : [0 , → [0 , β ( t ) = 0 for t ∈ [0 , /
5] and β ( t ) = 1 for t ∈ [4 / , x i to x i +1 by the linearpath in the chart U i , but pass along it with speed determined by β . That is, define h i ( t ) := ψ − i (cid:16) (1 − β ( t )) · ψ i ( x i ) + β ( t ) · ψ i ( x i +1 ) (cid:17) . Consider the composition (of isotopies) h = h ∗ · · · ∗ h k , defined so that h | [ i/k,i +1 /k ] corre-sponds to h i . Each h i is locally flat via the discs D i ( t ) = ψ − i ( B ( ψ i ( h i ( t )))), say. Also, h islocally flat at the times i/k , since it is the constant isotopy of a point in a neighbourhoodof time i/k . Since being locally flat is a local condition, we deduce that h is a locally flatisotopy from x to x k . Since y / ∈ U i for i = 0 , . . . , k −
1, the image of h is disjoint from y . Consequently, we can upgrade h to a locally flat isotopy S × I → M by declaring( − , t ) (cid:55)→ y and (1 , t ) (cid:55)→ h ( t ). The Isotopy Extension Theorem 2.10 yields an ambientisotopy H t : M → M with H t ( x ) = h ( t ) and H t ( y ) = y for all t ∈ I . HE FOUNDATIONS OF 4-MANIFOLD THEORY 7
Now we stretch the last chart U k all the way back to x = x . Define U = ( H ) − ( U k ) andnote that U contains both x and y . The homeomorphism φ = ψ k ◦ H : U → R n definesthe required chart on M .To obtain a locally flat arc connecting x and y , connect the points x and y by a straightline in the chart ( U, φ ). (cid:3) A related result [DV09, Proposition 4.5.1] shows that one can approximate a map of acompact polyhedra into a manifold by a (non-locally flat) embedding.3.
Smooth 4-manifolds and their intersection forms
In this chapter we consider some of the most famous 4-manifolds. Recall that not allsymmetric unimodular pairings over Z can be realised as the intersection forms of closed,smooth 4-manifolds. We discuss some of these limitations below.We start out with the definition of the intersection form. Definition 3.1. (1) Given a finitely generated abelian group H we write F H := H/ torsion subgroup.(2) Given an oriented compact 4-manifold M we refer to the map Q M : F H ( M ; Z ) × F H ( M ; Z ) → Z ( a, b ) (cid:55)→ Q M ( a, b ) := (cid:104) PD − M ( a ) Y PD − M ( b ) , [ M ] (cid:105) as the intersection form .Let E denote the even 8 × E = . Note that this is a symmetric integral matrix with determinant one.
Example . Here are some important closed, smooth 4-manifolds.(1) The 4-sphere S . This is simply connected and has H ( S ; Z ) = { } .(2) The complex projective plane C P , which comes with a canonical orientation. Thesame underlying manifold with the opposite orientation is C P . They are simply-connected manifolds with H ( C P ; Z ) ∼ = Z . The intersection form of C P is (1) andthe intersection form of C P is ( − RP , which is non-orientable and not simply-connected. S. FRIEDL, M. NAGEL, P. ORSON, AND M. POWELL (4) The products S × S and S × S . The manifold S × S is simply-connected and H ( S × S ; Z ) ∼ = Z ⊕ Z . The intersection form of S × S is represented by thestandard hyperbolic form H := (cid:18) (cid:19) .(5) The K K (cid:8) [ z : z : z : z ] ∈ C P (cid:12)(cid:12) z + z + z + z = 0 (cid:9) This is a simply connected, smooth, spin, closed 4-manifold with H ( K Z ) ∼ = Z .As is shown in [GS99, Theorem 1.3.8] or alternatively [MS17, p. 176], the intersectionform of K3 is isometric to E ⊕ E ⊕ H ⊕ H ⊕ H .In Theorem 11.2 we will see that any unimodular symmetric form occurs as the intersec-tion form of a closed oriented 4-manifold. In the following we survey results on intersectionforms of closed oriented smooth 4-manifolds. As we will see, the results in the smoothsetting differ dramatically from the results in the topological setting. Proposition 3.3.
Let M be a closed, oriented, spin 4-manifold. Then the signature sign( M ) is divisible by 8.Proof. Write Sq k for the k th Steenrod square operation. The k th Wu class v k ∈ H k ( M ; Z / k ( a ) = v k ∪ a , for every class a ∈ H − k ( M ; Z / a ∈ H ( M ; Z /
2) then v ∪ a = Sq ( a ) = a ∪ a . But the n th Stiefel-Whitney class of M is given by w n = (cid:80) i Sq i ( v n − i ) (see [MS74, Theorem 11.14]). Since M is oriented and spin, we have 0 = w = Sq ( v ) = v , and 0 = w = Sq ( v ) + Sq ( v ) = v . So for any a ∈ H ( M ; Z / a ∪ a = 0 ∪ a = 0 ∈ Z /
2. But this implies that for any x ∈ F H ( M ; Z ) we havethat Q M ( x, x ) = (cid:104) PD − ( x ) ∪ PD − ( x ) , [ M ] (cid:105) ≡ Q M is an evenform. It is then an algebraic fact, see e.g. [MH73, Theorem 5.1], that for any symmetricnonsingular bilinear even form Q the signature is divisible by 8. (cid:3) Rochlin’s Theorem [Roh52] gives an extra restriction on the signatures of intersectionforms of spin 4-manifolds that admit a smooth structure.
Theorem 3.4 (Rochlin) . Let M be a closed, oriented, spin, smooth 4-manifold. Then thesignature sign( M ) is divisible by 16.Remark . Let M be a closed oriented 4-manifold with an even intersection form andsuch that H ( M ; Z ) has no 2-torsion. This implies H ( M ; Z / ∼ = Hom( H ( M ; Z ) , Z / Q M is isomorphic to the pairing ( a, b ) = (cid:104) a ∪ b, [ M ] (cid:105) on H ( M ; Z / Q M is even, this implies that ( a, a ) = 0 ∈ Z / a ∈ H ( M ; Z / a ∪ a = v ∪ a , so we must have that v = 0 as this pairing isnondegenerate. We also saw above that v = w when M is oriented, so in fact w = 0 and M admits a spin structure.It is not true that simply having an even intersection form implies M is spin. Indeed,it is possible to construct a closed oriented 4-manifold M that has Q M = 0 (which is inparticular an even form), but has non-vanishing w [GS99, Exercise 5.7.7(a)]. In a similar HE FOUNDATIONS OF 4-MANIFOLD THEORY 9 spirit, by [Hab82, FS84] there exists a closed oriented 4-dimensional smooth manifold M with an even intersection form Q M that satisfies sign( M ) = 8. Hence this must also fail tobe spin, now by Rochlin’s theorem. Corollary 3.6.
There exists a closed orientable 4-manifold that does not admit a smoothstructure.Proof.
By Theorem 11.2 there exists a simply connected closed orientable 4-manifold M with Q M ∼ = E . By Rochlin’s Theorem 3.4 this manifold does not admit a smooth structure. (cid:3) In a remarkable twist, shortly after Freedman proved Theorem 11.2, Donaldson [Don83,Theorem A] [Don87, Theorem 1], proved the following result regarding intersection formsof smooth 4-manifolds.
Theorem 3.7 (Donaldson) . Let M be a closed oriented smooth 4-manifold. If Q M ispositive-definite, then Q M can be represented by the identity matrix. To understand the significance of Donaldson’s Theorem it is helpful consider the followingtable from [MH73, p. 28], which basically says that there are lots of isometry types ofnonsingular positive definite forms. Dimension: 8 16 24 32 40Number of isometry types of nonsingularpositive definite even symmetric forms: 1 2 24 ≥ ≥ It follows from [MH73, Theorem II.5.3] that every nonsingular indefinite odd symmetricform is isometric to k · (1) ⊕ (cid:96) · ( − k · C P (cid:96) · C P . Therefore weonly need to discuss the realisability of nonsingular indefinite even symmetric forms. Againby [MH73, Theorem II.5.3], every nonsingular even indefinite symmetric form is isometricto n · E ⊕ m · H for some ( m, n ) ∈ N × Z \ { (0 , } . The following theorem, proved byFuruta [Fur01], gives some restrictions on the possible values of m and n . Theorem 3.8 (Furuta’s 10/8 Theorem) . If M is a closed oriented connected smooth 4-manifold with indefinite even intersection form, then b ( M ) ≥ · | sign( M ) | + 2 . In particular Q M ∼ = n · E ⊕ m · H for some n ∈ Z and m ∈ N with m ≥ | n | + 1 . Furuta’s 10/8 Theorem is not yet optimal since it does not quite close the gap betweenthe forms we can realise by smooth manifolds and the forms we can exclude. More precisely,it follows from the calculation of the intersection form of the K3-surface and of S × S that for any n = 2 p ∈ Z and every m ≥ | p | there exists a closed oriented simply connected4-dimensional smooth manifold with intersection form isometric to n · E ⊕ m · H . In otherwords, we haveintersection form of p · K3 m − | p | ) · ( S × S ) ∼ = 2 p · E ⊕ m · H. The following conjecture predicts that this result is optimal.
Conjecture 3.9 (11/8-Conjecture) . If M is a closed oriented smooth 4-manifold withindefinite even intersection form, then b ( M ) ≥ · | sign( M ) | . Equivalently, if Q M ∼ = 2 p · E ⊕ m · H with p (cid:54) = 0 , then m ≥ | p | .Remark . (1) A proof of the 11/8-Conjecture would imply, by Freedman’s Theorem 11.2, thatany closed oriented simply connected smooth 4-manifold is homeomorphic to eithera connected sum of the form k · C P (cid:96) · C P or to a connected sum of the form n · K3 m · ( S × S ).(2) Currently the best known result in the direction of the 11/8-Conjecture is [HLSX18,Corollary 1.13], which says that if M is a closed oriented simply-connected 4-manifold that is not homeomorphic to S , S × S or the K3-surface and whoseintersection form is indefinite and even, then b ( M ) ≥ · | sign( M ) | + 4.4. CW structures
In this chapter we will discuss the existence of CW-structures on manifolds.
Definition 4.1 (CW complex) . A CW complex is a topological space X together with afiltration ∅ = X − ⊆ X ⊆ X ⊆ X ⊆ · · · with X = colim −−−−−→ X n , such that for each n ≥
0, the space X n arises as a pushout (cid:96) j ∈J n S n − (cid:47) (cid:47) (cid:15) (cid:15) X n − (cid:15) (cid:15) (cid:96) j ∈J n D n (cid:47) (cid:47) X n where J n indexes the discs D n . It is implicit in the statement X = colim −−−−−→ X n that thetopology of X agrees with the weak topology induced from the discs D n . The interiorsInt D n of the discs are called the n -cells . For n ≥
0, a CW complex X is said to be n -dimensional if X n \ X n − (cid:54) = ∅ and X i = X i +1 for all i ≥ n . A manifold M admits aCW-structure if M is homeomorphic to a CW complex.First we discuss the case of smooth manifolds. In [Mun66, Theorem 10.6] and [Whi57,Chapter IV.12] it is shown that every smooth manifold admits a simplicial structure, inparticular it admits a CW-structure. Alternatively, it is shown in [Mil63, Section 3] and[Hir94, Section 6.4] that every compact smooth manifold admits a handle decompositionwhich implies by [Mil63, Theorem 3.5] that every compact smooth manifold is homotopyequivalent to a compact CW-complex.It is natural to ask whether a similar result holds if we drop the smoothness hypothesis.The next theorem summarises what seems to be the state of the art. HE FOUNDATIONS OF 4-MANIFOLD THEORY 11
Theorem 4.2. (1)
For n ≤ every compact n -manifold admits the structure of a finite n -dimensionalCW complex. (2) Let n ≥ and let M be a compact n -manifold. Then M is homeomorphic to themapping cylinder of some map f : ∂M → X , where X is a finite CW complex. (3) For n ≥ , every closed n -manifold admits the structure of a finite n -dimensionalCW complex.Proof. Rad´o [Rad26] showed in 1926 that every compact 2-manifold has a simplicial struc-ture and so in particular has a CW structure. Hatcher’s exposition [Hat13] is well worthreading. Moise [Moi52, Moi77] proved the analogous result for 3-manifolds. See also [Ham76].Since a CW complex is finite if and only if it is compact we have shown (1).For M an n -manifold with boundary, Kirby-Siebenmann [KS77, Essay III.2, Theorem 2.1]showed for n ≥ M has a topological handlebody structure rel. ∂M × I . Quinn [Qui82,Theorem 2.3.1] extended this result to n = 5. Kirby-Siebenmann [KS77, Essay III.2,Theorem 2.2] then says that M is homeomorphic to the mapping cylinder of some map f : ∂M → X , where X is a finite CW complex. Thus (2) holds.In particular, if M is closed, it admits the structure of a finite n -dimensional CW complex,which shows (3). (cid:3) It is not clear to us whether Theorem 4.2 suffices to show that every compact high-dimensional manifold admits a CW structure. Put differently, to the best of our knowledgethe following question is still open for manifolds with nonempty boundary.
Question 4.3.
Let n ≥ . Does every compact n -manifold have a CW structure? Casson [AM90, p. xvi] showed in the 1980s that there exist closed 4-manifolds that donot have a simplicial structure. It is now known that in every dimension n ≥
5, thereexists a closed n -manifold that does not admit a simplicial structure. This question wasreduced to a problem about homology 3-spheres [Mat78, GS80], which was then solved byManolescu [Man16]. Note that a simplicial structure is not necessarily a PL structure; an n -dimensional PL structure satisfies the additional condition that the link of an m -simplexbe homeomorphic to an ( n − m − Question 4.4.
Does every compact -manifold have a CW structure? We have the following theorem regarding CW structures and manifolds. Given The-orem 4.2, the first statement is only of interest in dimension 4. The second statementappears in [Wal67, Theorem 2.2] for n ≥ Theorem 4.5. (1)
Every compact n -manifold is homotopy equivalent to an n -dimensional finite CWcomplex. (2) Every connected compact n -manifold with nonempty boundary is homotopy equiva-lent to an ( n − -dimensional finite CW complex. In the proof of Theorem 4.5 we will use the following theorem proved by Wall [Wal66,Corollary 5.1].
Theorem 4.6.
Let X be a finite connected CW complex. Suppose that there is an integer n ≥ such that H i ( X ; Z [ π ]) = 0 for all i > n . Then X is homotopy equivalent to an n -dimensional finite CW complex. The proof of Theorem 4.5 also makes use of the following definition. Recall that a spaceis metrisable if it admits the structure of a metric space inducing the given topology.
Definition 4.7 (Absolute Neighbourhood Retract (ANR)) . A space X is called an absoluteneighbourhood retract if X is metrisable and if whenever X ⊆ Y is a closed subset of ametrisable space Y , then X is a neighbourhood retract of Y . That is, there is an openneighbourhood U ⊆ Y containing X , with a map r : U → X such that the composition X → U r −→ X is equal to the identity on X . Proof of Theorem 4.5.
Hanner [Han51, Theorem 3.3] showed that every manifold is anANR, and West [Wes77] showed that every compact ANR is homotopy equivalent to afinite CW complex. (An alternative proof that every compact manifold has the homotopytype of a finite CW complex is given in Kirby-Siebenmann [KS69, Section 1 (III)].)Let M be a compact n -manifold. By the results above, M is homotopy equivalent to afinite CW complex X . By Theorem 4.2 we need to complete the proof of (1) only in thecase n = 4. But since the subsequent argument works for all n ≥ n ≥
4. Since M is n -dimensional it follows from Universal Poincar´e duality (Theorem A.15)that for any k > n and any Z [ π ( X )]-module Λ we have H k ( X ; Λ) ∼ = H n − k ( X ; Λ) = 0 . By Theorem 4.6, X is homotopy equivalent to an n -dimensional finite CW complex. Notethat to apply Theorem 4.6 we have used that n ≥ n = 1 , n -manifold admits a simplicial structure. It iswell known that a compact connected n -manifold with nonempty boundary and a simplicialstructure is homotopy equivalent to an ( n − n − n ≥
4. By (1) we know that if M is a connected n -manifold withnonempty boundary, then it admits the structure of a finite CW complex. Let k ≥ n andlet Λ be a Z [ π ( X )]-module. By Poincar´e-Lefschetz duality (Section A) we have H k ( X ; Λ) ∼ = H n − k ( X, ∂X ; Λ) = 0 . Here the last conclusion is obvious for k > n . For k = n the conclusion follows fromthe fact that ∂X (cid:54) = ∅ , that X is connected and the explicit calculation of 0-th twisted HE FOUNDATIONS OF 4-MANIFOLD THEORY 13 homology groups as given in [HS97, Chapter VI.3]. It follows from Theorem 4.6 that X is homotopy equivalent to an ( n − n ≥ (cid:3) Theorem 4.5 is strong enough to recover many familiar statements.
Corollary 4.8.
Let M be a compact connected manifold. (1) The group π ( M ) is finitely presented. (2) All homology groups H k ( M ; Z ) are finitely generated abelian groups, in particular itmakes sense to define the Euler characteristic χ ( M ) := (cid:80) n ( − n · b n ( M ) . (3) Let p : (cid:102) M → M be a finite covering. Then χ ( (cid:102) M ) = [ (cid:102) M : M ] · χ ( M ) . Remark . Let M be a compact, connected manifold. Borsuk’s theorem that M is aEuclidean Neighbourhood Retract shows that M is a retract of a finite CW complex;see [Hat02, Appendix A, Corollary A.9], [Bre97, Appendix E], and [Fri19, Proposition 65.22].This fact is nontrivial but it is much easier to prove than Theorem 4.5. Borsuk’s Theoremimplies immediately that the homology groups of M are finitely generated and that thefundamental group of M is finitely generated. In fact using a group theoretic lemma asin [Wal65, Lemma 1.3] or alternatively [FR01, Theorem 3.1], one actually obtains that π ( M ) is finitely presented. But it is not clear how Borsuk’s Theorem can be used to proveCorollary 4.8 (3). Proof.
The first two statements in the corollary are an immediate consequence of Theo-rem 4.5 and standard results on fundamental groups and homology groups of finite CWcomplexes. We turn to the final statement. Let X be a finite CW complex homotopyequivalent to M . Use the fact that the Euler characteristic is multiplicative for finite cov-ers of finite CW complexes and use that a k -fold cover (cid:102) M of M induces a k -fold cover (cid:101) X of X such that (cid:102) M and (cid:101) X are homotopy equivalent, to deduce the result. (cid:3) Remark . As pointed out above, every compact smooth manifold admits the structureof a finite CW complex. One can combine this fact with Theorem 8.6 below to obtain analternative proof of Corollary 4.8. Theorem 8.6 says that for any 4-manifold M there is a4-manifold N such that the connected sum M N admits a smooth structure.5. The annulus theorem
The annulus theorem is a fundamental result in the development of the theory of mani-folds. In high dimensions, it underpins the product structure theorem [KS77, Essay I, The-orem 5.1], which itself underpins all the results of [KS77]. We state the product structuretheorem in Section 5.4. In dimension four, the annulus theorem is needed for the proofs ofsmoothing theorems (Section 8), existence and uniqueness of normal bundles (Section 6), and transversality (Section 10). We discuss these developments in later sections. Laterin this section (Section 5.2), we will discuss a more immediate application: showing thatconnected sum is a well-defined operation on connected topological manifolds. Here is theannulus theorem.
Theorem 5.1 (Annulus theorem) . Let n ∈ N and let f, g : D n → R n be two orientation-preserving locally flat embeddings. If f ( D n ) ⊂ Int( g ( D n )) , then g ( D n ) \ Int( f ( D n )) ishomeomorphic to S n − × [0 , . For n = 2 , (cid:54) = 4 byKirby [Kir69] and in dimension 4 by Quinn [Qui82, p. 506]; see also [Edw84, p. 247].The known proofs of the annulus theorem deduce it from the stable homeomorphismtheorem. In the next section we will state the stable homeomorphism theorem 5.3 and wewill show how the annulus theorem can be deduced from that theorem.5.1. The stable homeomorphism theorem.
We reduce the annulus theorem to thestable homeomorphisms theorem stated in Theorem 5.3. This follows from work of Brownand Gluck [BG64b], but since it requires some work to find this deduction in [BG64b], wegive the details here.
Definition 5.2.
A homeomorphism f : R n → R n is said to be stable if there is a sequenceof homeomorphisms f , . . . , f n : R n → R n such that f n ◦ · · · ◦ f = f and such that for each i , the homeomorphism f i is somewhere the identity , which means that there is an opennonempty set U ⊆ R n such that f i | U is the identity on U .The key ingredient to the subsequent discussion is the following theorem. Theorem 5.3 (Stable homeomorphism theorem) . Let n ∈ N . Every orientation preservinghomeomorphism from R n to itself is stable. For n ≥ n = 4 the stablehomeomorphism theorem was proven by Quinn, see also [Edw84, p. 247].Before we discuss consequences of the Stable homeomorphism theorem 5.3 we recall thetwo versions of the Alexander trick. Lemma 5.4 (Alexander trick) . (1) Every homeomorphism of S n − can be extended radially to a homeomorphism of D n that sends 0 to 0. (2) Let f and g be two homeomorphisms of D n . If the restrictions of f and g to S n − are isotopic, then f and g are isotopic homeomorphisms of D n . HE FOUNDATIONS OF 4-MANIFOLD THEORY 15
The extension in the first statement can be obtained by coning: f ( t · x ) = t · f ( x ). Thesecond one is an amusing exercise; see [Han89, Lemma 5.6] for a proof. The same ideaextends to show that the topological group Homeo ∂ ( D n ) of homeomorphisms of D n fixingthe boundary pointwise is contractible.We can now prove the following almost immediate consequence of the Stable homeomor-phism theorem 5.3. Corollary 5.5.
Every orientation preserving self-homeomorphism of S n is isotopic to theidentity.Proof. We identify S n with R n ∪ {∞} . Let h be a self-homeomorphism of S n = R n ∪ {∞} .After an isotopy (using Theorem 2.10) we can assume that h ( ∞ ) = ∞ . By the Stablehomeomorphism theorem 5.3 we know that h is stable. Thus we only have to consider thecase that h fixes an open subset of R n ∪ {∞} = S n . After an isotopy we can assume that h fixes an open neighbourhood of ∞ , so in particular there exists C > h is theidentity on { x ∈ R n | (cid:107) x (cid:107) ≥ C } . It follows from Lemma 5.4 (2) that h is isotopic to theidentity. (cid:3) Denote the set of locally flat embeddings of D n into R n by Emb( D n , R n ); see Defini-tion 6.2 for the definition of locally flat. Definition 5.6.
We say that two elements f , f ∈ Emb( D n , R n ) are intertwined if thereexists an h ∈ Homeo( R n , R n ) with h ◦ f = f .We will need the following straightforward technical lemma. Lemma 5.7.
Let M be an n -dimensional manifold and let f : D n → M be a locally flatembedding into Int M = M \ ∂M . Then there exists a locally flat embedding F : R n → M such that the restriction of F to D n equals f .Proof. Let f : D n → M be a locally flat embedding. By definition f ( D n ) is a submanifoldof M . It is straightforward to see that W := M \ f (Int D n ) is also a submanifold of M . Bythe collar neighbourhood theorem 2.5 there exists a collar f ( S n − ) × [0 , F : R n → Mx (cid:55)→ (cid:26) f ( x ) , if x ∈ D n , (cid:0) f ( y ) , π arctan( t − (cid:1) if x = t · y with t ∈ [1 , ∞ ) and y ∈ S n − , is easily seen to be a locally flat embedding. (cid:3) Lemma 5.8.
Any two elements f , f ∈ Emb( D n , R n ) are intertwined.Proof. It suffices to show that any f ∈ Emb( D n , R n ) is intertwined with the standardembedding D n ⊂ R n . So let f ∈ Emb( D n , R n ). Apply Lemma 5.7 to extend f to a locallyflat embedding F : D n ( ) → R n . Note that F restricts to a locally flat embedding of S n − × [ , ] into S n = R n ∪ {∞} . Let (cid:101) D n be another copy of D n . By the generalised Schoenfliestheorem [Bro60, Theorem 5] there exists a homeomorphism g : (cid:101) D n → S n \ f (Int (cid:101) D n ). Sincethe homeomorphisms of D n act transitively on the interior of (cid:101) D n , arrange that g (0) = ∞ . Note that g − ◦ f : S n − → S n − is a homeomorphism. By Lemma 5.4 (1) this homeo-morphism extends to a homeomorphism φ of D n . Replace g by g ◦ φ if necessary to obtainthat f = g : S n − → f ( S n − ). Identify S n = R n ∪ {∞} = D n ∪ (cid:101) D n in such a way that0 ∈ (cid:101) D n corresponds precisely to ∞ . Consider the map F : S n = D n ∪ (cid:101) D n → S n x (cid:55)→ (cid:40) f ( x ) x ∈ D n g ( x ) x ∈ (cid:101) D n . The maps f and g agree on the overlap, so the map is well-defined and is a homeomorphism.Note that F restricts to a homeomorphism of R n which has the property that the restrictionto D n equals f . This shows that F ◦ Id = f so Id and f are intertwined. (cid:3) Definition 5.9.
Let f , f ∈ Emb( D n , R n ) with f ( D n ) ⊂ Int f ( D n ). We say f and f are strictly annularly equivalent if there exists a map F : S n − × I → M that is ahomeomorphism onto its image such that F ( x,
0) = f ( x ) and F ( x,
1) = f ( x ) for all x ∈ S n − . Theorem 5.10.
Let f , f ∈ Emb( D n , R n ) with f ( D n ) ⊂ Int f ( D n ) . If f and f areorientation preserving, then they are strictly annularly equivalent if and only if they areintertwined.Proof. If two such elements are strictly annularly equivalent, then they are intertwinedby [BG64b, Theorem 5.2].Now suppose that f and f are intertwined, that is there exists an h ∈ Homeo( R n , R n )with h ◦ f = f . By the Stable homeomorphism theorem 5.3 we know that h is stable.Thus we know from [BG64b, Theorem 5.4] that the embeddings are annularly equivalent,i.e. there exist h , . . . , h k ∈ Emb( D n , R n ) such that h = f , h k = f and for each i the maps h i and h i +1 are strictly annularly equivalent. Since f ( D n ) ⊂ Int f ( D n ), the embeddings ofthe boundary spheres f ( ∂D n ) and f ( ∂D n ) are disjoint. Therefore it follows from [BG64a,Theorem 3.5] that f and f are not only annularly equivalent, but are moreover strictlyannularly equivalent. (cid:3) Now we can easily prove the Annulus theorem 5.1.
Proof of the Annulus theorem 5.1.
Let f , f : D n → R n be two orientation-preserving lo-cally flat embeddings with f ( D n ) ⊂ Int( f ( D n )). By Lemma 5.8 and Theorem 5.10 thetwo maps f and f are strictly annularly equivalent. But this immediately implies that f ( D n ) \ Int( f ( D n )) is homeomorphic to S n − × [0 , (cid:3) The connected sum operation.
We recall the construction of the connected sumof two connected oriented n -manifolds M and N . Pick two orientation preserving locallyflat embeddings of n -balls Φ M : D n → M and Φ N : D n → N . Define the connected sum M N of M and N by M N := ( M \ Φ M (Int( D n ))) ∪ Φ M ( S n − )=Φ N ( S n − ) ( N \ Φ N (Int( D n ))) HE FOUNDATIONS OF 4-MANIFOLD THEORY 17 where we glue the left hand side to the right hand side via the mapΦ N ◦ Φ − M : Φ M ( S n − ) ∼ = −→ Φ N ( S n − ) . It follows from the Collar neighbourhood theorem 2.5 that the topological space M N inherits the structure of an n -manifold; see [Lee11, Proposition 6.6] for details. Furthermore M N can be oriented in such a way that M and N are oriented submanifolds. Theorem 5.11.
The connected sum M N of two connected oriented n -manifolds M and N is independent of the choice of embeddings of the n -balls.Remark . The manifolds C P C P and C P C P have different intersection forms,so they are not homeomorphic; see Proposition 5.15. Thus connected sum is not well-defined on orientable 4-manifolds, rather it depends on the choice of orientation. On theother hand for nonorientable manifolds, the connected sum is well-defined. As discussedin [BCF + Lemma 5.13.
Let D nr ( x ) and D ns ( y ) be two Euclidean balls in R n . There exists a homeo-morphism f : R n → R n with f ( D nr ( x )) = D ns ( y ) such that f is the identity outside of somecompact set. The next lemma is a consequence of the annulus theorem 5.1.
Lemma 5.14.
Let ϕ, ψ : D n → R n be two orientation-preserving locally flat embeddings. If ϕ ( D n ) ⊂ Int( ψ ( D n )) , then there exists a homeomorphism f of R n with f ( ϕ ( D n )) = ψ ( D n ) such that f is the identity outside of some compact set.Proof. By the annulus theorem 5.1 and the collar neighbourhood theorem 2.5 we can finda locally flat embedding θ : S n − × [ − ,
2] such that θ ( S n − × [ − , ϕ ( D n ),such that θ ( S n − × [0 , ψ ( D n ) \ ϕ (Int D n ) and such that θ ( S n − × [1 , R n \ ψ (Int D n ). It is now obvious that we can find a homeomorphism f with f ( ϕ ( D n )) = ψ ( D n ) which is the identity outside of θ ( S n − × [ − , (cid:3) The subsequent proof is partly based on the sketch given in [Rol90, p. 42].
Proof of Theorem 5.11.
We have to show that the connected sum is independent of thechoice of Φ M : D n → M and Ψ N : D n → N . By symmetry and transitivity it suffices toshow that the connected sum is independent of the choice of Φ M . So suppose we are giventwo orientation-preserving embeddings Φ : D n → M and Φ : D n → M and suppose weare given an orientation-preserving embedding Ψ : D n → N . We introduce the followingnotation.(1) Write D i := Φ i ( D n ). (2) Let X i := M \ Φ i (Int D n ) and let Y := N \ Ψ(Int D n ),(3) Denote the restriction of Φ i to S n − by ϕ i and denote the restriction of Ψ to S n − by ψ .Figure 1 hopefully makes it easier for the reader to internalise the notation. We have toshow that there exists a homeomorphism( X ∪ Y ) /ϕ ( x ) ∼ ψ ( x ) → ( X ∪ Y ) /ϕ ( x ) ∼ ψ ( x )where the gluing on both sides is given by taking x ∈ S n − . 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) M Ψ X ψ Cϕ D Φ Φ D Figure 1.
Claim.
There exists a homeomorphism h of M so that h ( D ) = D .First note that it follows from Lemma 5.7 and Lemma 5.13, together with our hypothesisthat M is path connected, that there exists a homeomorphism µ of M such that µ ( D ) ⊂ Int D . Then apply Lemma 5.7 and Lemma 5.14 to find a homeomorphism ν of M suchthat ν ( µ ( D )) = D . This concludes the proof of the claim.It follows from the claim that ϕ − ◦ ϕ is a homeomorphism of S n − . By Corollary 5.5we know that there exists an isotopy H : S n − × [0 , → S n − from ϕ − ◦ ϕ to the identity.We write C := Φ( S n − ). By the collar neighbourhood theorem 2.5 we can pick a collar C × [0 ,
1] for Y . It is straightforward to verify that( X ∪ Y ) /ϕ ( x ) ∼ ψ ( x ) → ( X ∪ Y ) /ϕ ( x ) ∼ ψ ( x ) P (cid:55)→ h ( P ) , if P ∈ X ,ψ ( H ( ψ − ( Q ) , t )) if P = ( Q, t ) ∈ C × [0 , P, if Y \ P ∈ C × [0 , and Φ give rise to homeomorphic manifolds. (cid:3) Further results on connected sums.
The definition of the intersection form wasgiven in Definition 3.1. The next proposition shows that the intersection form is wellbehaved under the connected sum operation.
Proposition 5.15.
Let M and N be two oriented compact -manifolds. Then Q M N isisometric to Q M ⊕ Q N . HE FOUNDATIONS OF 4-MANIFOLD THEORY 19
Proof.
In the smooth case this statement follows immediately from the fact that any class insecond homology can be represented by an embedded oriented submanifold [GS99, Propo-sition 1.2.3] and the fact that one can calculate the intersection form in terms of algebraicintersection numbers of embedded oriented surfaces [Bre97, Theorem VI.11.9]. To applythis approach to general manifolds, one needs to use topological transversality, which holds,as discussed in Section 10.On the other hand the statement for general manifolds (and thus also for smooth mani-folds) can be proved directly with the usual tools of algebraic topology, namely functorialityof the cup and cap products [Bre97, Theorem VI.5.2.(4)] for maps between pairs of topolog-ical spaces, a Mayer-Vietoris argument and the excision theorem. Full details are providedin [Fri19, Proposition 72.11]. (cid:3)
The product structure theorem.
The product structure theorem [KS77, Essay I,Theorem 5.1], is a key result for the development of topological manifold theory in highdimensions. It is a consequence of the stable homeomorphism theorem 5.3, and in turnis used in [KS77] to deduce handle structures, transversality, smoothing theory, and theexistence of a canonical simple homotopy type, for high dimensional ( n ≥
6) manifolds. Wewill give some examples of the use of the product structure theorem below. Even thoughit is only for high dimensional manifolds, it still appears in the development of the theoryof 4-manifolds.The product structure theorem will be stated for upgrading to either a smooth or PLstructure. A concordance of (smooth, PL) structures Σ , Σ (cid:48) on a manifold N is a (smooth,PL) structure Ω on N × I that restricts to Σ on N × { } and restricts to Σ (cid:48) on N × { } . Theorem 5.16 (Product structure theorem) . Let M be a manifold of dimension n ≥ .Let Σ be a ( smooth, PL ) structure on M × R s , with s ≥ . Let U be an open subset of M with a ( smooth, PL ) structure ρ on U such that ρ × R s = Σ | U × R s . If n = 5 then supposethat ∂M ⊂ U .Then there is a ( smooth, PL ) structure σ on M extending ρ , together with a concor-dance of (smooth,PL) structures from Σ to σ × R s , that is a product concordance in someneighbourhood of U × R s and that is a product near M × R s × { i } for i = 0 , .Remark . The statement of the product structure theorem was modelled on the Cairns-Hirsch theorem [KS77, Essay I, Theorem 5.3], which was proven in the early 1960s, andprovided the analogous upgrade from PL structures to smooth structures. See [HM74]for a comprehensive treatment of smoothing theory for PL manifolds. The Cairns-Hirschtheorem tells us that if M already has a PL structure (cid:36) , such that (cid:36) × R s is Whiteheadcompatible (see the discussion below [KS77, Essay I, Theorem 5.3] for details) with a smoothstructure Σ on M × R s , then the smooth structure σ on M produced by Theorem 5.16 isWhitehead compatible with (cid:36) .In Section 14.1 on the simple homotopy type of a manifold we make use of the followingstronger local version. Theorem 5.18 (Local product structure theorem) . Let M be a manifold of dimension n ≥ . (i) Let W be an open neighbourhood of M × { } in M × R s , for some s ≥ . (ii) Let Σ be a ( smooth, PL ) structure on W . (iii) Let C ⊆ M × { } be a closed subset such that there is a neighbourhood N ( C ) of C on which the ( smooth,PL ) structure Σ is a product Σ | N ( C ) = σ × R s for some ( smooth,PL ) structure σ on M . If n = 5 then suppose that ∂M ⊆ C . (iv) Let D ⊂ M × { } be another closed subset. (v) Let V ⊆ W be an open neighbourhood of D \ C .Then we have the following. (1) A ( smooth, PL ) structure Σ (cid:48) on W that equals Σ on ( W \ V ) ∪ (( C × R s ) ∩ W ) and isa product ( smooth,PL ) structure ρ × R s on ( N ( D ) × R s ) ∩ W for some neighbourhood N ( D ) of D and for some ( smooth, PL ) structure ρ on N ( D ) . (2) A concordance of ( smooth, PL ) structures from Σ to Σ (cid:48) , that is a product concor-dance on some neighbourhood of ( W \ V ) ∪ (( C × R s ) ∩ W ) and that is a productnear W × { i } for i = 0 , . Note that the concordance implies isotopy theorem [KS77, Essay I, Theorem 4.1] meansthat the concordances in Theorems 5.16 and 5.18 can be upgraded to isotopies of (smooth,PL) structures under the same hypotheses on dimensions, that is n ≥ n = 5 and thestructures already agree on ∂M .6. Tubular neighbourhoods
Submanifolds.
Every smooth submanifold of a smooth manifold admits a normalvector bundle and, by the smooth Tubular neighbourhood theorem, also admits a tubularneighbourhood [Hir94, Sections 5 & 6][Wal16, Chapter 2.5]. However in the topologicalcategory n -manifolds may not admit normal vector bundles, a general problem we discussfurther below and in Section 7 once we have developed the necessary language. Curiously,in the special case of 4-manifolds these general problems do not exist, and familiar smoothresults hold true using an appropriate notion of normal vector bundles (Definition 6.15).Before discussing tubular neighbourhoods and normal vector bundles in the topologicalcategory, we give our convention for submanifolds. Recall that E n − ⊂ R n is the hyper-plane { x n = 0 } . Definition 6.1.
Let M be an n -dimensional manifold. We say a subset X ⊂ M is a k -dimensional submanifold if given any P ∈ X one of the holds:(1) there exists a chart Φ : U → V of type (i) for M and P such thatΦ( U ∩ X ) ⊂ { (0 , . . . , , x , . . . , x k ) | x i ∈ R } , (2) there exists a chart Φ : U → V of type (ii) for M and P such that Φ( P ) lies in E n − and Φ( U ∩ X ) ⊂ { (0 , . . . , , x , . . . , x k ) ∈ R n | x k ≥ } , HE FOUNDATIONS OF 4-MANIFOLD THEORY 21 (3) or there exists a chart Φ : U → V of type (i) for M and P such that Φ( P ) lies in E n − and Φ( U ∩ X ) ⊂ { (0 , . . . , , x , . . . , x k ) ∈ R n | x k ≥ } . If for every P ∈ X we can find charts as in (1) and (2), then we call M a proper submanifold. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) M VX U x Φ Figure 2.
Definition 6.2.
A map f : X → M from a k -manifold to an m -manifold M is called a(proper) locally flat embedding if f is a homeomorphism onto its image and if the image isa (proper) submanifold of M .Given any submanifold X of M , the inclusion map X → M is a locally flat embedding.Conversely, if f : X → M is a locally flat embedding, then the image f ( X ) is a submanifold. Remark . (1) Note that if M is a k -manifold and U is an open subset of R k , then it follows fromthe invariance of domain that the image any injective map f : U → M is an opensubset of M . In particular f ( U ) is a submanifold of M . Put differently, f is locallyflat.(2) In point set topology, one often defines a topological embedding to be a map f : X → Y of topological spaces that is a homeomorphism to its image. The image of a topo-logical embedding is not necessarily a submanifold and such an image is sometimescalled wild due to the bizarre properties that such objects can exhibit. For exam-ple, the famous Alexander horned sphere [Ale24] is not a submanifold of S underDefinition 6.1, but it is the image of a wild topological embedding S → S .(3) In the literature a compact subset F of 4-manifold is often called a locally flat surface if F is homeomorphic to a compact 2-dimensional manifold with ∂F = F ∩ ∂M andif F has the following properties.(a) Given any P ∈ F \ ∂F there exists a topological embedding ϕ : D × D → M \ ∂M with ϕ ( D × D ) ∩ F = ϕ ( D × { } ) and with P ∈ ϕ ( D × { } ).(b) Given any P ∈ ∂F there exists a topological embedding ϕ : D ≥ × D → M suchthat ϕ ( D ≥ × D ) ∩ F = ϕ ( D ≥ × { } ), ϕ ( D ≥ × D ) ∩ ∂M = ϕ ( ∂ y =0 D ≥ × D ),and with P ∈ ϕ ( D ≥ × { } ). Here, we used the following abbreviations D ≥ = { ( x, y ) ∈ D | y ≥ } and ∂ y =0 D = { ( x, ∈ D } .It follows easily from the definitions that F ⊂ M is a locally flat surface if and onlyif F is proper submanifold of M . The following is a common justification for requiring slice discs to be locally flat in thetheory of knot concordance.
Proposition 6.4.
Given a knot K ⊂ S the corresponding cone Cone( K ) := { r · Q | Q ∈ K and r ∈ [0 , } ⊂ D . is locally flat if and only if K is the unknot.Proof. Consider the specific unknot U that is the equator of the equator U = S ⊂ S ⊂ S = ∂D . Taking the cone radially inwards to the origin of D exhibits cone( U ) as alocally flatly (properly) embedded disc. Any other unknotted K ⊂ S is related to U by a homeomorphism of S . By the Alexander trick 5.4(1), this homeomorphism extendsradially inwards to a homeomorphism of D fixing the origin. Thus the cone on any otherunknot K is locally flatly embedded, as we obtain a chart as in Definition 6.1(1) at theorigin of D .Conversely, suppose K ⊂ S is a knot such that C := Cone( K ) is locally flat. Thisimplies that there is a chart Φ : D → D such that Φ(0) = P and such that Φ( D × { } ) =Φ( D ) ∩ C . We introduce the following notation.(i) Given I ⊂ [0 ,
1] we write D I := { v ∈ D | (cid:107) v (cid:107) ∈ I } .(ii) Given I ⊂ [0 ,
1] we write N I := Φ( D I ).An elementary argument shows that there exist s < t < s < t < s such that D [0 ,s ] ⊂ N [0 ,t ] ⊂ D [0 ,s ] ⊂ N [0 ,t ] ⊂ D [0 ,s ] . We make the following observations:(1) For any I ⊂ [0 ,
1] we have homeomorphisms D I \ C ∼ = ( S \ K ) × I and N I \ C Φ − −−→ D I \ U ∼ = ( S \ U ) × I .(2) For any inclusion I ⊂ J of intervals the inclusion induced maps D I → D J and N I → N J are homotopy equivalences.We consider the following commutative diagram where all maps are induced by inclusions π ( S \ K ) ∼ = π ( D { t } \ C ) ∼ = (cid:42) (cid:42) (cid:47) (cid:47) π ( N [ t ,t ] \ C ) ∼ = Z (cid:117) (cid:117) π ( D [ t ,t ] \ C ) . Since the inclusion D { t } \ C → D [ t ,t ] \ C is a homotopy equivalence we see that the leftdiagonal map is an isomorphism. Thus we see that we have an automorphism of π ( S \ K )that factors through Z . Since the abelianisation of π ( S \ K ) is isomorphic to Z we seethat π ( S \ K ) ∼ = Z . It follows from the Loop Theorem that K is in fact the unknot [Rol90,Theorem 4.B.1]. (cid:3) In some applications one needs the following refinement of the Collar neighbourhoodtheorem 2.5.
HE FOUNDATIONS OF 4-MANIFOLD THEORY 23
Theorem 6.5 (Collar neighbourhood theorem for proper submanifolds) . Let M be a mani-fold and let X ⊂ M be a proper submanifold. There exists a collar neighbourhood ∂M × [0 , such that ( ∂M × [0 , ∩ X is a collar neighbourhood for ∂X ⊂ X .Proof. By the earlier Collar neighbourhood theorem 2.5 we can pick a collar neighbourhood ∂M × [0 ,
2] for ∂M and we can also pick a collar neighbourhood ∂X × [0 ,
2] for ∂X . Given t ∈ [0 ,
1] we consider the obvious homeomorphisms f t : M = ( M \ ( ∂M × [0 , ∪ ( ∂M × [0 , → ( M \ ( ∂M × [0 , ∪ ( ∂M × [ t, g t : X = ( X \ ( ∂X × [0 , ∪ ( ∂X × [0 , → ( X \ ( ∂X × [0 , ∪ ( ∂X × [ t, . Next we consider the following proper locally flat isotopy: h : X × [0 , → M ( x, t ) (cid:55)→ (cid:26) ( y, s ) ∈ ∂M × [0 , t ] , if x = ( y, s ) with y ∈ ∂X, s ∈ [0 , t ] ,f t ( g − t ( x )) , otherwise.Note that the collar neighborhood ∂M × [0 ,
1] is of the desired form for the proper sub-manifold h ( X ). By the Isotopy Extension Theorem 2.10 we can extend h to a isotopy H of M . Thus H − ( ∂M × [0 , M . (cid:3) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) ∂M × [0 , M∂X × [0 , h ( X ) X Figure 3.
Illustration of the proof of the Collar neighbourhood theorem 6.5.6.2.
Tubular neighbourhoods: existence and uniqueness.
In the literature one canfind many different definitions of tubular neighbourhoods for smooth manifolds. We willgive a definition for manifolds that is modelled on the definition provided by Wall [Wal16]for smooth manifolds. To do so we first need one extra definition.
Definition 6.6.
Let M be an n -dimensional manifold. We say a subset W ⊂ M is a k -dimensional submanifold with corners if given any P ∈ W there exists a chart of thetype (1), (2) or (3) as in Definition 6.1 above, or if(4) there exists a chart Φ : U → V of type (ii) for M such thatΦ( U ∩ W ) ⊂ { (0 , . . . , , x , . . . , x k ) | x i ∈ R with x k − ≥ x k ≥ } and with Φ( P ) ∈ { (0 , . . . , , x , . . . , x k − , , | x i ∈ R } . If W is an n -dimensional submanifold with corners we write ∂ W := W ∩ M \ W , ∂ W := W ∩ ∂M, and we note that Int W = W \ ∂ W. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) corner x k − x k M M ∂ W∂MW ∂ W Figure 4.
Definition 6.7.
Let M be an n -manifold and let X be a compact proper k -dimensionalsubmanifold. A tubular neighbourhood for X is a pair ( N, p : N → X ) with the followingproperties:(1) N is a codimension zero submanifold with corners.(2) The map p : N → X is a linear D n − k -bundle such that p ( x ) = x for all x ∈ X .(3) We have ∂ N = p − ( ∂X ).Here linear means that there exists an atlas of trivialisations such that the transitionmaps take values in O ( n − k ) instead of Homeo( D n − k ).In the topological category, tubular neighbourhoods do not always exist. Indeed it isshown in [Hir68, Theorem 4] that there exists a 4-dimensional submanifold of S that doesnot admit a tubular neighbourhood.Fortunately, for submanifolds of 4-manifolds, tubular neighbourhoods exist and they areunique in the appropriate sense. Theorem 6.8 (Tubular neighbourhood theorem) . Every compact proper submanifold X ofa -manifold M admits a tubular neighbourhood. Theorem 6.9. (Uniqueness of tubular neighbourhoods)
Let M be a -manifold and let X be a compact proper k -dimensional submanifold. Furthermore let p i : N i → X , i = 1 , betwo tubular neighbourhoods of X , with inclusion maps ι i : N i → M . Then there exists anisomorphism Ψ : N → N of linear disc bundles such that ι ◦ Ψ : N → M and ι : N → M are ambiently isotopic rel X . The proofs of the above two theorems rely on the existence and uniqueness results fornormal vector bundles in [FQ90, Section 9], which we discuss further in Section 6.3. Thuswe postpone the proofs of the above two theorems to Section 6.4.Right now, let us first observe some nice consequences of the existence and uniquenessof tubular neighbourhoods.
Remark . Let X be a compact proper submanifold of a 4-manifold M . By Theorem 6.8we can pick a tubular neighbourhood p : N → X . We refer to E X := M \ Int N as the exterior of X . By Theorem 6.9 the homeomorphism type of the exterior is well-defined. HE FOUNDATIONS OF 4-MANIFOLD THEORY 25
Lemma 6.11.
Let X be a compact proper submanifold of a -manifold M . The exterior E X of X is a deformation retract of the complement M \ X .Proof. Let p : N → X be a tubular neighbourhood for X . Using the fact that p is a linear bundle, introduce compatible radial coordinates in the fibres and isotope radially outwards.This implies that ∂ N is a deformation retract of N \ X . But this also implies that theexterior E X = M \ N is a deformation retract of M \ X . (cid:3) Corollary 6.12.
Let X be a submanifold of a compact -manifold M . If X is compact,then π ( M \ X ) and H ∗ ( M \ X ) are finitely generated.Proof. It follows from Lemma 6.11 that M \ X is homotopy equivalent to the exterior of X which in our case is a compact 4-manifold since we assume that M is compact. Thecorollary is now a consequence of Corollary 4.8. (cid:3) Proposition 6.13.
Let X ⊂ M be a -dimensional orientable submanifold of a compactorientable -manifold M , such that each connected component of X has nonempty boundary.Then the tubular neighbourhood of Theorem 6.8 is homeomorphic to X × D .Proof. A linear D n -bundle is the unit disc bundle of a vector bundle. Connected surfaceswith boundary are homotopy equivalent to wedges of circles. Every orientable vector spacebundle over a wedge of circles is trivial, and so are their unit disc bundles. (cid:3) Normal vector bundles.
The reader will be familiar with the definition of a normalvector bundle when working in the smooth category: if X ⊂ M is a smooth submani-fold of a smooth manifold, then the normal vector bundle is the quotient vector bundle T M | X /T X . This definition uses the smooth structure to ensure the existence of tangentvector bundles, and vector bundles are a strong enough bundle technology to ensure theexistence the perpendicular subspaces required to form the quotient bundle. While some(weaker) canonical tangential structures do exist in the topological category (see Section 7),the idea of ‘quotient bundle’ no longer makes sense for them.In the topological category, following [FQ90, Section 9], we will use a definition of normalvector bundle that is much closer to the geometry of tubular neighbourhoods. We beginwith a definition that is almost what we need but suffers from a slight technical problem,which we then remedy. Definition 6.14.
Let M be a n -manifold and let X be a proper k -dimensional submanifold.An internal linear bundle over X is a pair ( E, p : E → X ) with the following properties:(1) E is a codimension zero submanifold of M .(2) The map p : E → X is an ( n − k )-dimensional vector bundle such that p ( x ) = x forall x ∈ X .(3) We have ∂E = p − ( ∂X ).An internal linear bundle ( E, p : E → X ) is intended to recover, from the smooth cate-gory, an open tubular neighbourhood of X . As such, the definition as stands suffers from the potential technical problem that the closure of E in M , which should be a closed tubu-lar neighbourhood, may be an immersion; see Figure 5. As in [FQ90, p. 137], we use thefollowing additional idea to rule out this problem. Definition 6.15.
Let M be an n -manifold, let X be a proper k -dimensional submanifold,and let ( E, p : E → X ) be an internal linear bundle over X . Suppose that given any( n − k )-dimensional vector bundle ( F, q : F → X ), any radial homeomorphism from anopen convex disc bundle of F to E can be extended to a homeomorphism from the wholeof F to a neighbourhood of E . Then we say ( E, p : E → X ) is extendable .In Figure 5 we illustrate an example of a non-extendable internal linear bundle. 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radialhomeomorphism F MX disk bundle in F the closure of E touches itself XE Figure 5.
Now we can define the notion of a normal vector bundle.
Definition 6.16 ([FQ90, p. 137]) . Let M be a n -manifold and let X be a proper k -dimensional submanifold. A normal vector bundle for X is an internal linear bundle thatis extendable. Theorem 6.17 (Existence of normal vector bundles) . Every proper submanifold of a com-pact 4-manifold admits a normal vector bundle.Remark . Generally, the existence of normal vector bundles is peculiar to when thesubmanifolds has low dimension or low codimension. We refer the reader to [FQ90, Section9.4] for a discussion of the other known situations where these objects always exist. Here is asummary of the known cases. A submanifold of dimension at most 3 in a closed manifold ofdimension at least 5 has a normal bundle, and codimension one submanifolds have normalbundles [Bro62]. That every codimension two submanifold of a manifold of dimension notequal to four have normal bundles was shown in [KS75], and this was extended to includedimension four in [FQ90, Section 9.3]. It is striking that, while among smooth manifoldsdimension 4 exhibits worse than usual behaviour, in the topological category the existenceof normal vector bundles seems to show it is among the better behaved of the dimensions.For the proof of Theorem 6.17 we will essentially appeal to [FQ90, Theorem 9.3A]and [FQ90, Theorem 9.3D]; the former deals with existence while the latter deals withuniqueness. We reproduce this theorem here for the benefit of the reader.
HE FOUNDATIONS OF 4-MANIFOLD THEORY 27
Theorem 6.19.
Let N be a submanifold of a -manifold M , with a closed subset K ⊆ N \ ∂N and a normal bundle over some neighbourhood of K in N . Then there is a normalbundle over N that agrees with the given one over the neighbourhood of K . Moreover thisextension is unique up to ambient isotopy relative to some neighbourhood of K .Proof. Let X be a proper submanifold of a compact 4-manifold M . The case that X has noboundary is dealt with in [FQ90, Theorem 9.3A]. The case that X has nonempty boundaryfollows also from [FQ90, Theorem 9.3A] if we apply more care. We sketch the argument.First, in dimension three the topological and the smooth category are the same. Thuswe can view the submanifold ∂X ⊂ ∂M as a smooth submanifold. Hence it has a smoothnormal vector bundle e.g. [Kos93, Chapter III.2] or [Lan02b, Section IV.5].Next use the Collar neighbourhood theorem 6.5 to obtain a collar ∂M × [0 , ⊂ M that restricts to a collar ∂X × [0 ,
1] for the boundary of X . Extend the smooth tubularneighbourhood of ∂X ⊂ ∂M into the collar by taking a product with [0 , M (cid:48) := M \ ( ∂M × [0 , ]). Whatremains of X is a submanifold N := X \ ( ∂X × [0 , ]). The submanifold N already has apreferred normal vector bundle on the closed subset K := ∂X × (1 / , M (cid:48) , N, K ) to obtain a normal vector bundle E → N agreeingwith the given one on K . The normal vector bundles over N and ∂X × [0 ,
1] agree on theoverlap K . Thus they define a normal vector bundle on all of X . (cid:3) 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) XM collar ∂M × [0 , ∂X ⊂ ∂M M = M \ ∂M × [0 , ] N = X ∩ M Figure 6.
Illustration of the proof of Theorem 6.17.Next we turn to the uniqueness of normal vector bundles.
Theorem 6.20 (Uniqueness of normal vector bundles) . Let M be a compact 4-manifoldand let X be a proper submanifold of M . Suppose we are given two normal vector bundles p i : E i → X , i = 1 , for X . For i = 1 , let ι i : E i → M be the inclusion map. Then thereexists a bundle isomorphism f : E ∼ = −→ E such that ι ◦ f and ι are ambiently isotopic rel. X .Proof. If X has no boundary, then the theorem is an immediate consequence of [FQ90,Theorem 9.3D]. Now suppose that X has nonempty boundary. First we claim that any normal vector bundle of X is obtained by the constructionoutlined in the proof of Theorem 6.17. To see this, let p : E → X be a normal vectorbundle. Pick a collar neighbourhood ∂X × [0 , ⊂ X . Since p is extendable, we can view p as the interior of a disc bundle q : F → X in M . Write C := q − ( ∂X ) ⊆ ∂M . Thedisc bundle q : q − ( ∂X × [0 , → ∂X × [0 ,
2] defines a collar neighbourhood C × [0 , C of ∂M . By the collar neighbourhood theorem 2.5 we canextend the collar neighbourhood C × [0 ,
1] of C to a collar neighbourhood ∂M × [0 , ∂M , the construction in the proof of Theorem 6.17,with further appropriate choices, gives rise to the normal vector bundle p : E → X . Thiscompletes the proof of the claim.After this long preamble it suffices to prove the theorem for any two normal vectorbundles obtained as in the proof of Theorem 6.17. Uniqueness follows by arguing that eachstep in the proof of existence of normal vector bundles was essentially unique. The proofsof uniqueness in the three steps make use of the following ingredients.First, apply the uniqueness statement for normal vector bundles of submanifolds ofsmooth manifolds to ∂X ⊆ ∂M e.g. [Kos93, Chapter III.2] or [Lan02b, Section IV.5].Next use the uniqueness of collar neighbourhoods as formulated in Theorem 2.7, appliedto the two collar neighbourhoods of ∂M subordinate to the given normal bundles of X .Finally apply the full relative version of [FQ90, Theorem 9.3D] to extend the normalvector bundle uniquely over the rest of X . (cid:3) Tubular neighbourhoods: proofs.
Now we will use the results from the previoussection to prove the existence and uniqueness of tubular neighbourhoods. First we showhow one can obtain tubular neighbourhoods from normal vector bundles.
Definition 6.21.
Let p : E → X be a vector bundle. Given x ∈ X , write E x := p − ( x ). A positive definite form g = { g x } x ∈ X consists of a positive definite form g x for every E x suchthat g x changes continuously with x .The following lemma follows from standard techniques, so we leave it to the reader tofill in the details. Lemma 6.22.
Let X be a compact manifold and let p : E → X be an n -dimensionalvector bundle. Then the space of positive definite forms on E is nonempty and convex.Furthermore, let g = { g x } x ∈ X be a positive definite form on E and consider the map p : E ( g ) := (cid:91) x ∈ X { v ∈ E x | g x ( v, v ) ≤ } → X. This map has the following properties: (1)
The map p : E ( g ) → X is a linear D n -bundle. (2) Given two different positive definite forms g and h on E there exists an isotopy ofthe vector bundle E sending E ( g ) to E ( h ) , and restricting to the identity on the0-section and outside some compact subset of E . HE FOUNDATIONS OF 4-MANIFOLD THEORY 29
Let X be a compact manifold and let p : E → X be a vector bundle. Given any positivedefinite form g we refer to p : E (cid:48) := (cid:91) x ∈ X { v ∈ E x | g x ( v, v ) ≤ } → X as a corresponding disc bundle . It follows from Lemma 6.22 that for most purposes theprecise choice of g is irrelevant.We can now prove the existence of tubular neighbourhoods. Proof of the tubular neighbourhood theorem 6.8.
Let X be a compact proper submanifoldof a 4-manifold M . By Theorem 6.17 there exists a normal vector bundle p : N → X for X . A choice of corresponding disc bundle is easily seen to be a tubular neighbourhood. (cid:3) The uniqueness proof for tubular neighbourhoods also requires us to associate a normalvector bundle to a tubular neighbourhood.
Lemma 6.23.
Let M be a compact -manifold and let X be a compact proper k -dimensi-onal submanifold. Let p : N → X be a tubular neighbourhood for X . There exists a normalvector bundle q : E → X and a positive definite form g such that N = E ( g ) and p : N → X equals q : E ( g ) → X .Proof. Let p : N → X be a tubular neighbourhood for X . Recall that we have Int N = N \ ∂ N . Consider W := M \ Int N . This is a compact 4-manifold. Pick a collar neighbourhood ∂W × [0 ,
1] and set E := N ∪ ∂ N × [0 , ). We have an obvious projection map q : E → X turning q into a bundle map where the fibre is given by the open (4 − k )-ball of radius .We leave it to the reader to turn q : N → X into an internal linear bundle, to show that itis in fact extendable (at this point one has to use that in the definition of E we only used“half” of the collar neighbourhood ∂ N × [0 , N with a positive definiteform g such that N = E ( g ). (cid:3) We conclude the section with the proof of the uniqueness theorem for tubular neighbour-hoods.
Proof of Theorem 6.9.
Let M be a 4-manifold and let X be a compact proper k -dimensi-onal submanifold. Furthermore let p i : N i → X , i = 1 , X . For i = 1 ,
2, let q i : E i → X be two corresponding normal vector bundles and let g i be the positive definite forms provided by Lemma 6.23. It follows from Theorem 6.20that there exists a bundle isomorphism f : E ∼ = −→ E such that ι ◦ f and ι are ambientlyisotopic rel X . It follows from the definitions that N is equivalent to the disc bundledefined by f ∗ g on E . But Lemma 6.22 (3) implies that f ∗ g and g define equivalenttubular neighbourhoods. (cid:3) Background on bundle structures
In this section we recall the bundle technologies we will need to use in later sections.First we recall the three standard types of fibre bundle with fibre R n : O, PL and TOP. Definition 7.1.
Let TOP( n ) be the subgroup of homeomorphisms of R n that fix the origin,topologised using the compact open topology. A principal TOP( n )-bundle has an associatedfibre bundle with fibre R n and a preferred 0-section. Call such a bundle a topological R n -bundle . Let TOP be the colimit colim −−−−−→ TOP( n ) under the inclusions − × Id R : TOP( n ) → TOP( n + 1). Write BTOP( n ) and BTOP for the corresponding classifying spaces.Let O( n ) be the orthogonal homeomorphisms of R n that fix the origin. Similarly to thecase of homeomorphisms, define BO( n ), O, and BO.The definition of the analogous spaces for PL is a little more involved, using semi-simplicial groups. For a gentle introduction to simplicial sets, see [Fri12]. The canonicalreference for classifying spaces constructed using simplicial groups is [May75]. Definition 7.2.
A homeomorphism f : K → L between two simplicial complexes K and L is a PL -homeomorphism if there are subdivisions K (cid:48) of K and L (cid:48) of L such that f : K (cid:48) → L (cid:48) is a simplicial map.For background on piecewise linear topology see [RS72]. The next definition comes from[RS68]. Let ∆ k be the standard k -simplex. Definition 7.3.
Let PL( n ) • be the semi-simplicial group defined as follows.(i) The group PL( n ) k assigned to the k -simplex is the group of PL homeomorphisms f : R n × ∆ k → R n × ∆ k over ∆ k , such that f | R n ×{ t } fixes the origin in R n for every t ∈ ∆ k . That is, with p : R n × ∆ k → ∆ k the projection, the diagram R n × ∆ k p (cid:36) (cid:36) f (cid:47) (cid:47) R n × ∆ kp (cid:122) (cid:122) ∆ k commutes.(ii) The i th face map is given by restricting to the i th face of ∆ k .Now define BPL( n ) by first using the level-wise bar construction to obtain a semi-simplicialspace, and then geometrically realising to obtain a space BPL( n ). Define PL and BPL ascolimits analogously to Definition 7.1. Remark . It is interesting that we do not define PL( n ) using the subspace topology fromTOP( n ). Note that we also do not define Diff( n ) = Diff( R n ) in this way. But for definingDiff( n ) as a topological group, and using this topology to define BDiff( n ), we have thebespoke Whitney topology. In the absence of an analogous topology for PL( n ), we usethe simplicial strategy. In fact this simplicial method could be used to define all three ofTOP( n ), PL( n ) and Diff( n ), giving a uniform treatment. But only in the PL case do wereally know of no other method that works.All smooth manifolds have tangent vector bundles and all smooth submanifolds have nor-mal vector bundles. This is one reason that vector bundles, corresponding to the structure HE FOUNDATIONS OF 4-MANIFOLD THEORY 31 group O( n ), are the de facto bundle technology in the smooth category. A general difficultywe will face when talking about manifold transversality in Section 10 is that we will needto use some well-defined notion of normal structure for a submanifold and, outside of thesmooth category, submanifolds do not necessarily admit normal vector bundles. However,various weaker bundle technologies have been developed, which replace this crucial conceptin the topological category.The rest of this section is devoted to a discussion of microbundles [Mil64]. The existenceand uniqueness of microbundles leads to the existence and uniqueness of tangent and (sta-ble) normal R n -bundles for TOP( n ), as discussed below. Source material on microbundlesis not hard to find in the literature, but has been included here for the convenience of thereader, in order for this survey to be more self-contained.The interaction between the weaker fibre automorphism groups PL( n ) and TOP( n )for tangent and (stable) normal R n -bundles, and the topological/piecewise linear/smoothstructures on the manifold itself are the topic of smoothing theory , to which we turn inSection 8. Definition 7.5. An n -dimensional microbundle ξ consists of a base space B and a totalspace E sitting in a diagram B i −→ E r −→ B, such that r ◦ i = Id B , and that is locally trivial in the following sense: for every point b ∈ B ,there exists an open neighbourhood U , an open neighbourhood V of i ( b ) and a homeomor-phism φ b : V → U × R n such that V r (cid:38) (cid:38) φ b (cid:15) (cid:15) U i (cid:56) (cid:56) ×{ } (cid:37) (cid:37) UU × R n pr (cid:57) (cid:57) commutes.Note that we only require a neighbourhood of i ( b ) to be trivial, and not all of thefibre r − ( b ). In fact, we only care about neighbourhoods i ( B ) ⊂ E , and declare twomicrobundles B → E → B and B → E (cid:48) → B to be equivalent , if i ( B ) and i (cid:48) ( B ) havehomeomorphic neighbourhoods such that the homeomorphism commutes with both theinclusion map and the restriction of the retraction map. Definition 7.6.
Let r : E → B be a microbundle ξ and let f : A → B be a map. The pullback of ξ under f is the microbundle f ∗ ξ with total space f ∗ E = (cid:8) ( a, e ) ∈ A × E | f ( a ) = r ( e ) (cid:9) , retraction ( f ∗ r )( a, e ) = a , and section ( f ∗ s )( a ) = (cid:0) a, s ( f ( a )) (cid:1) . In the case that f isan inclusion, also consider the microbundle ξ | A , which has total space r − ( A ) ⊂ E , and retraction r A : r − ( A ) → A and section s A : A → r − ( A ) are both the restrictions of r , s .In this case, the map of total spaces ( a, e ) (cid:55)→ e gives a preferred isomorphism f ∗ ξ to ξ | A .A topological R n -bundle clearly has an underlying microbundle. Kister and Mazur provedindependently the surprising result that every microbundle is equivalent to such an under-lying microbundle [Kis64, Theorem 2]. Theorem 7.7 (Kister-Mazur) . Let B be a manifold and B i −→ E r −→ B be an n -dimensionalmicrobundle ξ . Then there exists an open set F ⊂ E containing i ( B ) such that r : F → B is the projection map of a topological R n -bundle, whose -section is i and whose underlyingmicrobundle is ξ . Moreover, if F and F are any two topological R n -bundles over B such that the underlying microbundles are isomorphic, then F and F are isomorphic astopological R n -bundles. Every manifold admits a tangent microbundle.
Definition 7.8 (Tangent microbundle) . The tangent microbundle of an n -dimensionalmanifold M is the microbundle M ∆ −→ M × M pr −−→ M where ∆ is the diagonal map.The Kister-Mazur theorem implies this corresponds to a unique topological tangent bun-dle τ M : M → BTOP( n ), with corresponding stable topological tangent bundle τ M : M → BTOP.More subtle is the concept of a normal microbundle.
Definition 7.9 (Normal microbundle) . A normal microbundle of a submanifold S of amanifold M is a microbundle S → E → S such that E is a neighbourhood of S in M .It is immediate from the definition of normal microbundle that the local flatness in thedefinition of a submanifold S is a necessary condition for the existence of a normal mi-crobundle. Wild submanifolds that are not submanifolds do not admit normal microbun-dles. Indeed, it is generally far from straightforward to prove the existence of normalmicrobundles at all. Here is an existence and uniqueness result due to Stern [Ste75, Theo-rem‘4.5]. See also [Hir66],[Hir68, p. 65], and [KS77, Essay IV, Appendix A]. Theorem 7.10.
Let M n + q be a manifold, and let N n ⊂ M n + q be a proper submanifold ofcodimension q . Suppose that n ≤ q + 1 + j and q ≥ j for some j = 0 , , . Then N admits a normal microbundle restricting to a normal microbundle of ∂N ⊂ ∂M .If in addition n ≤ q + j , then this normal microbundle is unique up to isotopy.Remark . For a submanifold N ⊂ M we say a normal mi-crobundle ν ( N ) is unique up to isotopy if whenever there is another normal microbundle ν (cid:48) ( N ), there exists a microbundle equivalence f between ν ( N ) and ν (cid:48) ( N ) such that π (cid:48) ◦ f is isotopic to π relative to N .We exploit these theorems to define a stable normal structure on any manifold, thatwill play an important role in Section 8. Consider that any closed n -manifold M can beembedded as a submanifold M ⊂ R m for large m [Hat02, Corollary A.9]. For large enough HE FOUNDATIONS OF 4-MANIFOLD THEORY 33 m , any two such embeddings are isotopic. For large enough m , Theorem 7.10 impliesthere is a normal microbundle ξ . After possibly increasing m further, the last sentence ofTheorem 7.10 implies this normal microbundle ξ is unique. By the Kister-Mazur Theoremthis defines a unique topological R m − n -bundle. We remove the dependence on m by passingto the stable bundle TOP( m − n ) ⊂ TOP. Thus the process described gives a well-definedclassifying map ν M : M → BTOP. Summarising, we have the following.
Definition 7.12.
Given any closed n -manifold, the topological R ∞ -bundle ν M : M → BTOP, described above, is called the stable topological normal bundle . It is well-definedand unique.The next example shows that outside the hypotheses of Theorem 7.10, we should expectthat normal microbundles can be very badly behaved.
Example . Normal microbundles do not necessarily exist. Rourke and Sanderson [RS67,Example 2] construct S as a submanifold of a certain 28-dimensional PL manifold M insuch a way that it does not admit a topological normal microbundle. The embedding iseven piecewise linear.Even when topological normal microbundles exist, they are not always unique: Rourkeand Sanderson consider the smooth standard embedding S ⊂ S [RS68, Theorem 3.12]and construct a certain normal microbundle ξ of S ⊂ S . The construction of ξ issuch that if ξ were concordant to the trivial normal microbundle, this concordance wouldinduce a normal microbundle structure back on the embedding S ⊂ M of the previousexample. As this is not possible, ξ is non-trivial. Note that the smooth normal bundle νS of the standard embedding is trivial, so S ⊂ S admits at least two different normalmicrobundles.The following theorem ensures the issues of the previous example are not seen in dimen-sion 4. Theorem 7.14.
Let X be a codimension n proper submanifold of a -manifold M . Then X admits a normal microbundle. Moreover, if ξ is a normal microbundle of X , it is theunderlying microbundle to a normal vector bundle.Proof. The existence of normal microbundles in ambient dimension 4 is an immediate con-sequence of the existence of normal vector bundles (Theorem 6.17).Given a normal microbundle ξ , we apply the Kister-Mazur theorem 7.7 to obtain anembedded R n -bundle with underlying microbundle ξ . For n ≤
2, the homotopy fibreTOP( n ) / O( n ) for the forgetful map BO( n ) → BTOP( n ) is contractible, and for k ≤ π k (TOP(3) / O(3)) = 0 [KS77, Essay V, Theorems 5.8 and 5.9]. Using these facts,and checking the obstructions in each of the cases n = 0 , , , ,
4, we see in each case theembedded topological R n -bundle can be upgraded to an embedded vector bundle. Choosesuch a vector bundle refinement. By restricting to an open disc bundle and rescaling wecan ensure this internal linear bundle is extendable and thus a normal vector bundle in thesense of Definition 6.15. (cid:3) We will make use of our discussion of normal microbundles in Section 10 on topologicaltransversality. 8.
Smoothing 4-manifolds
We present three theorems which associate a smooth manifold to a given 4-manifold.Often these theorems can be used to reduce proofs about 4-manifolds to the case of smooth4-manifolds, where the standard tools of differential topology are available.8.1.
Smoothing non-compact 4-manifolds.
The first of our smoothing theorems [Qui82,Corollary 2.2.3], [FQ90, p. 116] says that noncompact connected 4-manifolds admit asmooth structure.
Theorem 8.1.
Every connected, noncompact 4-manifold is smoothable. Thus every -manifold M has a smooth structure in the complement of any closed set that has at leastone point in each compact component of M . There are some related statements in the literature on smoothing 4-manifolds in thecomplement of a point, that appeared prior to Freedman’s work [Fre82] and prior to [Qui82].We discuss them briefly here. For the case of PL structures on noncompact 4-manifolds,given a lift of the (unstable) tangent microbundle classifying map M → BTOP(4) to the PLcategory M → BPL(4) (see Section 7), the proof can be found in [Las70, p. 54] and [KS77,Essay V, Addendum 1.4.1]. This result was stated for smooth bundle structures and smoothstructures on manifolds in [Las71]. Alternatively, [HM74], [FQ90, Theorem 8.3B] apply toimprove a PL structure to a smooth structure, unique up to isotopy, for any manifold ofdimension at most six. Again, in [Las71] Lashof assumes a lift of the (unstable) tangentmicrobundle classifying map M → BTOP(4) to a map M → BO(4). For noncompactconnected 4-manifolds, such a lift always exists, as was later shown by Quinn [Qui82,Qui84], [FQ90, p. 116] using the full disc embedding theorem [Fre82], and giving rise toTheorem 8.1.Due to the seminal nature of Freedman’s Field’s medal winning paper [Fre82], it is wellworth clarifying the details of some citations therein. In the proof of Corollary 1.2, in theproof of Theorem 1.5 on page 369, in the proof of Theorem 1.6, and at the start of Sec-tion 10, Freedman uses that smoothing theory is available for noncompact 4-manifolds. Inparticular, smoothing for noncompact contractible 4-manifolds plays a vital rˆole in Freed-man’s proof of the topological 4-dimensional Poincar´e conjecture [Fre82, Theorem 1.6].Freedman cites [KS77] for this fact, however [KS77, Essay V, Remarks 1.6 (A)] specificallyexcludes smooth structures (but for a stronger result). Nevertheless, as mentioned above,Lashof [Las71, p. 156] proved the smooth version of [KS77, Essay V, Addendum 1.4.1], orone can use PL smoothing theory [HM74], [FQ90, Theorem 8.3B] to improve a PL structurefrom [KS77, Essay V, Addendum 1.4.1] to a smooth structure, essentially uniquely.Freedman only applies smoothing theory in cases, such as for contractible M , that he canensure the existence of a lift of τ M : M → BTOP(4) to BO(4). Later, Quinn [Qui82, Corol-lary 2.2.3] showed that such a lift always exists for connected noncompact 4-manifolds. In
HE FOUNDATIONS OF 4-MANIFOLD THEORY 35 fact, he showed that the map TOP(4) / O(4) → TOP / O (cid:39) K ( Z / ,
3) is 5-connected [FQ90,Theorem 8.7A], where only 3-connected is needed for Theorem 8.1. In other words, it wasshown prior to Freedman’s work that homotopy 4-spheres admit a smooth structure in thecomplement of a point, so the results that Freedman required were indeed known. However,smoothing in the complement of a point was not known for general connected 4-manifoldsuntil after the work of Quinn in 1982. Further discussion can also be found in Quinn [Qui84]and Lashof-Taylor [LT84].Below we will give applications of Theorem 8.1, see e.g. the proof of Theorem 10.9.8.2.
The Kirby-Siebenmann invariant and stable smoothing of 4-manifolds.
Theformulation of the other two statements on smoothing 4-manifolds make use of the Kirby-Siebenmann invariant. The Kirby-Siebenmann invariant ks( M ) ∈ Z / / PL of the forgetful map BPL → BTOP has the homotopytype of a K ( Z / ,
3) and has the structure of a loop space, permitting the definition of thedelooping B(TOP / PL) [BV68, Theorem C], [BV73] which is an Eilenberg-Maclane spaceof type K ( Z / , / PL → BPL → BTOP → B(TOP / PL) , the unique obstruction to a lift of the classifying map τ M : M → BTOP of the stabletangent microbundle to BPL is therefore a homotopy class in[(
M, ∂M ) , (B(TOP / PL) (cid:124) (cid:123)(cid:122) (cid:125) = K ( Z / , , ∗ )] ∼ = H ( M, ∂M ; Z /
2) = Z / . Here, we used again that 4-manifolds have the homotopy type of a CW-complex. Werefer to the corresponding element of Z / M ) invariant of thecompact, connected manifold M . For disconnected compact 4-manifolds, M = (cid:70) ni =1 M i ,define ks( M ) = n (cid:88) i =1 ks( M i ) ∈ Z / . In the following theorem we summarise some key properties of the Kirby-Siebenmanninvariant.
Theorem 8.2.
Let M and N be compact 4-manifolds. (1) If M × R admits a smooth structure ( e.g. if M admits a smooth structure ) , then ks( M ) = 0 . (2) The Kirby-Siebenmann invariant gives rise to a surjective homomorphism Ω TOP4 → Z / . In particular for M a closed 4-manifold that bounds a compact 5-manifold, ks( M ) = 0 . (3) The Kirby-Siebenmann invariant is additive under the connected sum operation. (4)
The forgetful map Ω Spin4 → Ω TOPSpin4 fits into a short exact sequence → Ω Spin4 → Ω TOPSpin4 ks= σ/ −−−−→ Z / → with the last map given by the signature divided by 8, modulo 2, which equals theKirby-Siebenmann invariant Ω TOPSpin4 → Ω STOP4 ks −→ Z / . This sequence does notsplit, so Ω TOPSpin4 ∼ = Z . (5) If S ⊂ ∂M and T ⊂ ∂N are compact codimension zero submanifolds with S ∼ = T ,then ks( M ∪ S ∼ = T N ) = ks( M ) + ks( N ) . (6) If there exists a compact 5-manifold with ∂W = M ∪ ∂M = ∂N N , then ks( M ) = ks( N ) . We could not find explicit proofs of these facts in the literature, so we give some details.
Proof of Theorem 8.2.
Let us prove (1). The tangent bundle of M × R is isomorphic to τ M ⊕ ε , where τ M is the tangent microbundle of M and ε denotes a rank one trivial vectorbundle over M . If M × R admits a smooth structure, then there is a lift τ Diff M × R : M → BO(5),the smooth tangent bundle to M × R . Let p : BO(5) → BTOP(5) be the canonical map.Then τ M ⊕ ε = p ◦ τ Diff M × R . Passing to the stable classifying spaces, we obtain a lift M → BOwhose composition with the canonical map BO → BTOP agrees with τ M ⊕ ε ∞ , the stabletangent microbundle of M . Since the map BO → BTOP factors through BPL → BTOP,we have a stable lift of τ M and so ks( M ) = 0. This completes the proof of (1).Now to prove (2), suppose that a closed 4-manifold M = (cid:70) ki =1 M i bounds a compact5-manifold W (cid:48) . Perform 0 and 1-surgeries on W (cid:48) to obtain a path connected, simplyconnected, compact 5-manifold with ∂W = M . We prove that ks( M ) := (cid:80) ki =1 ks( M i ) = 0.Consider the diagram M i (cid:47) (cid:47) ks( M i ) (cid:39) (cid:39) M (cid:47) (cid:47) ks( M ) (cid:36) (cid:36) W τ W (cid:47) (cid:47) ks( W ) (cid:29) (cid:29) BTOP (cid:15) (cid:15)
B(TOP / PL) = (cid:47) (cid:47) K ( Z / , . The restriction M i → W → BTOP equals the stable tangent microbundle of M , since M has a collar M × [0 , ⊂ W by Theorem 2.5. Therefore the diagram commutes. It followsthat the top left horizontal map in the next diagram sends ks( W ) to (ks( M ) , . . . , ks( M k )), HE FOUNDATIONS OF 4-MANIFOLD THEORY 37 so the map H ( W ; Z / → Z / W ) to (cid:80) ki =1 ks( M i ) = ks( M ). H ( W ; Z / (cid:47) (cid:47) ∼ = (cid:15) (cid:15) H ( M ; Z / ∼ = (cid:47) (cid:47) ∼ = (cid:15) (cid:15) (cid:76) ki =1 H ( M i ; Z / ∼ = (cid:40) (cid:40) ∼ = (cid:15) (cid:15) (cid:76) ki =1 Z / (1 ,..., (cid:47) (cid:47) Z / H ( W, M ; Z / (cid:47) (cid:47) H ( M ; Z / ∼ = (cid:47) (cid:47) (cid:76) ki =1 H ( M i ; Z / ∼ = (cid:54) (cid:54) The left square of this diagram commutes by Poincar´e-Lefschetz duality. The middle squareand the triangle commute trivially. But since W is connected and simply connected, everyelement of H ( W, M ; Z /
2) can be represented by a (possibly empty) union of arcs withboundary on M . Thus the image of ks( W ) in (cid:76) ki =1 H ( M i ; Z /
2) is nonzero in evenly manysummands, and therefore its image in Z / M ) = 0, as desired.Now (2) follows. First note that the addition on Ω TOP4 is by disjoint union, so ks isadditive by definition. We have just shown that the map ks : Ω
TOP4 → Z / M a closed 4-manifold that bounds a compact 5-manifold, ks( M ) = 0. Thereforeks : Ω TOP4 → Z / E manifold, whose existence was established by Freedman [Fre82, Theorem 1.7]as a key step in the proof of the Classification Theorem 11.2, has ks( E ) = 1. To seethis note that E cannot be smoothed, even after adding copies of S × S , by Rochlin’stheorem (Theorem 3.4) that every closed spin smooth 4-manifold has signature divisible by16. Whereas if ks( E ) = 0, then E would be stably smoothable by Theorem 8.6. Thereforeks : Ω TOP4 → Z / M (cid:116) N is cobordant to M N via the cobordism ( M × I (cid:116) N × I ) ∪ S × D ( D × D ) , with {− }× D embedded in the interior of M ×{ } , and { }× D embedded in the interiorof N × { } . Then we have just shown that the Kirby-Siebenmann invariant vanishes on M N (cid:116) M (cid:116) N and therefore ks( M N ) = ks( M ) + ks( N ) ∈ Z / (cid:47) (cid:47) Ω Spin4 (cid:47) (cid:47) · (cid:15) (cid:15) Ω TOPSpin4 (cid:47) (cid:47) (cid:15) (cid:15) Z / (cid:47) (cid:47) = (cid:15) (cid:15) (cid:47) (cid:47) Ω SO4 (cid:47) (cid:47) Ω STOP4 (cid:47) (cid:47) Z / (cid:47) (cid:47) Recall that Ω
SO4 ∼ = Z given by the signature and generated by CP . The signature providesa splitting homomorphism, so Ω STOP4 ∼ = Z ⊕ Z /
2. Also Ω
Spin4 ∼ = Z given by the signaturedivided by 16 and generated by the K M ) implies smoothable after adding copies of S × S by Theorem 8.6 below. Since M S × S is (spin) bordant to M , the sequences are exact at their middle terms. Finally,a topological null bordism of a compact smooth 4-manifold can be smoothed by highdimensional smoothing theory, so the left hand maps are injective.We claim that the sequence in the upper row does not split. Consider the K Spin4 ∼ = Z . By the down-then-left route, [ K
3] maps to (16 , ∈ Z ⊕ Z / ∼ =Ω STOP4 . On the other hand the E -manifold represents a class in Ω TOPSpin4 and maps to(8 , ∈ Z ⊕ Z / ∼ = Ω STOP4 .It follows that 2 · [ E ] maps to 0 ∈ Z / Spin4 . Let N be a closed spin smooth 4-manifoldTOPSpin-bordant to E E . Since σ ( E E ) = 16 = σ ( K N ] = [ K ∈ Ω Spin4 .It follows that K
3, the generator of Ω
Spin4 ∼ = Z , maps to 2 · [ E ] ∈ Ω TOPSpin4 . Thus we havea diagram with exact rows:0 (cid:47) (cid:47) Z · (cid:47) (cid:47) = (cid:15) (cid:15) Z (cid:47) (cid:47) (cid:55)→ [ E ] (cid:15) (cid:15) Z / (cid:47) (cid:47) = (cid:15) (cid:15) (cid:47) (cid:47) Z (cid:55)→ · [ E ] (cid:47) (cid:47) Ω TOPSpin4 (cid:47) (cid:47) Z / (cid:47) (cid:47) . Since ks( E ) = 1, the diagram commutes. Then by the five lemma, Ω TOPSpin4 ∼ = Z , generatedby E , and the sequence does not split, as claimed. The diagramΩ TOPSpin4 ∼ = σ/ (cid:47) (cid:47) (cid:15) (cid:15) Z (cid:55)→ (8 , (cid:15) (cid:15) (cid:36) (cid:36) Ω STOP4 ∼ = (cid:47) (cid:47) Z ⊕ Z / ks (cid:47) (cid:47) Z / , which commutes by computing on the generator E of Ω TOPSpin4 ∼ = Z , shows that ks( M ) = σ ( M ) / ∈ Z / M . This completes the proof of (4).To prove (5), it was suggested by Jim Davis to consider the exact sequenceΩ O4 → Ω TOP4 → Ω { O → TOP } → Ω O3 = 0 . Here Ω { O → TOP } is represented by compact topological 4-manifolds with smooth boundary,up to 5-dimensional cobordism relative to a smooth cobordism on the boundary. Thatis, 4-manifolds with boundary ( M, ∂M ) and (
N, ∂N ) are equivalent if there is a compact5-manifold with boundary ∂W = M ∪ ∂M ∂ vert W ∪ ∂N N, HE FOUNDATIONS OF 4-MANIFOLD THEORY 39 for some smooth 4-dimensional cobordism ∂ vert W with boundary ∂M (cid:116) ∂N .By the exact sequence, Ω { O → TOP } is isomorphic to the cokernel of Ω O4 → Ω TOP4 . We claimthat this cokernel is isomorphic to Z / TOP4 → Z /
2. If ks( M ) = 0 then M is stablysmoothable by Theorem 8.6, so M is bordant to a smooth manifold and therefore lies in theimage of Ω O4 . If M is smooth, then ks( M ) is zero, so the sequence Ω O4 → Ω TOP4 ks −→ Z / → M (cid:116) N is bordant to M ∪ S = T N ,where S ⊂ ∂M and T ⊂ ∂N are compact codimension zero submanifolds with S ∼ = T .Here is a construction of such a bordism. For I = [0 , M × I ) (cid:116) ( S × I × [1 / , (cid:116) ( N × I ) , identify S × { } × [1 / , ∼ S × [1 / , ⊆ ( M × [1 / , , and, using the identification S ∼ = T , identify S × { } × [1 / , ∼ T × [1 / , ⊆ N × [1 / , . Let W be the result of this gluing and some rounding of corners. The boundary of W is( M (cid:116) N ) ∪ ∂M (cid:116) ∂N ∂ vert W ∪ ∂ ( M ∪ S = T N ) M ∪ S = T N, where ∂ vert W = ( ∂M × [0 , / ∪ ( ∂M \ S × [1 / , ∪ ( S × I × { / } ) ∪ ( ∂S × I × [1 / , ∂N × [0 , / ∪ ( ∂N \ T × [1 / , . This shows that M (cid:116) N and M ∪ S = T N are equal in Ω { O → TOP } , and therefore have thesame Kirby-Siebenmann invariants. Since ks( M (cid:116) N ) = ks( M ) + ks( N ), this completes theproof of (5).Finally we prove (6). If M ∪ ∂M = ∂N N bounds a compact 5-manifold, then by (2) we havethat ks( M ∪ ∂ N ) = 0. By (5), ks( M ) + ks( N ) = ks( M ∪ ∂ N ) ∈ Z /
2. Therefore ks( M ) =ks( N ) as required. This proves (6) and therefore completes the proof of Theorem 8.2. (cid:3) The following theorem says that the converse to Theorem 8.2 (1) holds.
Theorem 8.3. If M is a compact, connected 4-manifold with vanishing Kirby-Siebenmanninvariant, then M × R admits a smooth structure.Proof. The vanishing of the Kirby-Siebenmann invariant implies that there is a lift of τ M : M → BTOP to a map M → BPL. Since PL / O is 6-connected [FQ90, Theorem 8.3B],[HM74, Proof of 4.13], there is in fact a lift (cid:101) τ M : M → BO. This corresponds to a lift (cid:101) τ M ⊕ ε n : M → BO(4 + n ), for some n . This in turn corresponds to a lift (cid:101) τ M × R n : M × R n → BO(4 + n ) of the tangent microbundle τ M × R n : M × R n → BTOP. By [KS77, Essay V, Theorem 1.4],there exists a corresponding smooth structure on M × R n . Then apply the Product Struc-ture Theorem 5.16 [KS77, Essay I, Theorem 5.1], to deduce the existence of a smoothstructure on M × R , using that the dimension of M × R is at least five. (cid:3) Example . Here is an application of Theorem 8.3. By the classification of simply con-nected, closed 4-manifolds [FQ90, Section 10.1] (see also our Theorem 11.2), there is asimply connected, closed spin 4-manifold N with intersection form E ⊕ E . Since thisform is not diagonalisable over Z , by Donaldson’s theorem [Don83] (Theorem 3.7) this4-manifold does not admit a smooth structure. However the Kirby-Siebenmann invariantof N vanishes, since for a closed 4-manifold M with even intersection form, the Kirby-Siebenmann invariant ks( M ) coincides with σ ( M ) / E ⊕ E is rank 16 andpositive definite, with signature 16. Therefore N × R admits a smooth structure by Theo-rem 8.3, even though N does not. Construction . Here is a construction of the Chern manifold ∗ C P . This manifold wasfirst constructed in [Fre82, p. 370]. Attach a 2-handle D × D to D by identifying S × D with a +1-framed trefoil in ∂D = S . The boundary of the resulting manifoldis an integral homology sphere. Freedman proved that every integral homology spherebounds a contractible 4-manifold [Fre82, Theorem 1.4 (cid:48) ], [FQ90, Corollary 9.3C]. Cap off D ∪ D × D with this contractible 4-manifold, to obtain the closed 4-manifold ∗ C P . Bythe Rochlin invariant, every compact, smooth, spin 4-manifold with boundary +1-surgeryon the trefoil has σ/ ∗ C P ) = 1. The Chernmanifold ∗ C P is homotopy equivalent to C P but is not homeomorphic. For furtherdiscussion of the star construction, see [FQ90, Section 10.4] and [Tei97].The following theorem says in particular that given any compact 4-manifold M thereexists a closed orientable simply-connected 4-manifold N such M N is smoothable. Theorem 8.6.
Let M be compact -manifold. There exists a closed, orientable, simplyconnected -manifold N such M N admits a smooth structure. If moreover the Kirby-Siebenmann invariant of M is zero, then there exists a k ∈ N such that M k S × S admits a smooth structure.Proof. Let M be compact 4-manifold. Perform the connected sum with an appropriatenumber of copies of ∗ C P , the closed oriented simply-connected 4-manifold with nontrivialKirby-Siebenmann invariant, homotopy equivalent to C P , constructed on [FQ90, p. 167], inorder to obtain a manifold with every connected component having zero Kirby-Siebenmanninvariant. It follows from the discussion on [FQ90, p. 164] and the Sum-stable smoothingtheorem [FQ90, p. 125], that performing the connected sum with enough copies of S × S produces a manifold that admits a smooth structure. (cid:3) Remark . Given a lift of the classifying map of the (unstable) tangent microbundle of M to BO(4), Lashof-Shaneson [LS71] showed that there exists a k ∈ N such that M k S × S HE FOUNDATIONS OF 4-MANIFOLD THEORY 41 admits a smooth structure. The result quoted in the previous proof extended this to a liftof the corresponding stable maps.9.
Tubing of surfaces
As an example of the use of the technology we have discussed thus far, we show thatone can tube together two locally flat embedded surfaces in a 4-manifold, to obtain anembedding of the connected sum. This operation is standard in the smooth category, butas ever in the topological category one should take some care.The following situation is by no means the most general such result possible. We wishto illustrate two things. First, that operations on surfaces that can be performed in thesmooth category can usually also be performed in general 4-manifolds with locally flatsurfaces (although performing these operations in a parametrised way seems to be beyondcurrent knowledge). Second, we want to show the level of detail required to demonstratethat such operations work.
Proposition 9.1 (Tubing) . Let S and T be 2-dimensional proper submanifolds of a con-nected 4-manifold M , that is S and T are locally flat embedded surfaces. Pick a point P ∈ S \ ∂S and Q ∈ T \ ∂T . Let [ γ ] ∈ H ( M, { P, Q } ; Z ) be a relative homology class.There is a locally flat embedded arc C joining P and Q , satisfying the following. (i) We have [ C ] = [ γ ] ∈ H ( M, { P, Q } ; Z ) . (ii) The interior of C is disjoint from S ∪ T . (iii) The arc C extends to a neighbourhood C × D embedded in M such that E S := { P } × D ⊆ S and E T := { Q } × D ⊆ T . (iv) We have ( C × D ) \ ( E S ∪ E T ) ⊆ M \ ( S ∪ T ) . (v) The intersection of C × D with a normal disc bundle D ( S ) of S is such that forevery d , ( C × { d } ) ∩ D ( S ) is a ray in a single fibre of D ( S ) , and similarly for T .Moreover there is a trivialisation of the normal bundle over E S as E S × D suchthat for every c ∈ C with ( { c } × D ) ∩ D ( S ) (cid:54) = ∅ , we have that { c } × D = E S × { e } for some e ∈ D , and all such e that arise this way lie on a fixed ray from the originof D . These data allow us to perform tubing of surfaces ambiently.
Proposition 9.2.
Given data S , T , C × D , E S and E T as in Proposition 9.1, the subset ( S \ E S ) ∪ ( T \ E T ) ∪ C × S is a 2-dimensional submanifold abstractly homeomorphic to S T .Proof. The surfaces and the tube are locally flat by assumption, or by construction fromProposition 9.1. The circles where the tube is glued to the surface are locally flat points.To see this observe that we have arranged a coordinate system in which this gluing is acompletely standard attachment at angle π/ (cid:3) 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) C × D M C TS P Q
Figure 7.
Illustration of Proposition 9.1.
Proof of Proposition 9.1.
Since S and T are proper submanifolds, they have normal bundlesby Theorem 6.17. Pick normal disc bundles D ( S ) and D ( T ), and remove the interiors of D ( S ) and D ( T ) i.e. smaller disc bundles inside the normal disc bundles. We obtain amanifold with boundary X := M \ (cid:0) Int D ( S ) ∪ Int D ( T ) (cid:1) together with a collar neighbourhood of the boundary arising from D ( S ) \ Int D ( S ), andthe same with T replacing S , extended using Theorem 2.5 to a collar neighbourhood forall of ∂X . Choose a closed disc neighbourhood E S of P in S . We write ∂ S X for thefibrewise boundary of D ( S ), ∂ T X for the fibrewise boundary of D ( T ), and ∂ X for ∂ S X ∪ ∂ T X = ∂X \ ∂M .Choose a trivialisation of the normal bundle νS in a neighbourhood N ( E S ) of E S , as N ( E S ) × D . A ray in D from the origin to the boundary determines an embedding E S × [0 , ⊂ D ( S ). We obtain in particular a disc E S ×{ } ∈ N ( E S ) ×{ pt } ⊂ N ( E S ) × S .Choose a smooth structure on ∂X (which we may do since ∂X is a 3-manifold), and choosea smoothly embedded neighbourhood F S ∼ = D in ∂ S X that contains E S ×{ } in its interior.Make the analogous set of choices and constructions for T , to obtain E T , N ( E T ), E T × [0 , ⊂ D ( T ), and F T ∼ = D in ∂ T X that contains E T × { } in its interior.Remove a point r from X , and using Theorem 8.1 choose a smooth structure on X \ { r } extending the chosen smooth structure on ∂X . Choose a smoothly embedded path C X ⊂ X between the centres of E S × { } and E T × { } , such that C X extends along the previouslychosen rays inside the normal bundles to a path C between P and Q such that [ C ] = [ γ ] ∈ H ( M, { P, Q } ; Z ). Extend C X to a codimension zero submanifold N ( C X ) homeomorphic to I × D , with I × { } ⊂ I × D mapping to C X , and such that { } × D maps to F S ⊆ ∂ S X and { } × D maps to F T ⊂ ∂ T X .Now, for small ε , [0 , ε ] × D and [1 − ε, × D give rise to collar neighbourhoods of theclosed subsets F S and F T of ∂ X . Use Theorem 2.5 to extend this collar neighbourhood toa collar neighbourhood over all of ∂X .We now have two collar neighbourhoods of ∂X , the collar Ψ : ∂X × [0 , (cid:44) → X wehave just constructed which is compatible with N ( C X ), and the collar neighbourhood HE FOUNDATIONS OF 4-MANIFOLD THEORY 43 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(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) ∂ T X∂ S X X = M \ (Int D ( S ) ∪ Int D ( T )) TC X N ( C X ) ∼ = D × I disc bundle D ( T )disc bundle D ( S ) rE S S P E S ×{ } QE T E S ×{ } Figure 8.
Illustration for the proof of Proposition 9.1.Ψ : ∂X × [0 , (cid:44) → X constructed above from D ( S ) \ Int D ( S ) and D ( T ) \ Int D ( T ).By Theorem 2.7, there is an isotopy H t : M → M starting from the identity, such that H ◦ Ψ = Ψ , i.e. sends the first collar to the second.We now obtain a codimension zero submanifold C X × D homeomorphic to I × D suchthat, with respect to the collar neighbourhood Ψ , we have: • For all c ∈ C X such that { c } × D ∩ Ψ ( ∂X × [0 , (cid:54) = ∅ , we have that { c } × D ⊂ Ψ ( ∂X × { t } ) for some t ∈ [0 , • For every d ∈ D , ( C × { d } ) ∩ ( ∂X × [0 , ( { x } × [0 , x in either F S or F T .In addition, above we constructed two discs E S ⊂ F S and E T ⊂ F T . Any two embeddeddiscs in a 3-ball are ambiently isotopic: place this isotopy inside C X × D to obtain a locallyflat embedding C X × D ∼ = I × D ⊂ C X × D .Now consider X ⊂ M and take the union( E S × [0 , ∪ ( C X × D ) ∪ ( E T × [0 , ⊆ M to obtain an embedding C × D ∼ = I × D whose intersection with S equals E S andwhose intersection with T equals E T . The core C = C × { } is of course a locally flatembedded path in M from P to Q with interior in M \ ( S ∪ T ) and with the correctrelative homology class in H ( M, { P, Q } ; Z ). We may then perform the tubing S T :=( S \ E S ) ∪ ( T \ E T ) ∪ C × S as promised. (cid:3) Topological transversality
We turn to the subject of transversality in the topological category. Some discussion ofthis concept is in order. There are two important contexts for transversality: submanifoldtransversality and map transversality. In this article, map transversality will be deducedfrom submanifold transversality. Submanifold transversality when none of the manifoldsinvolved has dimension 4 is due to to Marin [Mar77]; cf. [KS77, Essay III, Section 1].
Transversality in the remaining cases is due to Quinn [Qui82, Qui88]; see also [FQ90,Section 9.5].A naive definition of submanifold transversality in the topological category is that man-ifolds are locally transverse if around any intersection point there is a chart in which thesubmanifolds appear as perpendicular planes. On the other hand, there are examples (inthe relative setting) of submanifolds which cannot be made locally transverse via ambientisotopy; see Remark 10.4. Thus one cannot generally use this definition.In light of this, in order to make general statements, one passes to some notion of globaltransversality . Global transversality means that transversality statements are made withrespect to a given choice of normal structure on one of the submanifolds involved. Ofcourse, this forces one to engage with the question of existence and uniqueness of whatevernormal structure is used, and the ‘correct’ choice of normal structure is still not fully settledin the topological category. We refer the reader to [FQ90, Sections 9.4, 9.6C] for a briefdiscussion of the competitors.The most general statement of transversality [Qui88, Theorem] uses microbundles todescribe normal structure, and this is the technology we will use. As discussed in Section7, for general manifolds, tangent microbundles always exist but normal microbundles donot (see Example 7.13). The case of dimension 4 is special, since here the normal vectorbundles of Section 6.3, which are a stronger notion than normal microbundles, always exist.In fact, the results obtained for these normal vector bundles in dimension 4 are a strongenough to ensure that submanifold transversality holds in ambient dimension 4 with thenaive, local transversality definition discussed above. The reader may therefore wonderwhy we even introduce normal microbundles into a discussion of 4-manifold transversality.The answer is that the ‘submanifold transversality implies map transversality’ argument ofSection 10.2 requires a bundle technology that works in all dimensions, and microbundlesappear to be the most convenient.10.1.
Transversality for submanifolds.Definition 10.1.
Consider proper submanifolds
X, Y of an ambient manifold and a normalmicrobundle νX for X , with retraction r X : E ( νX ) → X . The proper submanifold Y is transverse to νX if there exists a neighbourhood U ⊂ E ( νX ) of X such that Y ∩ U = r − X ( X ∩ Y ) ∩ U . Lemma 10.2.
Let
X, Y submanifolds of M . Let Y be transverse to a normal microbun-dle νX . Then X ∩ Y is a submanifold of Y with normal microbundle ( νX ) | X ∩ Y .Proof. Once we have established that ( νX ) | X ∩ Y is a normal microbundle of X ∩ Y in Y ,the subspace X ∩ Y will automatically be a submanifold since the trivialisation of themicrobundle ( νX ) | X ∩ Y gives the required charts for X ∩ Y .At least after shrinking the total space of E ( νX ), each fibre r − ( x ) for x ∈ X ∩ Y willbe contained in Y by the definition of transversality. That is E (( νX ) | X ∩ Y ) is a subset of Y and neighbourhood of X ∩ Y . This shows that ( νX ) | X ∩ Y is a normal microbundle of X ∩ Y ⊂ Y . (cid:3) HE FOUNDATIONS OF 4-MANIFOLD THEORY 45 νX Y
Figure 9.
Sketch of a transverse intersection of Y to νX .Transversality in high dimensions is due to to Marin [Mar77], cf. [KS77, Essay III, Sec-tion 1]. The formulation below is from Quinn [Qui88]. Recall Definition 6.1 of a propersubmanifold. Note that in the next theorem there is no restriction on dimensions. Themanifold M is allowed to be noncompact and have nonempty boundary. Theorem 10.3 (Transversality for submanifolds) . Let X and Y be proper submanifolds ofa compact manifold M . Let νY be a normal microbundle for Y . Let C be a closed subsetsuch that X is transverse to νY in a neighbourhood of C . Let U be a neighbourhood ofthe set (cid:0) M \ C (cid:1) ∩ X ∩ Y . Then there exists an isotopy of X supported in U to a propersubmanifold X (cid:48) such that X (cid:48) is transverse to νY .Proof. See Quinn [Qui88] for all cases but dim M = 4, dim X = 2 and dim Y = 2. Forthe remaining case, first establish local transversality using [FQ90, Section 9.5]. Note that X ∩ Y is a discrete collection of points. Therefore, the coordinate chart, witnessing localtransversality, defines a normal neighbourhood of Y near X ∩ Y . This normal vector bundlecan be extended to a normal vector bundle νY (cid:48) on all of Y by [FQ90, Theorem 9.3A]. Thesubmanifold X is now transverse to νY (cid:48) , but (possibly) not to νY . By Theorem 7.14,our microbundle νY comes from a normal vector bundle. By uniqueness of normal vectorbundles (Theorem 6.20), there is an isotopy from νY (cid:48) to νY . Apply this isotopy to X .Now X is transverse to νY . (cid:3) Remark . The analogous statement to Theorem 10.3 is false for local transversality.Examples of this failure even exist in the PL category: Hudson [Hud69] constructs, forcertain large n , closed PL submanifolds X, Y ⊂ R n , that are topologically unknottedEuclidean spaces of codimension ≥
3, in such a way that X and Y are PL locally transversenear a closed neighbourhood K of infinity but also so that it is impossible to move X and Y by isotopy relative to K to make them PL locally transverse everywhere.Although transversality for submanifolds (Theorem 10.3) is only stated for a pair ofsubmanifolds, it can be used to make collections of submanifolds transverse. X i and X j X i X j Yyu i u j Figure 10.
Displacing a triple point y in a microbundle chart. Lemma 10.5.
Let M be an m -dimensional manifold for m ≥ , and let X , . . . , X n be m -dimensional compact submanifolds with normal microbundles νX i . Then the submani-folds X i can be isotoped such that there are no triple intersection points and the submanifoldsintersect ( pairwise ) transversely.Proof. We give a proof by induction. When n = 1, there is nothing to show, since everysubmanifold is embedded. For the inductive step, denote X n by Y . The inductive hypoth-esis states that we can isotope any n − X , . . . , X n − so that there are notriple points and that they intersect pairwise transversely. We will prove that the subman-ifolds X , . . . , X n − can be further isotoped so that they are transverse to νY and Y is freeof triple points. Note that having no triple points on Y implies that there exists an openset U Y such that: X i ∩ X j ⊂ U Y for 1 ≤ i < j ≤ n −
1, and M \ U Y is a neighbourhood of Y . To obtain the lemma apply the inductive hypothesis, picking all further isotopies to besupported in U Y .We proceed by showing the inductive step: we can isotope every X i to be transverse to νY such that no triple points lie on Y . For each i = 1 , . . . , n −
1, apply Theorem 10.3to arrange that Y and X i intersect transversely. By compactness of the submanifolds, thesubset T Y = Y ∩ ( X ∪ · · · ∪ X n − )is compact. Pick disjoint open neighbourhoods V y ⊂ Y around each point y ∈ T Y . Pick achart φ of Y around φ (0) = y contained in V y , and a microbundle chart around y ∈ Y . Inthe local model, Y corresponds to R m ×{ } and the X i that intersect Y in y will be mappedto 0 × R m . For those X i , pick disjoint points u i ∈ R m (here we use m > η : R ≥ → [0 ,
1] with η ( t ) = 1 for 0 ≤ t ≤ η ( t ) = 0 for t ≥ HE FOUNDATIONS OF 4-MANIFOLD THEORY 47
Replace X i in the chart with the image of R m → R m × R m v (cid:55)→ (cid:0) η (cid:0) (cid:107) v (cid:107) (cid:1) u i , v (cid:1) . Call this new submanifold X (cid:48) i . It agrees with X i outside the ball of radius 2, and is isotopicto X i . In V y , the submanifold X (cid:48) i intersects Y only in φ ( u i ) and there it intersects Y transversely with respect to νY . The collection { X (cid:48) i } has no triple intersection points inthe set V y anymore. (cid:3) Here is another result on submanifold transversality. It might often happen that onecan find a continuous map of, for example, a disc D into a 4-manifold M , perhaps iffundamental group computations yield a null homotopy of a circle. Then this disc can beperturbed to locally flat immersion. If M were smooth, this would be a consequence ofstandard differential topology, an observation that we leverage. Theorem 10.6.
Let Σ be a connected 1 or 2 dimensional manifold and let f : Σ → X be a continuous map of Σ into a connected 4-manifold X . Let C be a closed subset of Σ such that f | C is a locally flat immersion. Then there is a perturbation of f to a locally flatimmersion f (cid:48) : Σ → X with f | C = f (cid:48) | C .Proof. Perturb f to a map that misses a point. Then smooth X in the complement ofthat point and C using Theorem 8.1. Now by [Hir94, Theorem 2.2.7] we can perturb f to a smooth immersion, which we can then perturb to be self-transverse f (cid:48) by [Hir94,Theorem 4.2.1], [Wal16, Theorem 4.6.6], with no triple points by general position [Wal16,Theorem 4.7.7]. Now add back in the point: f (cid:48) is a locally flat immersion. (cid:3) We are not sure how to give a purely topological proof of this.10.2.
Transversality for maps.Definition 10.7.
Let f : M → N be a continuous map and let X be a submanifold of N with normal microbundle νX . The map f is said to be transverse to νX if f − ( X ) is asubmanifold admitting a normal microbundle νf − ( X ) and f : νf − ( X ) → f ∗ νXm (cid:55)→ ( r ( m ) , f ( m ))is an isomorphism of microbundles.In the next theorem, we show how to reduce transversality for maps to transversality forsubmanifolds. Again, there are restrictions neither on dimensions nor codimensions. Theorem 10.8.
Let Y ⊂ N be a proper submanifold with normal microbundle νY . Let f : M → N be a map, and let U be a neighbourhood of the set graph f ∩ ( M × Y ) ⊂ M × N. Then there exists a homotopy F : M × I → N such that MNY graph( f ) T F − ( Y ) Figure 11.
Transversality for maps from transversality for submanifolds.(1) F ( m,
0) = f ( m ) for all m ∈ M ; (2) F : m (cid:55)→ F ( m, is transverse to νY ; and (3) for m ∈ M either(a) ( m, f ( m )) / ∈ U , in which case F ( m, t ) = f ( m ) for all t ∈ I , or(b) ( m, f ( m )) ∈ U , in which case ( m, F ( m, t )) ∈ U for all t ∈ I .Proof. Note that M × Y ⊂ M × N is a proper submanifold with normal microbundle M × νY = pr ∗ Y νY . Also graph f is a proper submanifold of M × N . By Theorem 10.3, thereexists an isotopy G : graph f × I → M × N, supported in U , of the submanifold graph f to a submanifold T ⊂ M × N such that T istransverse to M × νY over M × Y .Define the map F as the composition F : M × I → graph f × I G −→ M × N pr N −−→ N. Since the isotopy G is supported in U , statement (3) holds. By construction, F ( x,
0) =pr Y ( x, f ( x )) = f ( x ), which proves statement (1).Now we prove statement (2). Let F : M → N be the map that sends x (cid:55)→ F ( x, F ; seeFigure 11. By transversality of T to M × νY , we see that Z = T ∩ ( M × Y ) = pr − ( Y ) isa submanifold of T with normal bundle M × νY | Z , and that the projection to N induces amicrobundle isomorphism M × νY | Z ∼ −→ pr ∗ N νY . By definition, pr N : T → N is transverseto νY . HE FOUNDATIONS OF 4-MANIFOLD THEORY 49
We transport the submanifold Z back to M . Consider the commutative diagram NM graph f × { } T ∼ = F q ∼ = pr N , where q is the composition, which is a homeomorphism. Now F − ( Y ) = q − ( Z ) is asubmanifold with normal microbundle q ∗ (cid:0) M × νY | Z (cid:1) = q ∗ pr ∗ N νY = F ∗ νY, that is F : M → N is transverse to νY . (cid:3) Representing homology classes by submanifolds.
Our goal in this section is toprove the following theorem.
Theorem 10.9.
Let X be a compact orientable -manifold and let A be a union of com-ponents of ∂X . Let k = 2 or k = 3 and let σ ∈ H k ( X, A ; Z ) . (1) The class σ can be represented by a k -dimensional submanifold Y with ∂Y ⊂ A . (2) In the case k = 3 , the boundary of Y can be specified: if B ⊂ A is an oriented closed -dimensional smooth submanifold contained in A such that ∂ ( σ ) = [ B ] ∈ H ( A ) ,then σ can be represented by an oriented compact -dimensional submanifold Y with ∂Y = B . The submanifold B can be assumed to be smooth, since ∂X is a 3-manifold and so hasa unique smooth structure by [Moi52], [Moi77, p. 252–253].Note that Theorem 10.9 also holds for k = 0 and k = 1. This is trivial for k = 0.To see this for k = 1, remove a point from each connected component to get a smooth4-manifold by Theorem 8.1. Note that H ( X \ { pt } , A ; Z ) ∼ = H ( X, A ; Z ). Then by smoothapproximation and general position, every 1-dimensional homology class can be representedby a 1-dimensional submanifold of X . Example . (1) If we apply the theorem to A = ∂X , we see that any homology class in H ( X, ∂X ; Z )and H ( X, ∂X ; Z ) can be represented by a properly embedded submanifold. For A = ∅ we obtain the analogous statement for absolute homology groups.(2) Let F be a properly embedded 2-dimensional submanifold of D and let S be asurface in ∂D = S with ∂S = ∂F . Consider the 4-manifold X := D \ νF .In the boundary of X we have the surface B = S ∪ F × { } . It follows fromthe long exact sequence of the pair ( X, ∂X ) and Poincar´e duality that the map H ( X, ∂X ; Z ) → H ( ∂X ; Z ) is an epimorphism. It follows from Theorem 10.9,applied to A = ∂X , that there exists a 3-dimensional submanifold Y of D \ νF with ∂Y = B . This statement is folklore, and a proof using topological transversality formaps was recently written down by Lewark-McCoy [LM15].We will provide two proofs for Theorem 10.9. The first one use Theorem 8.1 to reducethe statement to the smooth case, and the second one uses the topological transversalityarguments from Section 10.For n = 4 the statement of the following theorem is precisely the statement of Theo-rem 10.9 in the smooth category. Proposition 10.11.
Let X be a compact orientable smooth n -manifold and let A be aunion of components of ∂X . Let (cid:96) = 1 or (cid:96) = 2 and let σ ∈ H n − (cid:96) ( X, A ; Z ) . Then thefollowing holds. (1) The class σ is represented by an ( n − (cid:96) ) -dimensional smooth orientable submani-fold Y with ∂Y ⊂ A . (2) Suppose (cid:96) = 1 and suppose we are given a oriented closed ( n − -dimensionalsmooth submanifold of A such that ∂ ( σ ) = [ B ] ∈ H n − ( A ) , then σ is represented byan oriented compact ( n − -dimensional smooth submanifold Y with ∂Y = B .Example . Let K ⊂ S be a knot. We write X = S \ νK . Let λ ⊂ ∂X be a longitudeof K . There exists a homology class σ ∈ H ( X, ∂X ; Z ) with ∂ ( σ ) = [ λ ]. It follows fromProposition 10.11 that there exists an orientable surface F in X with ∂F = K . Proof.
Let X be a compact orientable smooth n -manifold and let A be a union of compo-nents of ∂X . First we deal with the case (cid:96) = 1. Let σ ∈ H n − ( X, A ; Z ). Write (cid:101) A = ∂X \ A .Let PD : H ( X, (cid:101) A ; Z ) → H n − ( X, A ; Z ) be the Poincar´e duality isomorphism. We have H ( X, (cid:101) A ; Z ) ∼ = [ X/ (cid:101) A, S ] and any such class can be represented by a continuous map ϕ : X → S that is constant on (cid:101) A , and uniquely determined up to homotopy rel. (cid:101) A . Wecan and shall homotope ϕ to a smooth map. Furthermore, arrange that − ∈ S is aregular value of ϕ . Then ϕ − ( −
1) is an ( n − ∂X \ (cid:101) A , that is the boundary lies on A . This manifold has the desired property.Now suppose that we are given an oriented closed ( n − B of A such that ∂ ( σ ) = [ B ] ∈ H n − ( A ). Pick a collar neighbourhood ∂X × [0 ,
1] andchoose a continuous map ϕ : X \ (cid:0) ∂X × [0 , (cid:1) → S as above. Also choose a tubularneighbourhood B × [ − / , /
2] of B in A . Consider the map sending ( b, t ) to e πi ( t − , b ∈ B, t ∈ [ − / , /
2] and extend it to a smooth map ψ : A → S by sending all otherpoints into { e πit ∈ S : t ∈ [ − / , / } . Since ∂ ( σ ) = [ B ] ∈ H n − ( A ) ∼ = H ( A ; Z ), we seethat the restriction of ϕ to A × { } = A is homotopic to ψ : A → S . Therefore, using thishomotopy in the interval [ , ϕ to a function on X that restricts to ψ oneach A × { s } with s ∈ [0 , ]. Finally, smoothen ϕ without changing it on A × [0 , ] to obtaina smooth map X → S in the same homotopy class. This is possible since the original ϕ was already smooth on A × [0 , ]. Put differently, the new smooth map ϕ : X → S restricts to ψ on A = A × { } . HE FOUNDATIONS OF 4-MANIFOLD THEORY 51
Note that − ψ , and by changing ϕ outside A × [0 , ], we can alsoarrange − ϕ . The manifold Y = ϕ − ( −
1) satisfies [ Y ] = σ (thisfollows from Lemma 10.17 below, as explained in the second proof of Theorem 10.9) and ∂Y = B × { } = B .For (cid:96) = 2 the argument is similar: we have to replace the argument using S by theargument of [GS99, Proposition 1.2.3]. Recall from Theorem 4.5 that X is homotopyequivalent to finite CW complex. Therefore, we represent a codimension 2 homology class σ ∈ H n − ( X, ∂X ; Z ) ∼ = H ( X ; Z ) by a map X → CP ∞ , and homotope into the k -skeletonto a map f : X → CP k for k ≥
2. Now arrange f to be transverse to the codimension 2submanifold CP k − ⊂ CP k . The desired submanifold is the preimage Y = f − ( CP k − ). Weleave further details to the reader. (cid:3) Lemma 10.13.
Let W be a smooth n -manifold and let C be a compact subset. There existsa compact smooth n -dimensional submanifold X of W that contains C .Proof. By the Whitney embedding theorem (see e.g. [Lee11, Theorem 6.15]), there existsa proper embedding f : W → R n +1 . Recall that in this context proper means that thepreimage of a compact set is compact. Pick a point P ∈ R n +1 that does not lie in theimage of f . Denote the Euclidean distance to the point P by d : R n +1 → R ≥ . This mapis smooth outside P , so in particular d ◦ f : W → R ≥ is smooth. Since C is compact, thereexists an r ∈ R ≥ such that ( d ◦ f )( C ) ⊂ [0 , r ]. By Sard’s theorem, there exists a regularvalue x > r . Then X := ( d ◦ f ) − ([0 , x ]) has the desired properties. (cid:3) First proof of Theorem 10.9.
Let M be a compact orientable connected 4-manifold and let A be a union of components of ∂M . Let k = 2 or k = 3 and let σ ∈ H k ( X, A ; Z ).Pick a point P ∈ M \ ∂M and pick an open ball B ⊂ M \ ∂M containing P . It followsfrom a Mayer-Vietoris argument applied to M = ( M \ { P } ) ∪ B that the inclusion inducedmap H k ( M \ { P } , A ) → H k ( M, A ) is an isomorphism for k = 2 , σ ∈ H k ( M, A ). By the previous paragraph we can view σ as an element in H k ( M \ { P } , A ). By Theorem 8.1 the manifold M \ { P } is smooth. There exists a compactsubset K of M \ { P } such that σ lies in the image of H k ( K, A ) → H k ( M \ { P } , A ), sinceone can take the union of the images of the singular simplices in a singular chain represent-ing σ . By Lemma 10.13, there exists a compact 4-dimensional smooth submanifold X of M \ { P } that contains the compact set K ∪ ∂M . Note that A is again a union of compo-nents of ∂X . The desired statement of Theorem 10.9 is now an immediate consequence ofProposition 10.11 (1), with σ the image of σ ∈ H k ( K, A ) under the inclusion induced mapto H k ( X, A ). (cid:3) Now we collect the tools to conduct the proof of Theorem 10.9 in the topological category.We use transversality for submanifolds as a black box, but we endeavour to provide all theother details.
Definition 10.14.
Let ξ = S i −→ E π −→ S be a k -dimensional microbundle over S . A Thomclass of ξ is class τ ( ξ ) ∈ H k ( E, E \ i ( S ); Z ) that restricts to generator H k ( E x , E x \ i ( x ); Z ) ∼ = H k ( R k ; R k \ { } ; Z ) = Z for all x ∈ S . The microbundle ξ together with a Thom class isan oriented microbundle . Remark . As in the smooth case, consider the orientation bundle π : Or( ξ ) → S withfibre over x ∈ S the discrete setOr( ξ ) x = (cid:8) primitive classes of H k ( E x , E x \ i ( x ); Z ) (cid:9) . This is a Z / τ ( ξ ) determines a global section s ∈ Γ(Or( ξ )) by enforcing (cid:104) τ ( ξ ) , s ( x ) (cid:105) = 1 for every x ∈ S . By the same equation, a globalsection Γ(Or( ξ )) determines a Thom class. Remark . Let X be an oriented manifold. Let S be a submanifold with normalmicrobundle νS . An orientation of S determines a unique Thom class of νS compatiblewith the ambient orientation and vice versa.To prove Theorem 10.9, we will consider a map f : X → Y that is transverse to asubmanifold S . By the Remark 10.16, νS is an oriented microbundle and carrying theThom class τ . Note as f : νf − ( S ) → f ∗ νS is an isomorphic, also f ∗ τ is a Thom class of νf − ( S ) and we orient f − ( S ) accordingly. Before we proceed with the proof, we recall thefollowing compatibility between Thom classes and Poincar´e duality [Bre97, Definition 11.1,Corollary 11.6], interpreted for microbundles. Lemma 10.17.
Let X be a compact oriented manifold with fundamental class [ X ] , and let i : S → X be an oriented k -dimensional submanifold of X with normal microbundle νS .The composition H n − k (cid:0) νS, νS \ i ( S ) (cid:1) → H n − k ( X, X \ S ) → H n − k ( X ) PD X −−−→ H k ( X ) maps the Thom class τ of νS to the fundamental class i [ S ] .Proof. Recall that the Poincar´e duality map PD X is just X [ X ], capping with the fundamen-tal class. The composition of the last two maps factors through i : H k ( S ; Z ) → H k ( X ; Z ) byPoincar´e-Lefschetz duality [Bre97, Corollary 8.4]. Recall that the fundamental class [ S ] ∈ H k ( S, ∂S ; Z ) of the submanifold S is characterised by the property that for all x ∈ S theclass [ S ] is sent to the positive generator of H k ( S, S \ { x } ; Z ) ∼ = Z .Pick an x ∈ S and a k -disc U ⊆ S containing x such that the microbundle νS determinesa trivialisation E ( U ) = νS | U = U × D n − k . The fundamental class [ X ] restricts to afundamental class [ E ( U )] ∈ H n ( E ( U ) , ∂E ( U )), and Poincar´e duality can be computed HE FOUNDATIONS OF 4-MANIFOLD THEORY 53 locally: H n − k ( X, X \ S ) H k ( X ) H k ( νS ) H n − k ( E ( U ) , E ( U ) \ S ) H k ( E ( U ) , ( ∂U ) × D n − k ) H k ( U, ∂U ) . res X [ X ] X [ E ( U )] φ ∼ = We have to show that φ maps the Thom class τ U to the positive generator. For this, weclaim the following compatibility of the K¨unneth isomorphism with Poincar´e duality ofproducts. Claim.
The diagram below commutes: H n − k ( U × D n − k , U × ∂D n − k ) H ( U ) ⊗ H n − k ( D n − k , ∂D n − k ) H k ( U × D n − k , ∂U × D n − k )) H k ( U, ∂U ) ⊗ H ( D n − k ) PD E ( Dx ) PD U ⊗ PD Dn − k . We rewrite the diagram in terms of cap products and fundamental classes. Note thatthe fundamental class [ U × D n − k ] = [ U ] × [ D n − k ] is a cross product. Using compatibilityof the cross product with the cap product [Dol95, Section VII.12.17], we deduce that forarbitrary α ∈ H ( U ; Z ), and β ∈ H n − k ( D n − k , ∂D n − k ; Z ) the equality (cid:0) α × β (cid:1) X (cid:0) [ U ] × [ D n − k ] (cid:1) = ( − ( n − k ) ·| α | (cid:16)(cid:0) α X [ U ] (cid:1) × (cid:0) β X [ D n − k ] (cid:1)(cid:17) holds, where α × β = pr ∗ U α Y pr ∗ D n − k β denotes the external cohomological product. Thehorizontal arrows in the diagram above are exactly given by the cross product in homology,and the commutativity is exactly the formula above. This shows the claim.We deduce that PD E ( U ) τ U = [ U ] ⊗ pt since τ U = pt ⊗ ϑ , where ϑ ∈ H n − k ( D n − k , ∂D n − k ) isthe generator that evaluates positively on the orientation class. This implies that φ ( τ U ) =[ U ] ∈ H k ( U, ∂U ; Z ) is the positive generator, as desired. (cid:3) We give a proof of the main theorem above in the case of the codimension 1 and for acompact oriented manifold X , with A = ∂X . Second proof of Theorem 10.9.
Let α ∈ H ( X ; Z ) be the Poincar´e dual to σ ∈ H ( X, ∂X ).Recall the following correspondence between homotopy classes of maps to Eilenberg-Maclane spaces and cohomology classes of X ,[ X, S ] = [ X, K ( Z , ∼ = −→ H ( X ; Z ) f (cid:55)→ f ∗ θ, where θ is the Hom dual of the fundamental class of S . Note that we used here that X is homotopy equivalent to a CW complex. Pick an arbitrary point pt ∈ S and denotea tubular neighbourhood by ν (pt). Note that the Thom class τ pt for ν (pt) is mappedunder H ( S , S \ pt) → H ( S ) to θ . Let f : X → S be a map corresponding to α , so f ∗ θ = α . Make f transverse to a tubular neighbourhood of pt ∈ S using Theorem 10.8.Consequently, S := f − (pt) is an ( n − X . By definition, f induces an isomorphism f : νS → f ∗ ν (pt). We have, as elements in H ( X ; Z ), that α = f ∗ θ = f ∗ PD − X [pt] = f ∗ τ pt . Since f ∗ τ pt is the Thom class of νS , apply Lemma 10.17 and obtain α = f ∗ τ pt = τ S = PD − X [ S ] . (cid:3) Remark . We have seen that f ! [pt] = PD X ◦ f ∗ ◦ PD − S [pt] = [ f − (pt)], when f istransverse to pt. 11. Classification results for 4-manifolds
It is well-known (e.g. [CZ90, Theorem 5.1.1]) that any finitely presented group is thefundamental group of a closed orientable smooth 4-manifold. Markov [Mar58] used this factto show that closed 4-manifolds cannot be classified up to homeomorphism. To circumventthis group theoretic issue one aims to classify 4-manifolds with a given isomorphism typeof a fundamental group.In this section we present the known 4-manifold classification results that have beenobtained by the techniques of classical surgery theory in the topological category andusing Freedman’s disc embedding theorem [Fre82]. The use of this theorem requires thefundamental group of the 4-manifold be “good” [FQ90, Part II, Introduction], a conditionthat has a precise geometric description using the “ π -null disc property”. We will notreproduce that description here, but will instead note which groups are currently known tobe good. Freedman showed that the infinite cyclic group and finite groups are good [Fre82,pp. 658-659] (see also [FQ90, Section 5.1]). In addition, by [FT95, Lemma 1.2] the class ofgood groups is closed under extensions and direct limits. It follows that solvable groups aregood. Furthermore in [FT95, Theorem 0.1] and [FT95, KQ00] it was shown that groupswith subexponential growth are good.11.1. Simply connected -manifolds. The following theorem was the first noteworthyresult towards a classification of 4-dimensional manifolds.
Theorem 11.1.
Suppose M and N are two closed oriented simply-connected 4-dimensionalmanifolds. If the intersection forms are isometric, then M and N are homotopy equivalent. HE FOUNDATIONS OF 4-MANIFOLD THEORY 55
Proof.
This theorem was proved for smooth manifolds by Milnor [Mil58, Theorem 3], build-ing on work of Whitehead [Whi49]. A proof that works in the general case is given in [MH73,Chapter V, Theorem 1.5]. (cid:3)
We state Freedman’s classification for closed, simply connected 4-manifolds [Fre82, The-orem 1.5]. Note that a special case is the 4-dimensional topological Poincar´e conjecturethat every homotopy 4-sphere is homeomorphic to S . We give the statement as in [FQ90,Theorem 10.1]. The last sentence comes from [Qui86]. Theorem 11.2.
Fix a triple ( F, θ, k ) , where F is a finitely generated free abelian group, θ is a symmetric, nonsingular, bilinear form θ : F × F → Z , and k ∈ Z / . If θ is even, thatis θ ( x, x ) ∈ Z is even for every x ∈ F , then suppose that σ ( θ ) / ≡ k ∈ Z / .Then there exists a closed, simply connected, oriented -manifold M with H ( M ; Z ) ∼ = F ,with intersection form isometric to θ and with Kirby-Siebenmann invariant equal to k .Let M and M (cid:48) be two closed, simply connected, oriented -manifolds and let φ : H ( M ; Z ) ∼ = −→ H ( M (cid:48) ; Z ) be an isometry of the intersection forms. Suppose that ks( M ) = ks( M (cid:48) ) . Thenthere is an orientation preserving homeomorphism M ∼ = −→ M (cid:48) inducing φ on second homol-ogy. This homeomorphism is unique up to isotopy. In other words, every even, symmetric, integral matrix with determinant ± M = M (cid:48) implies that every automorphism ofthe intersection form of a closed, simply connected, oriented 4-manifold is realised by aself-homeomorphism of M .The following special case, when F = 0, is worth pointing out explicitly. Corollary 11.3 (4-dimensional Poincar´e conjecture) . Every homotopy 4-sphere is homeo-morphic to S .Proof. Let N be a homotopy 4-sphere. Then N and S are closed, simply connectedand oriented. Furthermore, H ( N ; Z ) = 0 and the zero map H ( N ; Z ) → H ( S ; Z ) isan isometry (between zero forms). By the last paragraph of Theorem 11.2, there is ahomeomorphism N ∼ = S realising this isometry. Note that since N has trivial and thereforeeven intersection form, ks( N ) = σ ( N ) / (cid:3) Non simply connected -manifolds. First, we present a classification result [FQ90,Theorem 10.7A] for closed, oriented 4-manifolds with fundamental group Z which is quitesimilar to Theorem 11.2. To state the theorem we need some extra definitions. Definition 11.4.
For a finitely generated free Z [ Z ] module F , a hermitian sesquilinearform θ : F × F → Z [ Z ] is called even if there is a left Z [ Z ]-module homomorphism q : F → Hom Z [ Z ] ( F, Z [ Z ]) with q + q ∗ : F → Hom Z [ Z ] ( F, Z [ Z ]) equal to the adjoint of θ . Otherwisewe call the form odd . Definition 11.5.
Two homeomorphisms h , h : M → N are pseudo-isotopic if there is ahomeomorphism H : M × I → N × I with H | M ×{ i } = h i : M × { i } → N × { i } for i = 0 , Theorem 11.6.
Fix a triple ( F, θ, k ) , where F is a finitely generated free Z [ Z ] -module, θ is a hermitian, nonsingular, sesquilinear form θ : F × F → Z [ Z ] , and k ∈ Z / . If θ is even,then suppose that σ ( R ⊗ θ ) / ≡ k ∈ Z / .Then there exists a closed, oriented -manifold M with π ( M ) ∼ = Z , with H ( M ; Z [ Z ]) isomorphic to F , whose equivariant intersection form λ M : H ( M ; Z [ Z ]) × H ( M ; Z [ Z ]) → Z [ Z ] is isometric to θ , and with ks( M ) = k .Let M and M (cid:48) be two closed, oriented -manifolds with π ( M ) ∼ = Z ∼ = π ( M (cid:48) ) andlet φ : H ( M ; Z [ Z ]) ∼ = −→ H ( M (cid:48) ; Z [ Z ]) be an isometry of the equivariant intersection forms.Suppose that ks( M ) = ks( M (cid:48) ) . Then there is an orientation preserving homeomorphism M ∼ = −→ M (cid:48) inducing φ on Z [ Z ] coefficient second homology. There are exactly two pseudo-isotopy classes of such homeomorphisms. The last sentence of this theorem is a correction to [FQ90, Theorem 10.7A] by Stong andWang [SW00].Here is another family of groups for which a complete classification of closed orientable 4-manifolds up to homeomorphism is known. This is the family of solvable Baumslag-Solitargroups B ( k ) := (cid:104) a, b | aba − b − k (cid:105) . Note that B (0) = Z and B (1) = Z . Baumslag-Solitar groups are solvable and, as wepointed out above, solvable groups are good. The next classification result was worked outby Hambleton, Kreck and Teichner in [HKT09]. Definition 11.7.
The w -type of a closed, oriented 4-manifold M is type I, II, III, asfollows: (I) w ( (cid:102) M ) (cid:54) = 0; (II) w ( M ) = 0; and (III) w ( M ) (cid:54) = 0 but w ( (cid:102) M ) = 0. Theorem 11.8.
Let B ( k ) be a solvable Baumslag-Solitar group and let M and N be closed,oriented 4-manifolds with fundamental group isomorphic to B ( k ) . Suppose that there is anisomorphism φ : H ( M ; Z [ B ( k )]) → H ( N ; Z [ B ( k )]) of Z [ B ( k )] -modules such that: (1) The map φ induces an isometry between the intersection form λ : H ( M ; Z [ B ( k )]) × H ( M ; Z [ B ( k )]) → Z [ B ( k )] and the corresponding intersection form on H ( N ; Z [ B ( k )]) . (2) The Kirby-Siebenmann invariants agree ks( M ) = ks( N ) . HE FOUNDATIONS OF 4-MANIFOLD THEORY 57 (3)
The w -types of M and N coincide.Then M and N are homeomorphic via an orientation preserving homeomorphism thatinduces φ : H ( M ; Z [ B ( k )]) → H ( N ; Z [ B ( k )]) . There is also a precise realisation result for these invariants [HKT09, Theorem B] and4-manifolds with fundamental group B ( k ).Next, 4-manifolds with finite cyclic fundamental groups were studied by Hambleton andKreck in [HK88, HK93]. Given a finitely generated abelian group G , let T G be its torsionsubgroup and let
F G := G/T G . Theorem 11.9.
Let G be a finite cyclic group and let M and N be closed, oriented 4-manifolds with fundamental group isomorphic to G . Suppose that there is an isomorphism φ : F H ( M ; Z ) → F H ( N ; Z ) such that the following hold. (1) The map φ induces an isometry between the intersection form λ M : F H ( M ; Z ) × F H ( M ; Z ) → Z and the intersection form λ N : F H ( N ; Z ) × F H ( N ; Z ) → Z . (2) The Kirby-Siebenmann invariants agree ks( M ) = ks( N ) . (3) The w -types of M and N coincide.Then M and N are homeomorphic via an orientation preserving homeomorphism thatinduces φ : F H ( M ; Z ) → F H ( N ; Z ) . To state a full realisation result for the invariants in Theorem 11.9 would require takingaccount of the interdependency between these invariants. However, a partial realisationresult is given by the following outline of a construction. For every finite cyclic group G ,[HK93, Proposition 4.1] produces rational homology spheres with w -type II and III, andwith fundamental group G . In w -type II, a rational homology sphere must have Kirby-Siebenmann invariant zero, since for spin manifolds M ks( M ) ≡ σ ( M ) / ∈ Z /
2, and thesignature of a rational homology sphere vanishes. In w -type III, [HK93, Proposition 4.1]gives two rational homology spheres, one with vanishing Kirby-Siebenmann invariant, andone with nonvanishing Kirby-Siebenmann invariant.(1) By taking the connected sum with a closed, spin simply connected manifold, wecan realise any even, nonsingular, symmetric, bilinear form as the intersection form λ M : F H ( M ; Z ) × F H ( M ; Z ) → Z of a closed, oriented 4-manifold M with fun-damental group G and with w type II. In this case ks( M ) is determined by thesignature of λ M .(2) Likewise we can realise every even λ M as the intersection form of a closed, oriented4-manifold M with fundamental group G and with w type III, with prescribedKirby-Siebenmann invariant.(3) Finally, by taking connected sum with a closed, oriented, simply connected 4-manifold, we can realise any odd, nonsingular, symmetric, bilinear form as the inter-section form λ M : F H ( M ; Z ) × F H ( M ; Z ) → Z of a closed, oriented 4-manifold M with fundamental group G and with w type I, with prescribed Kirby-Siebenmanninvariant. In his survey paper, Hambleton [Ham09, Theorem 5.2] also outlined a homeomorphismclassification for closed, spin 4-manifolds with finite odd order fundamental group. Somepartial results towards a classification for 4-manifolds whose fundamental groups are goodand have cohomological dimension 3 appear in [HH18].For nonorientable closed 4-manifolds, the homeomorphism classification results we areaware of are for fundamental group Z / Z in [Wan95].For nonorientable closed 4-manifolds with fundamental group Z /
2, the paper [HKT94]gives a complete list of invariants for distinguishing such manifolds up to homeomor-phism [HKT94, Theorem 2], and gives a list of the possible manifolds [HKT94, Theorem 3].Finally, simply-connected compact 4-manifolds with a fixed 3-manifold as boundary wereclassified by Boyer in [Boy86, Boy93] and independently by Stong [Sto93].12.
Stable smoothing of homeomorphisms
Wall [Wal64] proved that simply connected, closed, smooth 4-manifolds with isometricintersection forms are stably diffeomorphic. It follows that every pair of simply connected,closed, homeomorphic smooth 4-manifolds are stably diffeomorphic. We shall discuss theanalogous statement without the simply connected hypothesis.
Definition 12.1. (1) Let M and N be connected, smooth 4-manifolds. We say that M and N are stablydiffeomorphic if there is an integer k such that the connected sums M k S × S and N k S × S are diffeomorphic.(2) Let M and N be connected 4-manifolds. We say that M and N are stably home-omorphic if there is an integer k such that the connected sums M k S × S and N k S × S are homeomorphic.The next theorem is due to Gompf [Gom84]. Theorem 12.2.
Every homeomorphic pair of compact, connected, orientable, smooth 4-manifolds with diffeomorphic boundaries are stably diffeomorphic.
One might imagine a stronger statement, that given a homeomorphism f : M → N we can smoothen it stably. However such a statement is only known, given a lift of thestable tangent microbundle classifying map to BO , for simply connected 4-manifolds [FQ90,Chapter 8].The proof of Theorem 12.2 that we shall give using Kreck’s modified surgery [Kre99] wasoutlined in Teichner’s thesis [Tei92, Theorem 5.1.1].Gompf also proved that for every pair of compact, connected, nonorientable, smooth4-manifolds M and N that are homeomorphic, M S (cid:101) × S and N S (cid:101) × S are stably dif-feomorphic. We shall slightly improve on this statement. Theorem 12.3.
Let M and N be compact, connected, nonorientable, smooth 4-manifolds M and N . Suppose that M and N are homeomorphic via a homeomorphism restricting to HE FOUNDATIONS OF 4-MANIFOLD THEORY 59 a diffeomorphism ∂M ∼ = ∂N . If w ( (cid:102) M ) (cid:54) = 0 (cid:54) = w ( (cid:101) N ) , that is the universal covers of M and N are not spin, then M and N are stably diffeomorphic. The hypothesis that w ( (cid:102) M ) (cid:54) = 0 (cid:54) = w ( (cid:101) N ) cannot be dropped in general. Cappell andShaneson found an example of a smooth 4-manifold R that is homotopy equivalent to RP but that is not stably diffeomorphic to R P [CS71, CS76]. When these papers werepublished, it was not possible to prove that the fake RP manifold R is homeomorphicto RP , but this was later established [Rub84, p. 221] as a consequence of the work ofFreedman and Quinn [FQ90], and the fact that the Whitehead group of Z / K b ( K
3) = 22. Here is Kreck’s result from [Kre84].
Theorem 12.4.
Let π be a finitely presented group with a surjective homomorphism w : π → Z / . Then there exists a closed, smooth, connected -manifold W with fundamental group π and orientation character w , with the property that W K and W S × S are homeo-morphic -manifolds that are not stably diffeomorphic. One part of this is easy to see: if W is nonorientable then there are homeomorphisms W K ∼ = W E E S × S ∼ = W E E S × S ∼ = W S × S S × S ∼ = W S × S . Here we used Theorem 11.2 that simply connected closed 4-manifolds with Kirby-Siebenmanninvariant vanishing are determined by their intersection forms, and we used that the con-nected sum M N of an oriented manifold M with a nonorientable manifold N is homeo-morphic to M N .Gompf’s statement for the nonorientable case, given in the next corollary, follows easilyfrom Theorem 12.3. However note that Theorem 12.3 shows that for many nonorientable4-manifolds, the extra summand given by the twisted bundle S (cid:101) × S is not necessary. Corollary 12.5.
Let M and N be compact, connected, nonorientable, smooth 4-manifolds M and N . Suppose that M and N are homeomorphic via a homeomorphism restricting toa diffeomorphism ∂M ∼ = ∂N . Then M S (cid:101) × S and N S (cid:101) × S are stably diffeomorphic.Proof. Taking the connected sum of any 4-manifold with S (cid:101) × S ∼ = CP CP gives riseto a 4-manifold whose universal cover is not spin. The corollary therefore follows fromTheorem 12.3. (cid:3) In the following three sections we will prove Theorem 12.2 and Theorem 12.3. To keepthe notation manageable we will only provide a proof for closed manifolds.12.1.
Kreck’s modified surgery.
Below we will state a theorem due to Kreck that relatesstable diffeomorphisms of 4-manifolds with bordism theory. This came as a corollary ofKreck’s modified surgery theory [Kre99]. First we need some definitions from [Kre99].
Definition 12.6. A normal 1-type of a closed, connected, smooth 4-manifold M is a 2-coconnected fibration ξ : B → BO for which there is a 2-connected lift (cid:101) ν M : M → B of thestable normal bundle ν M : M → BO such that ξ ◦ (cid:101) ν M : M → BO is homotopic to ν M . Wecall such a choice of lift (cid:101) ν M : M → B a normal 1-smoothing . Remark . (1) Here by definition a 2-coconnected map induces an isomorphism on homotopygroups π i for i ≥ i = 2. A 2-connected map in-duces a surjection on π and an isomorphism on π and π .(2) The data of a normal 1-type is ξ : B → BO. The existence of (cid:101) ν M is a condition onthat data. Definition 12.8. A normal 1-type of a closed, oriented, connected 4-manifold M is a 2-coconnected fibration ξ : B → BTOP for which there is a 2-connected lift (cid:101) ν M : M → B ofthe stable normal microbundle ν M : M → BTOP (Definition 7.12) such that ξ ◦ (cid:101) ν M : M → BTOP is homotopic to ν M . We call such a choice of lift (cid:101) ν M : M → B a normal TOP .Normal 1-types ξ : B → BO of a closed, connected smooth 4-manifold are fibre homotopyequivalent over BO, and thus we may speak of the normal 1-type of a smooth 4-manifold,and similarly for the topological version. Here are some of the key examples in the orientedcase. We will give the details of the nonorientable case in Section 12.3.Write π = π ( M ) and let w ∈ H ( M ; Z /
2) be the second Stiefel-Whitney class of M .There are three main cases for the normal 1-types of oriented, closed smooth 4-manifolds.For more details, see [KLPT17, Sections 2 and 3]. Lemma 12.9.
Let M be a closed, oriented, connected, smooth 4-manifold. (1) Suppose that we have w ( (cid:102) M ) (cid:54) = 0 . Then ξ : B = B π × BSO → BO is the normal1-type of M , with the map ξ given by projection to BSO followed by the canonicalmap
BSO → BO . (2) Suppose that M is spin. Then ξ : B = B π × BSpin → BO is the normal 1-typeof M , with the map ξ given by projection to BSpin followed by the canonical map
BSpin → BO . (3) Suppose that we have w ( M ) (cid:54) = 0 but w ( (cid:102) M ) = 0 . Then ξ : B = B π × BSpin → BSO is the normal 1-type of M , with B as in the spin case, but the map ξ twisted usinga complex line bundle on B π defined in terms of w ( M ) . Similarly there are three main cases for the normal 1-types of closed, oriented, connected4-manifolds. Let STOP( n ) be the group of orientation preserving homeomorphisms of R n fixing the origin and let STOP be the corresponding colimit of STOP( n ). Let TOPSpinbe the universal (2-fold) cover of STOP. Here π (STOP) ∼ = π (TOP) ∼ = π (O) ∼ = Z / π (TOP) ∼ = π (O) we use that there is a 6-connected map TOP / O → K ( Z / ,
3) [KS77, Essay V, Section 5].
Lemma 12.10.
Let M be a closed, oriented, connected 4-manifold. HE FOUNDATIONS OF 4-MANIFOLD THEORY 61 (1)
Suppose that we have w ( (cid:102) M ) (cid:54) = 0 . Then ξ : B = B π × BSTOP → BTOP is thenormal 1-type of M , with the map given by projection to BSTOP followed by thecanonical map
BSTOP → BTOP . (2) Suppose that M is spin. Then ξ : B = B π × BTOPSpin → BTOP is the normal 1-type of M , with the map given by projection to BTOPSpin followed by the canonicalmap
BTOPSpin → BTOP . (3) Suppose that we have w ( M ) (cid:54) = 0 but w ( (cid:102) M ) = 0 . Then ξ : B = B π × BTOPSpin → BTOP is the normal 1-type of M , with B as in the spin case, but the map ξ twistedusing a complex line bundle on B π defined in terms of w ( M ) . For more details on these assertions, see [KLPT17, Sections 2 and 3]. Here is the relevanttheorem of Kreck, which relates bordism over the normal 1-type to stable diffeomorphism.
Theorem 12.11.
Two closed, connected, smooth 4-manifolds M and N with χ ( M ) = χ ( N ) and fibre homotopy equivalent normal 1-types are stably diffeomorphic if and only if [( M, (cid:101) ν M )] = [( N, (cid:101) ν N )] ∈ Ω ( B, ξ ) for some choices of normal 1-smoothings (cid:101) ν M and (cid:101) ν N .Sketch of the proof. One direction is quite easy: one has to check that M and M S × S are bordant over the normal 1-type of M .For the other direction, start with a 5-dimensional bordism W over ( B, ξ ) and performsurgery on W below the middle dimension to make the map to B π ( W ) → π ( B )) by framed embedded spheres, and removethickenings of these spheres. Also remove tubes of these to either M or N , tubing enough2-spheres to either side so as to preserve the Euler characteristic equality. This operationof removing copies of S × D , tubed to the boundary, has the effect of adding copies of S × S to M and N giving rise to M (cid:48) and N (cid:48) respectively, and it converts W to an s -cobordism W (cid:48) . That ( W (cid:48) ; M (cid:48) , N (cid:48) ) is an s -cobordism means by definition that the inclusionmaps M (cid:48) → W (cid:48) and N (cid:48) → W (cid:48) are simple homotopy equivalences. The stable s -cobordismtheorem [Qui83] states that every 5-dimensional s -cobordism becomes diffeomorphic to aproduct after adding copies of S × S × I along a smoothly embedded interval I ⊂ W (cid:48) withone endpoint on each of M and N . This completes the sketch proof of Theorem 12.11. (cid:3) The proof of the topological version is similar.
Theorem 12.12.
Two closed, topological 4-manifolds M and N with χ ( M ) = χ ( N ) andfibre homotopy equivalent normal 1-types are stably homeomorphic if and only if [( M, (cid:101) ν M )] = [( N, (cid:101) ν N )] ∈ Ω TOP4 ( B TOP , ξ
TOP ) for some choices of normal 1-smoothings (cid:101) ν M and (cid:101) ν N . From now on, to ease notation, we will abbreviate Ω
TOP4 ( B TOP , ξ
TOP ) with Ω
TOP4 ( B, ξ ). Proof.
One direction is again quite easy: we need that homeomorphic manifolds are bordantover B , and that M and M S × S are bordant in Ω TOP4 ( B, ξ ). For the other direction, apply the same argument as above to improve a cobordism W to an s -cobordism. The stable s -cobordism theorem applies to topological s -cobordisms as well as to smooth s -cobordisms.This is not written in [Qui83], but the same proof applies, with the following additions(see the Exercise on [FQ90, p. 107]). First, 5-dimensional cobordisms admit a topologicalhandlebody structure [FQ90, Theorem 9.1]. The proof of [Qui83] consists of simplifying ahandle decomposition, and tubing surfaces in 4-manifolds around and into parallel copiesof one another to remove intersections. This is possible in the topological category by usingtransversality (Theorem 10.3) to arrange that intersections between surfaces are isolatedpoints, and the existence of normal bundles (Theorem 6.17) to take parallel copies usingsections. (cid:3) Stable diffeomorphism of homeomorphic orientable 4-manifolds.
Now wewill explain the proof of Theorem 12.2. For the convenience of the reader, we recall thestatement.
Theorem 12.13.
Every homeomorphic pair of closed, connected, orientable, smooth 4-manifolds are stably diffeomorphic.
The proof will rest on the following proposition.
Proposition 12.14.
Let ( B, ξ ) be one of the oriented smooth normal 1-types from Lemma 12.9,and let ( B TOP , ξ
TOP ) be the corresponding topological normal 1-type from Lemma 12.10obtained by replacing BSO with
BSTOP or BSpin with
BTOPSpin as appropriate. Theforgetful map F : Ω ( B, ξ ) → Ω TOP4 ( B TOP , ξ
TOP ) = Ω
TOP4 ( B, ξ ) is injective. The combination of this proposition with Theorem 12.11 and Theorem 12.12 implies thefollowing corollary, which is the closed version of Theorem 12.2, with a slightly more precisestatement concerning orientations.
Corollary 12.15.
Every pair of smooth, closed, connected, oriented 4-manifolds that arehomeomorphic via an orientation preserving homeomorphism are stably diffeomorphic viaan orientation preserving diffeomorphism.Proof.
We prove the corollary assuming Proposition 12.14. Homeomorphic 4-manifoldsare in particular stably homeomorphic and have the same normal 1-types. Therefore twohomeomorphic smooth 4-manifolds as in the statement of the corollary are bordant overthe normal 1-type, so give rise to equal elements in Ω
TOP4 ( B, ξ ). By Proposition 12.14, theygive rise to equal elements of Ω ( B, ξ ). Then by Theorem 12.11, the two 4-manifolds arestably diffeomorphic, as asserted. (cid:3)
Proof of Proposition 12.14.
Let S be SO in case (1) of the smooth list of 1-types above,and let S denote Spin in cases (2) and (3).Let ST be STOP in case (1) of the topological list of 1-types above, and let ST denoteTOPSpin in cases (2) and (3). HE FOUNDATIONS OF 4-MANIFOLD THEORY 63
The James spectral sequence [Tei92, Theorem 3.1.1],[KLPT17, Section 3] is of the form: E p,q = H p (B π ; Ω Sq ) ⇒ Ω p + q ( B, ξ ) . We have that Ω S ∼ = Z , detected by the signature. Indeed, the signature is a Z -valuedinvariant for stable diffeomorphisms. This arises on the E page as H (B π ; Ω S ) ∼ = Z . Claim.
This term H (B π ; Ω S ) survives to the E ∞ page. That is, all differentials with thisas codomain are trivial.Let us prove the claim. Since Ω Sq is torsion for q = 1 , ,
3, no terms from those q -linescan map to H (B π ; Ω S ) under a differential.Aside from H (B π ; Ω S ), there is one other potentially infinite term on the E ∞ page,namely the subgroup of H (B π ; Ω S ) arising as the kernel of relevant differentials. Theimage of a 4-manifold in here is the image c ∗ ([ M ]) of the fundamental class under theclassifying map c ∗ : H ( M ; Z ) → H (B π ; Z ) ∼ = H (B π ; Ω S ).There could be a differential H (B π ; Ω S ) → H (B π ; Ω S ). However if there were a nonzerodifferential, then only finitely many signatures would occur for 4-manifolds with normal 1-type B and fixed invariant in H (B π ; Ω S ). But we can add copies of either CP or the K c ∗ ([ M ]) the same, but change the signature by1 or 16, for each copy of CP or K
3, respectively. This completes the proof of the claimthat the term H (B π ; Ω S ) survives to the E ∞ page.We note that in case (3), the entry in Ω Spin4 is not necessarily the signature of the manifold;this entry could be a multiple of the signature.Since H (B π ; Ω S ) survives to the E ∞ term, we have a short exact sequence:0 → Ω S → Ω ( B, ξ ) → (cid:101) Ω ( B, ξ ) → , where (cid:101) Ω ( B, ξ ) denotes the quotient. That is, there is a filtration with iterated gradedquotients given by the E ∞ page:0 ⊆ E ∞ , = Ω S ⊆ · · · ⊆ Ω ( B, ξ ) , and it is the quotient by the E ∞ , subgroup that we denote (cid:101) Ω ( B, ξ ).Similarly, for the topological case, we have0 → Ω ST → Ω TOP4 ( B, ξ ) → (cid:101) Ω TOP4 ( B, ξ ) → . The only difference in the proof from the smooth case is that we use E in place of K H (B π ; Ω ST ), and we also haveto argue that the Kirby-Siebenmann invariant Z / ⊂ Ω ST survives to the E ∞ page. Butthe Kirby-Siebenmann invariant is additive, and realised on simply connected manifolds,either by the Chern manifold ∗ CP (Construction 8.5) or the E manifold depending onthe normal 1-type. Thus there exist bordism classes (i.e. stable homeomorphism classes)realising both trivial and nontrivial Kirby-Siebenmann invariants within a normal 1-type,and so this Z / Since the structure forgetting map Ω Sq → Ω STq is an isomorphism for 0 ≤ q ≤
3, wehave an isomorphism (cid:101) Ω ( B, ξ ) ∼ = (cid:101) Ω TOP4 ( B, ξ ). This uses that the differentials agree, bynaturality of the James spectral sequence with respect to homology theories. Indeed, notethat the differentials depend only on the classifying space B π , and on the complex linebundle E → B π in case (3). Both are category independent.Then there is a map of short exact sequences:0 (cid:47) (cid:47) Ω S (cid:47) (cid:47) (cid:15) (cid:15) Ω ( B, ξ ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:101) Ω ( B, ξ ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) (cid:47) (cid:47) Ω ST (cid:47) (cid:47) Ω TOP4 ( B, ξ ) (cid:47) (cid:47) (cid:101) Ω TOP4 ( B, ξ ) (cid:47) (cid:47) . The left vertical map is injective, either inclusion into the first summand Z → Z ⊕ Z / Z → Z in the spin case. Since the left and right vertical maps areinjective, it follows from a diagram chase that the central vertical map is also injective, asrequired. (cid:3) Non-orientable 4-manifolds and stable diffeomorphism.
For the convenienceof the reader, we recall the statement of Theorem 12.3.
Theorem 12.16.
Let M and N be closed, connected, nonorientable, smooth 4-manifolds M and N . Suppose that M and N are homeomorphic. If w ( (cid:102) M ) (cid:54) = 0 (cid:54) = w ( (cid:101) N ) , that is theuniversal covers of M and N are not spin, then M and N are stably diffeomorphic. Here is the normal 1-type for nonorientable manifolds with a certain w -type [Tei92,Chapter 2]. Lemma 12.17.
Let M be a nonorientable closed, connected smooth 4-manifold with w ( (cid:102) M ) (cid:54) = 0 . Then the normal 1-type of M is ξ : B = B π × BSO → BO with the map ξ = w ⊕ Bi given by the Whitney sum of a bundle on B π determined by w : π → Z / andthe canonical map Bi : BSO → BO induced by the inclusion i : SO → O . Lemma 12.18.
Let M be a nonorientable closed, connected 4-manifold with w ( (cid:102) M ) (cid:54) = 0 .Then the normal 1-type of M is ξ : B = B π × BSTOP → BTOP with the map ξ = w ⊕ Bi given by the Whitney sum of a bundle on B π determined by the orientation character w : π → Z / and the canonical map Bi : BSTOP → BTOP induced by the inclusion i : STOP → TOP . These normal 1-types gives rise to a James spectral sequence governing the bordismgroups of (
B, ξ ) E p,q = H p (B π ; Ω w q ) ⇒ Ω p + q ( B, ξ ) . Note that the coefficients are twisted using Z w ⊗ Ω q , where by definition, g ∈ π acts on Z w by multiplication by ( − w ( g ) . The corresponding topological James spectral sequence is: E p,q = H p (B π ; (Ω STOP q ) w ) ⇒ Ω TOP p + q ( B, ξ ) . As in the previous section, here we abbreviate Ω
TOP4 ( B TOP , ξ
TOP ) with Ω
TOP4 ( B, ξ ). HE FOUNDATIONS OF 4-MANIFOLD THEORY 65
By Kreck’s theorem (Theorem 12.11) and the argument in the proof of Theorem 12.2,in order to prove Theorem 12.3 it suffices to prove the next injectivity statement.
Proposition 12.19.
Let ( B, ξ ) be one of the normal 1-types in Lemma 12.17 and let ( B TOP , ξ
TOP ) be the corresponding topological normal 1-type over BTOP . The forgetfulmap F : Ω ( B, ξ ) → Ω TOP4 ( B TOP , ξ
TOP ) = Ω
TOP ( B, ξ ) is injective.Proof of Theorem 12.3 assuming Proposition 12.19. Since homeomorphic 4-manifolds havethe same normal 1-types and so are trivially TOP bordant over this normal 1-type, injectiv-ity of F implies that homeomorphic nonorientable, closed, connected, smooth 4-manifoldsare smoothly bordant over their normal 1-type, and therefore by Theorem 12.11 are stablydiffeomorphic. (cid:3) Proof of Proposition 12.19.
The structure of the proof is very similar to that of the proofof Proposition 12.14. This proof is therefore somewhat terse. In the smooth James spectralsequence computing Ω ( B, ξ ), we consider the term on the E page H (B π ; Ω w ) ∼ = Z / CP (note that for nonorientable manifolds connected sumwith C P and C P is the same), both mod 2 Euler characteristics are realised by bordismclasses over ( B, ξ ). Also note that adding C P does not change the normal 1-type when w ( (cid:102) M ) (cid:54) = 0. Therefore H (B π ; Ω w ) ∼ = Z / E ∞ page.In the topological case, the corresponding term in the James spectral sequence computingΩ TOP4 ( B, ξ ) is H (B π ; (Ω STOP4 ) w ) ∼ = Z / ⊕ Z / . We can add copies of C P and ∗ C P to show that this term survives to the E ∞ page.The structure forgetting map Z / ∼ = H (B π ; Ω w ) → H (B π ; (Ω STOP4 ) w ) ∼ = Z / ⊕ Z / ( B, ξ ) and Ω
TOP4 ( B, ξ ) arising from the spectral sequencegive rise to short exact sequences, that form the rows of the following commutative diagram:0 (cid:47) (cid:47) Z / (cid:47) (cid:47) (cid:15) (cid:15) Ω ( B, ξ ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:101) Ω ( B, ξ ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) (cid:47) (cid:47) Z / ⊕ Z / (cid:47) (cid:47) Ω TOP4 ( B, ξ ) (cid:47) (cid:47) (cid:101) Ω TOP4 ( B, ξ ) (cid:47) (cid:47) . We noted above that the left vertical map is injective. Since Ω q → Ω STOP q is an isomorphismfor 0 ≤ q ≤
3, the right vertical map is an isomorphism and is therefore injective. It followsfrom a diagram chase that the central vertical map is also injective, as required. (cid:3)
Twisted intersection forms and twisted signatures
Let M be a compact, orientable, connected 4 m -dimensional manifold. We write π := π ( M ). Let α : π → U ( k ) be a unitary representation. We view the elements of C k as rowvectors. Given g ∈ π and v ∈ C k , define v · g := v · α ( g ). Thus we can view C k as a right Z [ π ]-module. Denote this module by C kα . Define the twisted intersection form of ( M, α ) tobe the form Q M : H m ( M ; C kα ) × H m ( M ; C kα ) PD − × PD − −−−−−−−−→ H m ( M, ∂M ; C kα ) × H m ( M, ∂M ; C kα ) ↓ Y H m ( M, ∂M ; C kα ⊗ C kα ) ↓ (cid:104) , (cid:105) H m ( M, ∂M ; C ) ↓ PD H ( M ; C ) = C . Here the first and the last map are given by the isomorphisms from the Poincar´e dualitytheorem A.15 and the second map is given by Lemma A.11. Note that in the bottom weview C as a trivial Z [ π ]-module. The third map is induced by the following homomorphismof right Z [ π ]-modules: C kα ⊗ C kα → C ( v, w ) (cid:55)→ (cid:104) v, w (cid:105) = vw T . It follows easily from the definitions that Q M is sesquilinear, namely C -conjugate linear inthe first entry and C -linear in the second entry. The usual proof for the (anti-) symmetryof the cup product e.g. [Hat02, Theorem 3.14], can be modified to show that the abovepairing is hermitian, that is for every v, w ∈ H m ( M ; C kα ) we have Q M ( v, w ) = Q M ( w, v ).Since Q M is hermitian, its signature is defined as the difference in the number of positiveand negative eigenvalues. We refer to the signature of Q M as the twisted signature σ ( M, α ).Finally, for a group homomorphism γ : π ( M ) → Γ, denote the corresponding L -signature by σ (2) ( M, γ ), as defined in say [Ati76, L¨uc02] and [COT03, Chapter 5].
Theorem 13.1.
Let M be a closed oriented -manifold. (1) For every finite cover p : (cid:102) M → M we have σ ( (cid:102) M ) = [ (cid:102) M : M ] · σ ( M ) . (2) For every unitary representation α : π ( M ) → U ( k ) we have σ ( M, α ) = k · σ ( M ) . (3) For every group homomorphism γ : π ( M ) → Γ we have σ (2) ( M, γ ) = σ ( M ) .Remark . (1) The same statement does not hold for 4-dimensional Poincar´e complexes in gen-eral. More precisely, Wall [Wal67, Corollary 5.4.1] gave examples of 4-dimensionalPoincar´e complexes for which the signature is not multiplicative under finite covers.(2) Alternative proofs for the first statement and the third statement are provided bySchafer [Sch70, Theorem 8] and L¨uck-Schick [LS03, Theorem 0.2]. In fact thesepapers are also valid for manifolds of any dimension 4 m . HE FOUNDATIONS OF 4-MANIFOLD THEORY 67
Proof.
For smooth manifolds, the first statement is a consequence of the Hirzebruch signa-ture theorem (see e.g. [MS74]) while the second statement was proven in [APS75] (in factthe second statement contains the first statement as a special case). The third statementwas proven in [Ati76, p. 44].We will prove the second statement of the theorem. The other statements can be provedin a similar fashion. We refer to [COT03, Lemma 5.9] for a proof of (3).So let M be a closed oriented 4-manifold and let α : π ( M ) → U ( k ) be a unitaryrepresentation. By Theorem 8.6, there exists a closed orientable simply-connected 4-manifold N such that M N is smooth. We have π ( M N ) = π ( M ) ∗ π ( N ) ∼ = π ( M )since π ( N ) = { } . Let β : π ( N ) → U ( k ) be the trivial representation. We also write α ∗ β : π ( M N ) = π ( M ) → U ( k ) for the representation uniquely determined by α on π ( M ).By Proposition 5.15, we have σ ( M N ) = σ ( M ) + σ ( N ). Furthermore a slight general-isation of Proposition 5.15 shows that σ ( M N, α ∗ β ) = σ ( M, α ) + σ ( N, β ). Finally, wehave σ ( N, β ) = k · σ ( N ). The desired statement follows from these equalities and from theformula for twisted signatures of the closed smooth manifold M N . (cid:3) Reidemeister torsion in the topological category
The simple homotopy type of a manifold.
In the following we need the notionof a simple homotopy equivalence. We will not give a definition, instead we refer to [Tur01,p. 40] for details. Roughly, a simple homotopy equivalence between CW complexes is asequence of elementary expansions and collapses of pairs of cells whose dimension differsby one.The following definition allows us to define a simple homotopy type even for topologicalspaces which are not homeomorphic to a CW complex.
Definition 14.1.
Let (
W, V ) be a pair of topological spaces. Consider tuples (
W, V, f, X, Y ),where (
X, Y ) is a finite CW complex pair with Y ⊆ X , and f : W → X and f | V : V → Y are homotopy equivalences. Two such tuples ( W, V, f, X, Y ) and (
W, V, f (cid:48) , X (cid:48) , Y (cid:48) ) with( X (cid:48) , Y (cid:48) ) another finite CW pair and f (cid:48) : ( W, V ) → ( X (cid:48) , Y (cid:48) ) are equivalent if there exists asimple homotopy equivalence of pairs s : ( X, Y ) → ( X (cid:48) , Y (cid:48) ) such that s ◦ f is homotopic to f (cid:48) and s | Y ◦ f | V : V → Y (cid:48) is homotopic to f (cid:48) | V . Such an equivalence class of ( W, V, f, X, Y )is called a simple homotopy type of (
W, V ). In particular, a simple homotopy type of ( W, ∅ )is called a simple homotopy type of W .Consider now a compact connected n -manifold M . If M admits a smooth structure,then M admits in particular a CW structure [Hir94, Section 6.4], and we equip M with thesimply homotopy type given by ( M, Id). By Chapman’s theorem [Cha74, p. 488] below,this simple homotopy type is independent of the choice of CW structure on M . Theorem 14.2 (Chapman) . Let W be a compact topological space. Any two CW structureson W are simple homotopy equivalent. As we pointed out in Section 4, it is unknown whether every compact manifold admitsa CW structure. In the remainder of this section, we will nonetheless introduce the simplehomotopy type of a compact manifold M following [KS77, Essay III, Section 4]. The firststep is to construct a disc bundle D ( M ) → M together with a PL structure on the totalspace D ( M ). We will work with a compact m -dimensional manifold M with boundary ∂M , and seek to construct the simple homotopy type of ( M, ∂M ). (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) B ( M ) R n triangulation of B ( M ) M Figure 12. 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(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) X × [ − , R n R n B ( ∂M ) ∂MM B ( M ) XB ∂ ( M ) ∂B ( M ) X∂B × [ − , Figure 13.
Construction . We deal with the case ∂M = ∅ first, and then later address the addi-tional complications arising from having nonempty boundary.As a first step to constructing the disc bundle D ( M ) ⊂ R n , we need an embedding of M into R n − for some large integer n − > m + 5. For a closed m –manifold M suchan embedding is readily available [Hat02, Corollary A.9]. It follows from Theorem 7.10that for n − > m + 5 all such embeddings of M are isotopic, and that they admit anormal microbundle ν R n − ( M ) that is unique up to isotopy. By Theorem 7.7 this normalmicrobundle ν R n − ( M ) can be upgraded to a topological R n − − m –bundle. By taking theproduct with R , construct an embedding M ⊂ R n whose normal microbundle is ν ( M ) = ν R n − ( M ) × R . Since we stabilised once, the normal microbundle ν ( M ) contains a normaldisc bundle B ( M ) [KS77, Essay III, Proposition 4.4].The next big step will be to upgrade B ( M ) ⊂ R n from a submanifold to a PL submani-fold. Since the interior is codimension 0, the interior of B ( M ) is automatically also a PLsubmanifold. However, we have to arrange ∂B ( M ) to be a PL submanifold of R n itself.In the next paragraphs, we modify the PL structure on R n such that ∂B ( M ) becomes aPL submanifold and then isotope this new PL structure on R n back to the standard PLstructure. HE FOUNDATIONS OF 4-MANIFOLD THEORY 69
Using the Collar Neighbourhood Theorem 2.5, pick a collar W ∂ = ∂B ( M ) × ( − ,
1) and D ∂ = ∂B ( M ) × [ − , ]. The local product structure theorem 5.18 [KS77, Essay I, Theo-rem 5.2] gives a PL structure σ ∂ on R n such that ∂B ( M ) is a PL submanifold and σ ∂ isconcordant to the standard PL structure σ std .Now we will isotope the pair ∂B ( M ) ⊂ B ( M ) so that they become PL submanifolds of( R n , σ std ). The PL structure σ ∂ is concordant to σ std . Since concordance implies isotopy[KS77, Essay I, Theorem 4.1] in dimension m ≥
6, there is an isotopy φ t ∈ Homeo( R n )such that φ = Id, and φ ∗ σ std = σ ∂ . Consequently, D ( M ) := φ ( B ( M )) and D ( ∂M ) := φ ( B ( ∂M )) are PL submanifolds of ( R n , σ std ), which defines a simple type of M .Having finished the case ∂M (cid:54) = ∅ , next we discuss the procedure for a manifold M withnon-empty boundary.Take the union of M with an external open collar ∂M × [0 ,
1) of its boundary. Write M (cid:48) := M ∪ ∂M ∂M × [0 , M (cid:48) into R n as in the closed case [Hat02, Corollary A.9].Note that M (cid:48) has empty boundary and so it is properly embedded. As in the closed case,obtain a disc bundle B ( M (cid:48) ) and let B ( M ) be the restriction of this disc bundle to M .Now we have to take much more care. Note that ∂B ( M ) decomposes as ∂B ( M ) = B ( ∂M ) ∪ X B ∂ ( M ). Here B ∂ ( M ) denotes the fibrewise boundary and X = ∂ (cid:0) B ( ∂M ) (cid:1) denotes the intersection of B ( ∂M ) and B ∂ ( M ). As above, we will find a PL structure σ ∂ of R n such that each subset B ( ∂M ), X and B ∂ ( M ) is PL–submanifold of R n .Our first goal is to modify the PL structure on R n so that the corners X become aPL submanifold of R n . Denote the standard PL structure on R n by σ std . Pick a bicollar ∂B ( M ) × [ − , ⊂ R n of the boundary of the codimension 0 submanifold B ( M ). Again bythe Collar Neighbourhood Theorem 2.5, we can pick a bicollar X × [ − , ⊂ ∂B ( M ). Weconsider the open set W X := X × ( − , ⊂ ∂B ( M ) × ( − , ⊂ R n , and D X = X × [ − , ] .The local product structure theorem 5.18 [KS77, Essay I, Theorem 5.2] gives a PL structureon X and a PL structure σ X on R n , which is concordant to σ std rel. R n \ (cid:0) X × ( − , ) (cid:1) .This PL structure σ X has the property that it agrees with the product PL structure on X × ( − , in a neighbourhood of D X . Thus X is a PL submanifold of ( R n , σ X ).Now we arrange the next stratum ∂B ( M ) ⊃ X to be a PL submanifold of R n . Near D X = X × [ − , ], the PL structure σ X is the product PL structure, and therefore ∂B ( M ) ∩ Int D X = X × ( − , ) × { } is already a PL submanifold of ( R n , σ X ). Fur-thermore, σ X is a product along ( − , ) near X × [ − , ]. Pick W ∂ = ∂B ( M ) × ( − , C ∂ = X × [ − , ] × [ − , ] and D ∂ = ∂B ( M ) × [ − , ]. As above, the local structure the-orem 5.18 [KS77, Essay I, Theorem 5.2] gives a PL structure σ ∂ on R n such that ∂B ( M )is a PL submanifold and σ ∂ is concordant to σ X rel. (cid:0) R n \ ∂B ( M ) × ( − , ) (cid:1) ∪ C ∂ . Since X ⊂ C ∂ the submanifold X is still a PL submanifold of ( R n , σ ∂ ).As in the closed case, use a concordance from σ ∂ to σ std to obtain an isotopy φ t ∈ Homeo( R n ) such that φ = Id, and φ ∗ σ std = σ ∂ . Define D ( M ) := φ ( B ( M )) and D ( ∂M ) := φ ( B ( ∂M )), which are both PL submanifolds of ( R n , σ std ). This finishes the case where M has nonempty boundary. In both cases, ∂M empty and non-empty, our construction involved many choices. Let D (cid:48) ( ∂M ) ⊂ D (cid:48) ( M ) be obtained by other choices. Following the discussion [KS77, p. 123],we can suitably stabilise the bundles and find a commutative diagram of PL maps: D ( M ) × D s D (cid:48) ( M ) × D r D ( ∂M ) × D s D (cid:48) ( ∂M ) × D r , ∼ = ∼ = where D k denotes the disc with its standard PL structure and the horizontal maps are PLisomorphisms that preserve the zero sections up to homotopy. Definition 14.4.
The simple homotopy type of a compact connected n -manifold M is givenby ( M, s ), where s : M → D ( M ) is the inclusion of the 0-section. The simple homotopytype of the pair ( M, ∂M ) is given by the square
M D ( M ) ∂M D ( ∂M ) , ss | ∂M where D ( ∂M ) ⊂ D ( M ) are the disc bundles from Construction 14.3, with CW structuresarising from a choice of PL triangulations corresponding to the PL structures.By the commutative square at the end of Construction 14.3, the simple homotopy typeof ( M, ∂M ) is well-defined. Here we use that PL isomorphisms are simple: for any choiceof triangulations underpinning the PL structures, the resulting homeomorphism is a simplehomotopy equivalence. Also stabilising by D s does not change the simple homotopy type,since as PL manifolds D s ∼ = D s − × [ − , D s − × { } → D s − × [ − ,
1] is a simpleequivalence.
Remark . Why is the simply homotopy type of ∂M obtained in this way the same asthat obtained by applying Construction 14.3 with ∂M considered as a manifold withoutboundary?For suitably high n , we may assume that the embedding of ( M, ∂M ) into R n is isotopic,and thus by Theorem 2.10 ambiently isotopic, to an embedding with i : ∂M (cid:44) → { (cid:126)x ∈ R n | x = 0 } ∼ = R n − and an (interior) collar ∂M × [0 ,
1] embedded as a product in { (cid:126)x ∈ R n | ≤ x ≤ } with ( x, t ) (cid:55)→ ( i ( x ) , t ), as in Theorem 6.5. Such an isotopy doesnot affect the simple homotopy type obtained, by the argument sketched above, whichcan also be found on [KS77, p. 123]. The simple homotopy type of ∂M obtained fromConstruction 14.3, via an embedding of ∂M into R n − , uses a disc bundle D ( ∂M ) thatstabilises using the x direction to a disc bundle D (cid:48) ( ∂M ), with fibre a disc of one dimensionhigher, for ∂M embedded in R n . This latter disc bundle gives rise to the canonical simplyhomotopy type of ∂M from Definition 14.4. HE FOUNDATIONS OF 4-MANIFOLD THEORY 71
Remark . If M is a smooth manifold, then M has an underlying PL structure, andwith a bit more care in Construction 14.3, we can arrange that the bundle D ( M ) is a PLbundle. Note that this is stronger than just a PL structure on the total space. For PLbundles, the bundle projection D ( M ) → M is a simple homotopy equivalence. Indeed, fortrivial bundles this is discussed above, and in general the projection is an α -equivalence (anotion defined in loc. cit.) for any cover α of M and so is simple [Fer77, Corollary 3.2]. Itfollows that the simple homotopy type defined by ( M, Id) agrees with the one of (
M, s ),and the same holds for the relative simple homotopy type of the pair (
M, ∂M ).According to [KS77, Essay III, Theorem 5.11], if a manifold has a triangulation, thenthe simple homotopy type of the manifold agrees with the simple homotopy type of thattriangulation. It is not clear to us whether the analogous statement holds if M has aCW-structure not coming from a triangulation.14.2. The cellular chain complex and Poincar´e triads.
Throughout this section let M be a compact connected n -manifold. Furthermore assume that we are given a decomposition ∂M = R − ∪ R + into codimension zero submanifolds such that ∂R − = R − ∩ R + = ∂R + .The following proposition follows from the argument of Construction 14.3, applied witheven more iterations to deal with corners of corners. See also the proof of [KS77, Es-say III, Theorem 5.13]. Proposition 14.7.
There exists a finite CW-complex triad ( X, X − , X + ) and a homotopyequivalence of triads f : ( M, R − , R + ) → ( X, X − , X + ) such that the following two statementshold: (1) The restrictions of f to M , R ± and R − ∩ R + give the simple homotopy types ofthese manifolds, as defined in the previous section. (2) The restrictions of f to the pairs ( M, ∂M ) , ( ∂M, R ± ) and ( R ± , R − ∩ R + ) give thesimple homotopy types of these pairs of manifolds, as defined in the previous section. We continue with a general definition regarding CW complexes.
Definition 14.8.
Let (
X, Y ) be a pair of CW-complexes such that X is connected. Wewrite π = π ( X ) and we denote by p : (cid:101) X → X the universal covering. We define C cell ∗ ( X, Y ; Z [ π ]) := Z [ π ] ⊗ Z [ π ] C cell ∗ ( (cid:101) X, p − ( Y )) C ∗ cell ( X, Y ; Z [ π ]) := Hom right- Z [ π ] ( C cell ∗ ( (cid:101) X, p − ( Y )) , Z [ π ]) . The group π acts freely on the left on the cells of the CW-complex ( (cid:101) X, p − ( Y )). For eachcell in X , pick a lift to (cid:101) X . This turns C cell ∗ ( X, Y ; Z [ π ]) and C ∗ cell ( X, Y ; Z [ π ]) into based Z [ π ]-module (co-) chain complexes.Now we can state the main theorem of this section. Theorem 14.9.
The finite CW-complex triad ( X, X − , X + ) is a simple Poincar´e triad,meaning that there is a chain level representative σ ∈ C cell n ( X, X − ∪ X + ) of the fundamental class [ X ] ∈ H n ( X, X + ∪ X − ; Z ) = H n ( M, ∂M ; Z ) such that − X σ : C n − r cell ( X, X − ; Z [ π ( X )]) → C cell r ( X, X + ; Z [ π ( X )]) is a simple chain homotopy equivalence. The theorem is proved in [KS77, Essay III, Theorem 5.13]. In the Universal Poincar´eDuality Theorem A.16 we will prove that there exists a homotopy equivalence betweenthe two chain complexes. But we will not prove that there exists a simple homotopyequivalence; for that the reader will need to consult [KS77].14.3.
Reidemeister torsion.
In this section we introduce Reidemeister torsion invariantsfor compact manifolds and discuss some of these key properties of these invariants.Let M be a compact connected n -manifold and write π = π ( M ). Let R − be a compactcodimension 0 submanifold of ∂M . In many applications R − = ∅ or R − = ∂M . Wewrite R + = ∂M \ R − . Let F be a field and let α : π → GL( d, F ) be a representation ofthe fundamental group of M . With respect to this representation, we consider the twistedhomology H k ( M, R − ; F d ), as defined in Section A.1. Assumption 14.10.
Suppose that H k ( M, R − ; F d ) = 0 for all k . Pick a homotopy equivalence of triads f : ( M, R − , R + ) → ( X, X − , X + ) as in Proposi-tion 14.7. We use the homotopy equivalence f to make the identification π ( X ) = π . Bya serious abuse of notation, we refer to the cellular chain complex C cell ∗ ( X, X − ; Z [ π ]) of( X, X − ) as the cellular chain complex C cell ∗ ( M, R − ; Z [ π ]) of ( M, R − ). As in Section 14.2 weview C cell ∗ ( M, R − ; Z [ π ]) as a based left Z [ π ]-module chain complex. Equip the F -modulechain complex C cell ∗ ( M, R − ; F d ) = F d ⊗ Z [ π ] C cell ∗ ( M, R − ; Z [ π ]) with the basing given by thetensor products of the Z [ π ]-bases of C cell ∗ ( M, R − ; Z [ π ]) and the canonical F -basis for F d .We write ∼ α for the equivalence relation on F × := F \ { } that is given by the subgroup {± det( αg ) | g ∈ π ( M ) } ⊂ F × . We define τ ( M, R − , α ) ∈ F × / ∼ α to be the Reidemeistertorsion of the above acyclic, based F -module chain complex. We refer to [Tur01, Section 6]for the definition of the Reidemeister torsion of an acyclic, based F -module chain complex.It follows from a slight generalisation of [Tur01, Theorem 9.1] that τ ( M, R − , α ) ∈ F × / ∼ α is well-defined, in that it is independent of the choice of the representative of the simplehomotopy type of ( X, X − , X + ) and it is independent of the choice of the lifts of the cells.The following two theorems give the two arguably most important properties of torsion. Theorem 14.11.
Let M be a compact connected n -manifold and let R − be a compactcodimension 0 submanifold of ∂M . Let α : π ( M ) → GL( d, F ) be such that H ∗ ( R − ; F d ) =0 = H ∗ ( M ; F d ) . By abuse of notation we also write α for the composition α : π ( ∂M ) → π ( M ) → GL( d, F ) defined for each connected component of ∂M using the basing paths asdescribed above. Then we have τ ( M, α ) = τ ( R − , α ) · τ ( M, R − , α ) ∈ F × / ∼ α . HE FOUNDATIONS OF 4-MANIFOLD THEORY 73
Proof.
We have the following short exact sequence of chain complexes with compatiblebases: 0 → C cell ∗ ( X − ; F d ) → C cell ∗ ( X ; F d ) → C cell ∗ ( X, X − ; F d ) → . Given such a short exact sequence, the multiplicativity of the torsion is proven in [Tur01,Theorem 3.4]. (cid:3)
Definition 14.12.
Let F be a field with (possibly trivial) involution. Given a representa-tion α : π → GL( d, F ) we denote the representation g (cid:55)→ α ( g − ) T by α † . We say that α is unitary if α = α † . Example . Let φ : π → Z be a group homomorphism. Equip Q ( t ) with the usualinvolution given by t = t − . The representation α : π → GL(1 , Q ( t )) given by g (cid:55)→ t φ ( g ) isunitary. Theorem 14.14.
Let M be a compact n -manifold with ( possibly empty ) boundary. Assumethat we are given a decomposition ∂M = R − ∪ R + into codimension zero submanifolds suchthat ∂R − = R − ∩ R + = ∂R + . Furthermore let F be a field with ( possibly trivial ) involution.Let α : π ( M ) → GL( d, F ) be a representation such that H ∗ ( ∂M ; F d ) = 0 = H ∗ ( M ; F d ) .Then τ ( M, R − , α ) = τ ( M, R + , α † ) ( − n +1 in F × / ∼ α . In particular, if α is unitary we have τ ( M, R − , α ) = τ ( M, R + , α ) ( − n +1 in F × / ∼ α . Proof.
We write π = π ( M ). Write C ±∗ = C cell ∗ ( M, R ± ; Z [ π ]), recalling the conventiondescribed below Assumption 14.10.It follows from Theorem 14.9 that the torsion of the based F -module chain complex F d ⊗ Z [ π ] C −∗ agrees with the torsion of the based F -module chain complex F d ⊗ Z [ π ] Hom right- Z [ π ] ( C + n −∗ , Z [ π ]) . Consider the following isomorphism of based left F -module chain complexes F dα ⊗ Z [ π ] Hom right- Z [ π ] ( C + n −∗ , Z [ π ]) → Hom left- F ( F dα † ⊗ Z [ π ] C + n −∗ , F ) v ⊗ ϕ (cid:55)→ (cid:32) F dα † ⊗ Z [ π ] C + n −∗ → F ( w ⊗ σ ) (cid:55)→ vα ( ϕ ( σ )) w T (cid:33) Using this isomorphism τ ( M, R − , α ) also equals the torsion of the chain complex on theright hand side. It follows from algebraic duality for torsions [Tur01, Theorem 1.9] that thetorsion of the based chain complex on the right hand side equals τ ( M, R + , α † ) ( − n +1 . (cid:3) Obstructions to being topologically slice
The Fox-Milnor Theorem.
In this section we provide an example of the use ofmany of the theorems described above by applying them to obtain an obstructions for aknot to be topologically slice.
Definition 15.1.
Let Y be a homology 3-sphere that is the boundary of an integral ho-mology 4-ball X .(1) We say a knot K in Y is topologically slice in X if K bounds a slice disc , that is aproper submanifold of X homeomorphic to a disc.(2) Suppose X is equipped with a smooth structure, e.g. X = D . We say a knot K in Y is smoothly slice in X if K bounds a smooth slice disc , that is a proper smoothsubmanifold of X diffeomorphic to a disc.There are many classical obstructions to a knot being smoothly slice. For example,there are obstructions based on the Alexander polynomial [FM66] and the Levine-Tristramsignatures [Tri69, Lev69] and there are the more subtle Casson-Gordon [CG78, CG86]obstructions. Even though these results, having appeared prior to the work of Freedmanand Quinn, were formulated as obstructions to being smoothly slice, it has been understoodfor many years that the original proofs can be modified to prove that these are in factobstructions to being topologically slice.In this section we will prove the following sample theorem on the Alexander polynomialof a topologically slice knot. Theorem 15.2 (Fox-Milnor) . Suppose that K is a knot in a homology -sphere Y thatbounds an integral homology -ball X . If K is slice in X , then the Alexander polynomial ∆ K ( t ) of K factors as ∆ K ( t ) = ± t k f ( t ) f ( t − ) for some k ∈ Z and for some f ( t ) ∈ Z [ t, t − ] such that f (1) = ± . Even though this result is very well known we want to provide a detailed proof. Inparticular we want to highlight where some of the results discussed in this article are used.The reader is encouraged to go through the other papers mentioned above and to modifythe proofs to deal with topologically slice knots.15.2.
A proof of the Fox-Milnor Theorem.
For the proof of the Fox-Milnor Theo-rem 15.2 we adopt the following notation.(1) Let Y be a homology 3-sphere Y bounding some integral homology 4-ball X .(2) Given a knot K in Y , denote its zero framed surgery by N K .(3) Given an oriented knot K let µ K be an oriented meridian.(4) For a slice disc D in X , let N ( D ) be a tubular neighbourhood provided by Theo-rem 6.8. We refer to W D = X \ N ( D ) as the exterior of D .(5) The ring of integral Laurent polynomials in one variable is denoted Z [ t, t − ] or Z [ t ± ]. HE FOUNDATIONS OF 4-MANIFOLD THEORY 75
Many topological slicing obstructions, such as knot signatures [Tri69], the Fox-Milnor con-dition [FM66], the Blanchfield form [Kea75], Casson-Gordon invariants [CG78, CG86], L -signature defects [COT03] and L (2) -von Neumann ρ -invariants [CT07], rely implicitlyand explicitly on the next three propositions or slight variations thereof. Proposition 15.3.
Let K be an oriented knot in Y and let D be a slice disc in Y . (1) We have ∂W D = N K . (2) The inclusion map µ K → W D induces a Z -homology equivalence. In the remainder of this section, given an oriented knot, we use φ : π ( N K ) → (cid:104) t (cid:105) and φ : π ( W D ) → (cid:104) t (cid:105) to denote the unique homomorphisms that send the oriented meridianto t . These homomorphisms allow us to view Z [ t ± ] and Q ( t ) as a Z [ π ( N K )]-module anda Z [ π ( W D )]-module. Proof.
For the first statement, we have to check that the framing of K induced by theunique trivialisation νD ∼ = D × R is the 0–framing. Consider S with an equatorial S ,which contains K and splits S into two 4–balls. Let D be contained in the one 4–ball,and push a Seifert surface Σ into the other 4–ball. Pick a normal bundle νS = S × I ,and arrange using Theorem 6.5 that D ∩ νS = K × [ − , ∩ νS = K × [0 , F = Σ ∪ − D ⊂ S . We compute the Euler number e ( F ) ∈ Z in two ways. First, note that e ( F ) = [ F ] · [ F ] = 0, since H ( S ; Z ) = 0. On the other hand, the number e ( F ) is alsothe difference between the induced framings of ν Σ | K and νD | K . Consequently, the twoframings agree and νD induces the 0–framing, which by definition is the framing inducedby ν Σ | K .We turn to the proof of the second statement. By Proposition 6.13 the tubular neighbour-hood of D is trivial, thus we can identify it with D × D . Let µ K → W D be the inclusionof the meridian µ K of K . Then we have H ∗ ( W D , ∗ × µ K ; Z ) = H ∗ ( W D , D × S ; Z ) = H ∗ ( X, D × D ) = 0 by excision and the hypothesis that X be a homology 4-ball. Bythe homology long exact sequence for the pair ( W D , µ K ), the meridional map µ K → W D induces a homology equivalence, so W D is a homology circle. (cid:3) Proposition 15.4.
The exterior W D of a slice disc D is homotopy equivalent to a finite -dimensional CW complex. In particular the homology groups H ∗ ( W D ; Z [ t ± ]) , H ∗ ( W D , N K ; Z [ t ± ]) and H ∗ ( N K ; Z [ t ± ]) are all finitely generated.Proof. Note that W D is a compact 4-manifold with nonempty boundary. It follows fromTheorem 4.5 that W D is homotopy equivalent to a 3-dimensional CW complex. The state-ments regarding the homology groups follow from Proposition A.9. (cid:3) Proposition 15.5. (1)
For any knot K in a homology 3-sphere the modules H ∗ ( N K ; Z [ t ± ]) are Z [ t ± ] -torsion. (2) If D is a slice disc, then all the modules H ∗ ( W D ; Z [ t ± ]) are Z [ t ± ] -torsion. Proof.
We start out with the proof of the second statement. Let P ⊂ Z [ t ± ] be the multi-plicative subset of Laurent polynomials that augment to ±
1, that is p (1) = ± p ∈ P . We shall prove the slightly stronger statement, that H k ( W D ; Z [ t ± ]) is P -torsionfor k >
0. Since H ( W D ; Z [ t ± ]) ∼ = Z [ t ± ] / ( t −
1) is Z [ t ± ]-torsion, the result will follow.We write π = π ( W D ). Let Q := P − Z [ t ± ] be the result of inverting the polynomials in P . By Proposition A.5 there exists a chain complex C ∗ of finite length consisting of finitelygenerated free left Z [ π ]-modules such that for any ring R and any ( R, Z [ π ])-bimodule A wehave H k ( W D , µ K ; A ) ∼ = H k ( A ⊗ Z [ π ] C ∗ ) . By Proposition 15.3 we know that H k ( Z ⊗ Z [ π ] C ∗ ) = H k ( W D , µ K ; Z ) = 0. Since C ∗ isa chain complex of finite length consisting of finitely generated free left Z [ π ]-modules weobtain from chain homotopy lifting [COT03, Proposition 2.10], see also [NP17, Lemma 3.1],that H k ( Q ⊗ Z [ π ] C ∗ ) = 0. A straightforward calculation shows that H ∗ ( S , pt; Q ) = 0. Itfollows that H ∗ ( W D , pt; Q ) = 0, so that H k ( W D ; Z [ t ± ]) is P -torsion for k > S → Y \ νK is a homology equivalence to show thatthe modules H ∗ ( Y \ νK ; Z [ t ± ]) are torsion. A basic Mayer-Vietoris argument then showsthat the modules H ∗ ( N K ; Z [ t ± ]) are also torsion. (cid:3) We want to recall the definition of the Alexander polynomial of a knot. To do so weneed the notion of the order of a module.
Definition 15.6.
Let H be a finitely generated free abelian group and let M be a finitelygenerated Z [ H ]-module. By [Lan02a, Corollary IV.9.5] the ring Z [ H ] is Noetherian whichimplies that M admits a free resolution Z [ H ] r · A −→ Z [ H ] s → M → . Without loss of generality we can assume that r > s . Since Z [ H ] is unique factorisationdomain, see [Lan02a, Lemma IV.2.3], the order ord( M ) is defined as the greatest commondivisor of the s × s -minors of A . By [Tur01, Lemma 4.4] the order is well-defined, i.e.independent of the choice of the free resolution, up to multiplication by a unit in Z [ H ].In the proof of the Fox-Milnor theorem we will need the following lemma, collecting basicfacts about orders of finitely generated Z [ H ]-modules. Lemma 15.7.
Let H be a finitely generated free abelian group. (1) If → A → B → C → is a short exact sequence of finitely generated Z [ H ] -modules, then ord( B ) = ord( A ) · ord( C ) . (2) If → C k → · · · → C → is an exact sequence of finitely generated Z [ H ] -torsionmodules, then the alternating product of the orders is a unit in Z [ H ] . (3) For any finitely generated Z [ H ] -module A we have ord( A ) = ord( A ) . (4) For any finitely generated torsion Z [ H ] -module A we have ord(Ext Z [ H ] ( A, Z [ H ])) =ord( A ) . HE FOUNDATIONS OF 4-MANIFOLD THEORY 77
Proof.
Statement (1) is proved for H ∼ = Z in [Lev67, Lemma 5]. The general case followsfrom [Hil12, Theorem 3.12]. Note that (2) is an immediate consequence of (1), by separatingthe long exact sequence into short exact sequences such at 0 → Im C j → C j − → Im C j − →
0, applying (1), and performing substitutions using the resulting equations involving orders.Next (3) follows immediately from the definition. Finally (4) is well-known to the experts,but we could not find a reference, therefore we sketch the key ingredients in the proof. Weintroduce the following notation.(a) Given any prime ideal p of Z [ H ], let Z [ H ] p be the localisation at p , that is we invertall elements that do not lie in p . We view Z [ H ] as a subring of Z [ H ] p .(b) Given a ring R and f, g ∈ R we write f . = R g if f and g differ by multiplication bya unit in R .Now we sketch the proof of (4). We will use the following five observations.(i) First note that since Z [ H ] is a unique factorisation domain, for any prime element p ∈ Z [ H ] the ideal ( p ) is a prime ideal.(ii) Note that being unique factorisation domain and being Noetherian is preservedunder localisation, see [Pes96, Theorem 7.53] and [Row06, Corollary 8.8’]. In par-ticular each Z [ H ] p is a noetherian unique factorisation domain. This allows us, bythe same definitions as above, to define the order of a finitely generated moduleover Z [ H ] p .(iii) Localisation is flat [Lan02a, Proposition XVI.3.2]. It follows that for any finitelygenerated Z [ H ]-module M and any prime element p ∈ Z [ H ] one has ord( M ) . = Z [ H ] ( p ) ord( Z [ H ] ( p ) ⊗ Z [ H ] M ) and Z [ H ] ( p ) ⊗ Z [ H ] Ext Z [ H ] ( M, Z [ H ]) ∼ = Ext Z [ H ] ( p ) ( Z [ H ] ( p ) ⊗ Z [ H ] M, Z [ H ] ( p ) ) as Z [ H ] ( p ) -modules.(iv) By [Osb00, Corollary A.14] any commutative ring that has the property that eachprime ideal is principal, is a PID. It follows easily that for each prime element p ,the localisation Z [ H ] ( p ) is a PID.(v) Let L be a torsion Z [ H ] ( p ) -module. Since Z [ H ] ( p ) is a PID every two elements havea greatest common divisor. We can therefore perform row and column operationsto find a resolution for L such that the presentation matrix is diagonal. Fromthis observation one easily deduces that L ∼ = Ext Z [ H ] ( p ) ( L, Z [ H ] ( p ) ) as left Z [ H ] ( p ) -modules. Since L is torsion the presentation matrix is injective and so its transposepresents the Ext group. To convert the Ext group to a left module, we use thetrivial involution, which we may do since Z [ H ] ( p ) is a commutative ring.(vi) Suppose that f and g are in Z [ H ]. If f . = Z [ H ] ( p ) g for all prime elements p ∈ Z [ H ],then since Z [ H ] is a unique factorisation domain we must have f . = Z [ H ] g .Now with L = Z [ H ] ( p ) ⊗ Z [ H ] A a finitely generated Z [ H ]-torsion module, we have Z [ H ] ( p ) ⊗ Z [ H ] A ∼ = Ext Z [ H ] ( p ) ( Z [ H ] ( p ) ⊗ Z [ H ] A, Z [ H ] ( p ) ) for every prime element p , by (iv). On the other hand, again for each prime element p , wehave ord( A ) . = Z [ H ] ( p ) ord( Z [ H ] ( p ) ⊗ Z [ H ] A )by (iii). Combining these two observations yieldsord( A ) . = Z [ H ] ( p ) ord(Ext Z [ H ] ( p ) ( Z [ H ] ( p ) ⊗ Z [ H ] A, Z [ H ] ( p ) )) . By the second part of (iii) we have thatord(Ext Z [ H ] ( p ) ( Z [ H ] ( p ) ⊗ Z [ H ] A, Z [ H ] ( p ) )) . = Z [ H ] ( p ) ord( Z [ H ] ( p ) ⊗ Z [ H ] Ext Z [ H ] ( A, Z [ H ])) . By the first part of (iii) again we haveord( Z [ H ] ( p ) ⊗ Z [ H ] Ext Z [ H ] ( A, Z [ H ])) . = Z [ H ] ( p ) ord(Ext Z [ H ] ( A, Z [ H ])) . Thus combining the last three equalities we haveord( A ) . = Z [ H ] ( p ) ord(Ext Z [ H ] ( A, Z [ H ]))for all prime elements p . Now (4) follows by applying (vi). (cid:3) We use the notion of order to define the Alexander polynomial of a knot in a homology3-sphere.
Definition 15.8.
The
Alexander polynomial ∆ K ( t ) of a knot K is defined as the order ofthe Alexander module H ( N K ; Z [ t ± ]). Note that this polynomial is only well-defined upto units in Z [ t ± ].After these preparations we turn to the actual proof of the Fox-Milnor Theorem 15.2.We need the following elementary lemma. Lemma 15.9.
Let π be a group, let C ∗ be a chain complex of left free Z [ π ] -modules and let φ : π → (cid:104) t (cid:105) be a homomorphism. The map Hom right- Z [ π ] ( C ∗ , Z [ t ± ]) → Hom left- Z [ t ± ] ( Z [ t ± ] ⊗ Z [ π ] C ∗ , Z [ t ± ]) f (cid:55)→ ( p ⊗ σ (cid:55)→ p · f ( σ )) is well-defined and is an isomorphism of left Z [ t ± ] -cochain complexes. (cid:3) First proof of the Fox-Milnor theorem 15.2.
In this proof we abbreviate Λ := Z [ t ± ]. Westart out with the following three observations.(a) We have H ( W D ; Λ) ∼ = H ( N K ; Λ) ∼ = Λ / ( t − H ( W D , N K ; Λ) = 0,(c) By Proposition 15.5 and Proposition 15.4 we know that for all k Ext ( H k ( W D , N K ; Λ) , Λ) = Hom Λ ( H k ( W D , N K ; Λ) , Λ) = 0 . Claim.
For any i ∈ N we have H i ( N K ; Λ) ∼ = Ext ( H − i ( N K ; Λ) , Λ) H i ( W D ; Λ) ∼ = Ext ( H − i ( W D , N K ; Λ) , Λ) . HE FOUNDATIONS OF 4-MANIFOLD THEORY 79
We prove the second statement of the claim. The proof of the first statement is almostidentical. By the Poincar´e duality theorem A.15 we have an isomorphism H i ( W D ; Λ) ∼ = H − i ( W D , N K ; Λ) of Λ-modules. By Lemma 15.9, applied to C ∗ = C ∗ ( W D , N K ; Z [ π ]), weknow that H − i ( W D , N K ; Λ) ∼ = H − i (Hom Λ (Λ ⊗ Z [ π ] C ∗ ( W D , N K ; Z [ π ]) , Λ)) . Finally we apply the universal coefficient spectral sequence [Lev77, Theorem 2.3] to theΛ-module chain complex C ∗ ( W D , N K ; Λ). It follows from the above observations (b) and(c) that the spectral sequence collapses and that we have an isomorphism H − i (cid:0) Hom Λ (Λ ⊗ Z [ π ] C ∗ ( W D , N K ; Λ) (cid:1) ∼ = Ext ( H − i ( W D , N K ; Λ) , Λ) . This concludes the proof of the claim.Next we consider the long exact sequence of the pair ( W D , N K ) of twisted homology withΛ-coefficients: · · · → H ( W D ; Λ) → H ( W D , N K ; Λ) → H ( N K ; Λ) → H ( W D ; Λ) → H ( W D , N K ; Λ) → . . . It follows from Propositions 15.4 that all the above modules are finitely generated modules.Thus it makes sense to consider their orders. Also note that in Proposition 15.5 we sawthat the modules for N K and W D are all Λ-torsion. It follows from the long exact sequencethat the relative homology groups H ∗ ( W D , N K ; Λ) are also Λ-torsion. By Lemma 15.7 (3)the alternating product of the orders equals ± t k .By the above claim and Lemma 15.7 (3) and (4) the orders are anti-symmetric around H ( N K ; Λ). More precisely, we haveord( H ( W D , N K ; Λ)) = ord (cid:0) Ext ( H ( W D ; Λ) , Λ) (cid:1) = ord( H ( W D ; Λ)) , and the same type of relation holds as we progress further from the middle term H ( N K ; Λ)in the above long exact sequence. But this implies that there exist non-zero polynomials f, g ∈ Λ with f · f = ∆ K ( t ) · g · g . By considering irreducible factors, we obtain the desiredresult. (cid:3) We conclude with an alternative argument for the Fox-Milnor theorem in the topologicalcategory using Reidemeister torsion. The advantage of the Reidemeister torsion invariantis that proofs are often easier, and it has in general a smaller indeterminacy than the orderof homology, although this will not manifest itself in the upcoming proof.
Second proof of Theorem 15.2.
We continue with the notation introduced above. As beforewe have a homomorphism α : π ( W D ) → H ( W D ; Z ) (cid:39) −→ Z , sending an oriented meridianof K to 1 ∈ Z . As usual Q ( t ) denotes the field of fractions of the Laurent polynomialring Z [ t, t − ]. We take d = 1, and so obtain a representation φ : π ( W D ) → GL(1 , Q ( t )),that sends g (cid:55)→ ( t α ( g ) ). In the previous proof we had already seen that the modules H ∗ ( N K ; Z [ t ± ]), H ∗ ( W D ; Z [ t ± ]) and H ∗ ( N K ; Z [ t ± ]) are Z [ t ± ]-torsion. Since Q ( t ) is flatover Z [ t ± ] it follows that the corresponding twisted homology groups with Q ( t )-coefficientsare zero. By the discussion in Section 14.3 we can consider the Reidemeister torsions τ ( W D , φ ), τ ( N K , φ ) and τ ( W D , N K , φ ). By Theorem 14.14, τ ( W D , N K , φ ) = τ ( W D , φ ) ( − = τ ( W D , φ ) − .Since the torsion is multiplicative in short exact sequences by Theorem 14.11, we have that τ ( W D , φ ) = τ ( N K , φ ) · τ ( W D , N K , φ ) = τ ( M K , φ ) · τ ( W D , φ ) − . By [Tur01, Theorem 14.12] the torsion of the zero surgery of a knot is equal to ∆ K ( t ) / (( t − t − − K ( t ) is a norm as claimed. (cid:3) Remark . The two proofs presented above avoid the use of the smooth category,and so are in keeping with the spirit of this article. However, one can give a furtheralternative proof by allowing smooth techniques. First one can use Theorem 8.6 to finda simply connected 4-manifold W (cid:48) such that W := V W (cid:48) is smoothable. Then one cantriangulate W and apply Reidemeister torsion machinery without appealing to [KS77,Essay III]. The disadvantage of this approach is that typically H ( W (cid:48) ; Z ) will be nontrivial,so that W is not acyclic over Q ( t ). One can proceed by choosing a self-dual basis forhomology, so that one can still obtain a torsion invariant that is well-defined up to norms.Apply [CF13, Theorem 2.4], and argue that since the intersection form of W is nonsingular,the contribution of W (cid:48) to the torsion is a norm. Appendix A. Poincar´e Duality with twisted coefficients
Surveying the literature, we felt it would be of benefit to have a more detailed proofof Poincar´e duality with twisted coefficients for manifolds with boundary, but without asmooth or PL structure, so we offer one in this appendix. One can find other proofs ofPoincar´e duality for some subsets of these conditions.A.1.
Twisted homology and cohomology groups.
We start out with the followingnotation.
Notation
A.1 . Given a group π and a left Z [ π ]-module A , write A for the right Z [ π ]-modulethat has the same underlying abelian group but for which the right action of Z [ π ] is definedby a · g := g − · a for a ∈ A and g ∈ π . The same notation is also used with the rˆoles ofleft and right reversed and g · a := a · g − .We recall the definition of twisted homology and cohomology groups. Definition A.2.
Let X be a connected topological space that admits a universal cover p : (cid:101) X → X . Write π = π ( X ). Let Y be a subset of X , let A be a right Z [ π ]-module. Let π act on (cid:101) X by deck transformations, which is naturally a left action. Thus, the singularchain complex C ∗ ( (cid:101) X, p − ( Y )) becomes a left Z [ π ]-module chain complex. Define the twistedchain complex C ∗ ( X, Y ; A ) := (cid:0) A ⊗ Z [ π ] C ∗ ( (cid:101) X, p − ( Y )) , Id ⊗ ∂ ∗ (cid:1) . HE FOUNDATIONS OF 4-MANIFOLD THEORY 81
The corresponding twisted homology groups are H k ( X, Y ; A ). With δ k = Hom( ∂ k , Id) definethe twisted cochain complex to be C ∗ ( X, Y ; A ) := (cid:0) Hom right- Z [ π ] (cid:0) C ∗ ( (cid:101) X, p − ( Y )) , A (cid:1) , δ ∗ (cid:1) . The corresponding twisted cohomology groups are H k ( X, Y ; A ).Note that if R is some ring (not necessarily commutative) and if A is an ( R, Z [ π ])-bimodule, then the above twisted homology and cohomology groups are naturally left R -modules.Given a CW complex one can similarly define twisted cellular (co-) chain complexesand twisted cellular (co-) homology groups. The following proposition implies that twistedsingular (co-) homology groups are isomorphic to twisted cellular (co-) homology groups. Proposition A.3.
Let ( X, Y ) be a CW complex pair and write π = π ( X ) . The singu-lar chain complexes C sing ∗ ( X, Y ; Z [ π ]) and C cell ∗ ( X, Y ; Z [ π ]) are chain equivalent as chaincomplexes of left Z [ π ] -modules. The proof of Proposition A.3 relies on the following very useful lemma.
Lemma A.4.
Let f : C ∗ → D ∗ be a chain map of free Z [ π ] -module chain complexes ( herechain complexes are understood to start in degree 0 ) that induces an isomorphism on ho-mology. Then f is a chain equivalence.Proof. Since f induces an isomorphism of homology groups we know that the cone( f ) ∗ isacyclic. By assumption C ∗ and D ∗ are free Z [ π ]-modules and so is cone( f ) ∗ . But thisguarantees the existence of a chain homotopy Id cone f ∗ (cid:39) P
0, since we can view cone( f ) ∗ asa free resolution of 0 and any two such resolutions are chain homotopic. Recall that chainhomotopy means ∂ cone f ∗ ◦ P + P ◦ ∂ cone f ∗ = Id cone f ∗ (A.1)If we write P as a matrix P n = (cid:18) P n P n P n P n (cid:19) : C n − D n → C n D n +1 then one easily verifies using Equation (A.1), that P ∗ : D ∗ → C ∗ is a chain homotopyinverse of f ∗ , where the chain homotopies are given by P ∗ and P ∗ . (cid:3) Proof of Proposition A.3.
In [Sch68, p. 303] (see also [L¨uc98, Lemma 4.2]) it is shown thatthere exists a natural chain homotopy equivalence C sing ∗ ( (cid:101) X, p − ( Y )) → C cell ∗ ( (cid:101) X, p − ( Y )) of Z -modules, where p : (cid:101) X → X denotes the universal cover. Since the chain homotopy isnatural it is in particular π -equivariant. In other words, it is a chain homotopy equivalenceof Z [ π ]-modules. Now the proposition follows from Lemma A.4. (cid:3) Proposition A.5.
Let M be a compact n -manifold and let N ⊂ M be a subspace that is acompact manifold in its own right. Write π = π ( M ) . There exists a chain complex C ∗ of finite length consisting of finitely generated free left Z [ π ] -modules such that for any ring R and for any ( R, Z [ π ]) -bimodule A we have left R -module isomorphisms H k ( M, N ; A ) ∼ = H k ( A ⊗ Z [ π ] C ∗ ) and H k ( M, N ; A ) ∼ = H k (Hom Z [ π ] ( C ∗ , A )) . Remark
A.6 . Note that we do not demand that N be a submanifold of M . For example N could be a union of boundary components of M , or N could be a submanifold of theboundary. Evidently N could also be the empty set. Proof.
By Theorem 4.5 the manifolds M and N are homotopy equivalent to finite CWcomplexes X and Y respectively. Let i : N → M be the inclusion map. By the cellularapproximation theorem there exists a cellular map j : X → Y such that the followingdiagram commutes up to homotopy N i (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) M (cid:39) (cid:15) (cid:15) Y j (cid:47) (cid:47) X. Next we replace M and X by the mapping cylinders of i and j respectively, to createcofibrations. Given a map f : U → V between topological spaces let cyl( f ) be the mappingcylinder. We view U as a subset of cyl( f ) in the obvious way. With this notation we have H k ( M, N ; A ) ∼ = H k (cyl( i : N → M ) , N ; A ) ∼ = H k (cyl( j : Y → X ) , Y ; A ) . The mapping cylinder Z := cyl( j : Y → X ) admits the structure of a finite CW com-plex such that Y is a subcomplex. Thus we can compute the twisted homology groups H k (cyl( j : X → Y ); A ) using the relative twisted cellular chain complex, and similarly forcohomology. Put differently, C ∗ = C cell ∗ ( Z, Y ; Z [ π ]) has the desired properties. (cid:3) In order to give a criterion for twisted homology modules to be finitely generated, weneed the notion of a Noetherian ring.
Definition A.7.
A ring R is said to be left Noetherian if for any descending chain R ⊇ I ⊇ I ⊇ I ⊇ . . . of left R -ideals the inclusions eventually become equality. Example
A.8 . The following rings are left Noetherian:(1) The ring Z is Noetherian.(2) Any (skew) field is left Noetherian.(3) If A is a commutative Noetherian ring, then the multivariable Laurent polynomialring A [ t ± , . . . , t ± k ] is also Noetherian [Lan02a, Corollary IV.9.5].The following theorem is often implicitly used. HE FOUNDATIONS OF 4-MANIFOLD THEORY 83
Proposition A.9.
Let M be a compact n -manifold, let N ⊂ M be a subspace that is acompact manifold in its own right, let R be a ring and let A be an ( R, Z [ π ]) -bimodule. If R is Noetherian and if A is a finitely generated R -module, then all the twisted homologymodules H ∗ ( M, N ; A ) are finitely generated left R -modules. In the proof of Proposition A.9 we will need the following lemma; cf. [Lam91, Proposi-tion 1.21] or [Lan02a, Proposition X.1.4].
Lemma A.10.
Let R be a Noetherian ring. If P is a finitely generated left R -module, thenany submodule of P is also a finitely generated left R -module.Proof of Proposition A.9. By Proposition A.5, there exists a chain complex C ∗ of finitelength consisting of finitely generated free left Z [ π ]-modules such that H k ( M, N ; A ) ∼ = H k ( A ⊗ Z [ π ] C ∗ ) . Given k ∈ N we denote the rank of C k as a free left Z [ π ]-module by r k . Then we have A ⊗ Z [ π ] C k ∼ = A ⊗ Z [ π ] Z [ π ] r k ∼ = A r k . In particular H k ( M, N ; A ) is isomorphic to a quotientof a submodule of a finitely generated R -module. The desired statement follows fromLemma A.10. (cid:3) A.2.
Cup and cap products on twisted (co-) chain complexes.
Throughout thissection let X be a connected topological space admitting a universal cover, and write π = π ( X ). We want to introduce the cup product and the cap product on twisted (co-)chain complexes. Given an n -simplex σ , define the p -simplices σ (cid:98) p and σ (cid:99) p by σ (cid:99) p ( t , . . . , t p ) := σ ( t , . . . , t p , , . . . , ,σ (cid:98) p ( t , . . . , t p ) := σ (0 , . . . , , t , . . . , t p ) . Given two right Z [ π ]-modules A and B we view A ⊗ Z B as a right Z [ π ]-module via thediagonal action of π .First we introduce the cup product on twisted cohomology. The following lemma can beverified easily by hand, say along the lines of the proof of [Hat02, Lemma 3.6]. Lemma A.11.
Let Y be a subset of X . We consider the map Y : C p ( X ; A ) × C q ( X ; B ) −→ C p + q ( X ; A ⊗ Z B )( φ, ψ ) (cid:55)−→ (cid:0) σ (cid:55)→ ϕ ( σ (cid:98) p ) ⊗ Z ψ ( σ (cid:99) k − p ) (cid:1) . ( Note that the right-hand side is indeed Z [ π ] -homomorphism, i.e. it defines an element C p + q ( X ; A ⊗ Z B ) . ) Furthermore the map descends to a well defined map Y : H p ( X, Y ; A ) × H q ( X, Y ; B ) −→ H p + q ( X, Y ; A ⊗ Z B ) . We refer to this map as the cup product . Next we introduce the cap product. As with cup product, first we define it on the chainlevel.
Lemma A.12.
Let X be a topological space and let S, T ⊂ X be subsets. We write C k ( X, { S, T } ) = C k ( X ) / ( C k ( S ) + C k ( T )) . The map X : C p ( X, S ; A ) × C k ( X, { S, T } ; A ) −→ C k − p ( X, T ; A ⊗ Z B )( ψ, b ⊗ Z [ π ] σ ) (cid:55)−→ ( ψ ( σ (cid:98) p ) ⊗ Z b ) ⊗ Z [ π ] σ (cid:99) k − p . is well-defined. We refer to this map as the cap product .Proof. We verify that the given map respects the tensor product. Thus let ψ ∈ C p ( X ; A ), σ ∈ C k ( (cid:101) X ), γ ∈ π and b ∈ B . We calculate that ψ X b ⊗ Z [ π ] γσ = ( ψ ( γσ (cid:98) p ) ⊗ Z b ) ⊗ Z [ π ] γσ (cid:99) k − p = ( γψ ( σ (cid:98) p ) ⊗ Z b ) · γ ⊗ Z [ π ] σ (cid:99) k − p = ( γ − γψ ( σ (cid:98) p ) ⊗ Z bγ ) ⊗ Z [ π ] σ (cid:99) k − p = ψ X bγ ⊗ Z [ π ] σ. It follows easily from the definitions that the cap product descends to the given quotient(co-) chain complexes. (cid:3)
Lemma A.13.
Let f ∈ C p ( X ; A ) and let c ∈ C k ( X ; B ) . We have ∂ ( f X c ) = ( − p · ( − δ ( f ) X c + f X ∂c ) ∈ C k − ( X ; A ⊗ Z B ) . Proof.
The lemma follows from a calculation using the definition of the cap product andthe boundary maps, see e.g. [Fri19, Lemma 59.1] for details. Note that the precise signsdiffer from similar formulas in some textbooks in algebraic topology since there are manydifferent sign conventions in usage. (cid:3)
Corollary A.14.
Let X be a connected topological space, let S, T ⊂ X be subsets, let R be a ring and let A be an ( R, Z [ π ( M )]) -bimodule. For any cycle σ ∈ C n ( X, { S, T } ; Z ) thecap product X [ σ ] : H k ( X, S ; A ) → H n − k ( X, T ; A ) = H n − k ( X, T ; A ⊗ Z Z )[ ϕ ] (cid:55)→ [ ϕ X σ ] is well-defined. Furthermore this map only depends on the homology class [ σ ] ∈ H n ( X, S ∪ T ; Z ) . A.3.
The Poincar´e duality theorem.
The following theorem is a generalisation of thefamiliar Poincar´e duality for untwisted coefficients to the case of twisted coefficients.
Theorem A.15 (Twisted Poincar´e duality) . Let M an compact, oriented, connected n -dimensional manifold. Let S and T be codimension 0 compact submanifolds of ∂M suchthat ∂S = ∂T = S ∩ T and ∂M = S ∪ T . Let [ M ] ∈ H n ( M, ∂M ; Z ) be the fundamentalclass of M . If R is a ring and if A is an ( R, Z [ π ( M )]) -bimodule, then the map − X [ M ] : H k ( M, S ; A ) → H n − k ( M, T ; A ) defined by Lemma A.13 is an isomorphism of left R -modules. We also have the following Poincar´e Duality statement on the (co-) chain level.
HE FOUNDATIONS OF 4-MANIFOLD THEORY 85
Theorem A.16 (Universal Poincar´e duality) . Let M an compact, oriented, connected n -dimensional manifold. Let S and T be codimension 0 compact submanifolds of ∂M suchthat ∂S = ∂T = S ∩ T and ∂M = S ∪ T . Let σ ∈ C n ( M, { S, T } ; Z ) be a representative ofthe fundamental class of M . If R is a ring and if A is an ( R, Z [ π ( M )]) -bimodule, then themap − X σ : C k ( M, ∂M ; Z [ π ( M )]) → C n − k ( M ; Z [ π ( M )]) defined by Lemma A.13 is a chain homotopy equivalence of left R -chain complexes. Note that Theorem 14.9 can be used to give an alternative proof that the chain complexesof the theorem are chain homotopy equivalent.
Proof.
The Universal Poincar´e Duality Theorem A.16 follows immediately from the TwistedPoincar´e Duality Theorem A.15 together with Lemma A.4. (cid:3)
Below we will provide a proof of the Twisted Poincar´e Duality Theorem A.15. Butjust for fun we would like to show that the Universal Poincar´e Duality Theorem A.16 alsoimplies the Twisted Poincar´e Duality Theorem A.15.
Proof of Theorem A.15 using Theorem A.16.
Let M be an compact, oriented, connected n -dimensional manifold. We pick a representative σ for [ M ] and we write π = π ( M ). Let A be an ( R, Z [ π ])-bimodule. Given a chain complex D ∗ of right Z [ π ]-modules we considerthe cochain mapΞ : A ⊗ Z [ π ] Hom right- Z [ π ] ( D ∗ ; Z [ π ])) → Hom right- Z [ π ] ( D ∗ , A ) a ⊗ f (cid:55)→ ( σ (cid:55)→ a · f ( σ )) . Note that Ξ is an isomorphism if each D k is a finitely generated free Z [ π ]-module. But ingeneral Ξ is not an isomorphism.Furthermore we consider the following diagram H k ( A ⊗ Z [ π ] C ∗ ( M, ∂M ; Z [ π ])) Id A ⊗ ( X σ ) (cid:47) (cid:47) Ξ ∗ (cid:15) (cid:15) H n − k ( A ⊗ Z [ π ] C ∗ ( M ; Z [ π ])) = H n − k ( M ; A ) H k ( C ∗ ( M, ∂M ; A )) = H k ( M, ∂M ; A ) . X [ M ] (cid:49) (cid:49) One easily verifies that the diagram commutes. The top horizontal map is an isomorphismby the Universal Poincar´e Duality Theorem A.16. It remains to show that the vertical mapis an isomorphism. As we had pointed out above, on the chain level Ξ is in general not anisomorphism.As in the proof of Proposition A.5 we can use Theorem 4.5 to find a pair (
X, Y ) of finiteCW-complexes and a homotopy equivalence f : ( X, Y ) → ( M, ∂M ). By Proposition A.3there exists a homotopy equivalence Θ : C cell ∗ ( X, Y ; Z [ π ]) → C ∗ ( X, Y ; Z [ π ]) of Z [ π ]-chain complexes. We consider the following diagram where all tensor products and homomor-phism are over Z [ π ]: A ⊗ C ∗ cell ( X, Y ; Z [ π ]) Ξ ∗ (cid:15) (cid:15) Θ ∗ (cid:47) (cid:47) A ⊗ C ∗ ( X, Y ; Z [ π ]) Ξ ∗ (cid:15) (cid:15) f ∗ (cid:47) (cid:47) A ⊗ C ∗ ( M, ∂M ; Z [ π ]) Ξ ∗ (cid:15) (cid:15) C ∗ cell ( X, Y ; A ) Θ ∗ (cid:47) (cid:47) C ∗ ( X, Y ; A ) f ∗ (cid:47) (cid:47) C ∗ ( M, ∂M ; A ) . One easily verifies that the diagram commutes. As pointed out above, the horizontal mapsare chain homotopy equivalences over Z [ π ]. Since X is a finite CW-complex we see thateach C cell k ( X, Y ; Z [ π ]) is a finitely generated free Z [ π ]-module. Thus we obtain from theabove that the left vertical map is an isomorphism. Therefore we see that the right verticalmap is a chain homotopy equivalence. In particular it induces an isomorphism of homologygroups. (cid:3) The remainder of this appendix is dedicated to the proof of the Twisted Poincar´e DualityTheorem A.15. Even though the theorem is well-known and often used, there are not manysatisfactory proofs in the literature. The proof which is closest to ours in spirit is the proofof Sun [Sun17]. For closed manifolds Kwasik-Sun [KS18] provide a proof by using the workof Kirby-Siebenmann to reduce the proof to the case of triangulated manifolds.The proof of the Twisted Poincar´e Duality Theorem A.15 is modelled on the proof ofuntwisted Poincar´e Duality that is given in Bredon’s book [Bre97, Chapter VI Section 8].The logic of his proof is unchanged, but some arguments and definitions have to be adjustedfor the twisted setting.A.4.
Preparations for the proof of the Twisted Poincar´e Duality Theorem A.15.
We fix some notation that we will use for the remainder of this appendix. Let M be a connected manifold and denote by π := π ( M, x ) the fundamental group. Finally let R bea ring and let A be an ( R, Z [ π ])-bimodule.We write p : (cid:102) M → M for the universal cover of M . For a subset X ⊂ M (not necessarilyconnected) we consider the (co)-homology of X with respect to the coefficient systemcoming from M by setting C ∗ ( X ; A ) := A ⊗ Z [ π ] C ∗ ( p − ( X ); Z ) ,C ∗ ( X ; A ) := Hom Z [ π ] (cid:16) C ∗ ( p − ( X ); Z ) , A (cid:17) , with generalisation to pairs Y ⊂ X ⊂ M by C ∗ ( X, Y ; A ) := A ⊗ Z [ π ] C ∗ ( p − ( X ) , p − ( Y ); Z ) ,C ∗ ( X, Y ; A ) := Hom Z [ π ] (cid:16) C ∗ ( p − ( X ) , p − ( Y ); Z ) , A (cid:17) . We summarise the basic properties of twisted coefficients in the following theorem, whichshould be compared to the untwisted case.
HE FOUNDATIONS OF 4-MANIFOLD THEORY 87
Theorem A.17.
Let M be a connected manifold with fundamental group π , and let A bean ( R, Z [ π ]) -bimodule. (1) Given Y ⊂ X ⊂ M there is a long exact sequence of pairs in homology · · · H k ( Y ; A ) H k ( X ; A ) H k ( X, Y ; A ) H k − ( Y ; A ) · · · and cohomology · · · H k ( X, Y ; A ) H k ( X ; A ) H k ( Y ; A ) H k +1 ( X, Y ; A ) · · · . (2) Suppose we have a chain of subspaces Z ⊂ Y ⊂ X ⊂ M such that the closure of Z iscontained in the interior of Y . Then the inclusion ( X \ Z, Y \ Z ) → ( X, Y ) inducesan isomorphism in homology and cohomology i.e. H k ( X \ Z, Y \ Z ; A ) ∼ −→ H k ( X, Y ; A ) and H k ( X \ Z, Y \ Z ; A ) ∼ ←− H k ( X, Y ; A )(3) If U ⊂ U ⊂ M and V ⊂ V ⊂ M are open subsets in M , then there are long exactsequences in homology . . . H k ( U ∩ V , U ∩ V ; A ) H k ( U , U ; A ) ⊕ H k ( V , V ; A ) H k ( U ∪ V , U ∪ V ; A ) H k − ( U ∩ V , U ∩ V ; A ) . . . and cohomology . . . H k ( U ∪ V , U ∪ V ; A ) H k ( U , U ; A ) ⊕ H k ( V , V ; A ) H k ( U ∩ V , U ∩ V ; A ) H k − ( U ∪ V , U ∪ V ; A ) . . . (4) Suppose the inclusion Y → X is a homotopy equivalence, then the inclusion inducedisomorphisms H k ( Y ; A ) ∼ −→ H k ( X ; A ) and H k ( Y ; A ) ∼ ←− H k ( X ; A ) . (5) Let U ⊂ U ⊂ . . . be a sequence of open sets in M and let U = (cid:83) i ∈ N U i , then inclusionsinduce an isomorphism lim −→ i ∈ N C ∗ ( U i ; A ) ∼ = −→ C ∗ ( U ; A ) . The proofs are essentially the same as in the classical case. Therefore we will only sketchthe arguments and focus on what is different. We also warn the reader that we give the“wrong proof” of statement (4). This is due to the fact that we developed the theory of twisted coefficients only for inclusions and hence a homotopy inverse does not fit in ourtheory. Therefore statement (4) will be deduced in a slightly round-about way using thefollowing elementary lemma [Bre97, Chapter III Theorem 3.4 & remark after proof].
Lemma A.18 (The Covering Homotopy Theorem) . Given a covering p : (cid:101) X → X , a ho-motopy H : Y × I → X , and a lift (cid:101) h : Y → (cid:101) X of H ( − , , there exists a unique lift (cid:101) H : Y × I → (cid:101) X of H with (cid:101) h = (cid:101) H ( − , .Proof of Theorem A.17. Recall that p : (cid:102) M → M denotes the universal cover.For statement (1) we consider the short exact sequence 0 → C ∗ ( Y ; Z [ π ]) → C ∗ ( X ; Z [ π ]) → C ∗ ( X, Y ; Z [ π ]) → free Z [ π ]-modules. Since the modules are free the sequence staysexact after applying the functors A ⊗ Z [ π ] − and Hom Z [ π ] ( − , A ).Recall the proof of statement (2) and (3) in the classical case as it is done for examplein Bredon’s book [Bre97, Chapter IV Section 17]. The main ingredient is to show thatthe inclusion of chain complexes C U∗ ( X ; Z [ π ]) → C ∗ ( X ; Z [ π ]) induces an isomorphism onhomology [Bre97, Theorem 17.7]. Here U is an open cover of X and C U∗ ( X ; Z [ π ]) is thefree abelian group generated by simplices σ for which there is a U ∈ U such that σ : ∆ ∗ → p − ( U ). This is done by defining the barycentric subdivision Υ ∗ : C ∗ ( (cid:101) X ; Z ) → C ∗ ( (cid:101) X ; Z )and a chain homotopy T between Υ ∗ and the identity [Bre97, Lemma 17.1]. The importantthing for us to observe is that both maps are natural [Bre97, Claim (1) in proof of Lemma17.1]. Hence for a twisted chain Υ( e ⊗ Z [ π ] σ ) := e ⊗ Z [ π ] Υ( σ ) is well-defined, becauseΥ( e ⊗ Z [ π ] γσ ) = e ⊗ Z [ π ] Υ( γσ )= e ⊗ Z [ π ] γ Υ( σ ) (naturality of Υ)= eγ ⊗ Z [ π ] Υ( σ ) = Υ( eγ ⊗ Z [ π ] σ )The same holds for T and from now on one can follow the classical proofs. Alternatively,one could invoke Lemma A.4.Next we prove statement (4). Let f : X → Y ⊂ X be a homotopy inverse of theinclusion and H : X × I → X a homotopy between Id X and f . Since p : (cid:101) X → X is acovering and Id (cid:101) X is a lift of H ( p ( − ) , (cid:101) H : (cid:101) X × I → (cid:101) X of the homotopy H . One easily verifies that the inclusion (cid:101) Y → (cid:101) X induces a homotopyequivalence where a homotopy inverse is given by (cid:101) H ( − , H k ( C ∗ ( (cid:101) Y ; Z )) → H k ( C ∗ ( (cid:101) X ; Z )) is an isomorphism for every k . Thus the claim followsfrom Lemma A.4.The proof of Statement (5) is almost verbatim the same proof as in the classical case. (cid:3) A.5.
The main technical theorem regarding Poincar´e Duality.
Given a group π wecan view Z as a Z [ π ]-module with trivial π -action. We denote this module by Z triv . Let p : (cid:102) M → M be the covering projection. We have the following useful lemma, concerningthe chain map C ∗ ( X ; Z triv ) → C ∗ ( X ; Z ) defined by k ⊗ Z [ π ] (cid:101) σ (cid:55)→ k · p ( σ ). HE FOUNDATIONS OF 4-MANIFOLD THEORY 89
Lemma A.19.
Given any subset X ⊂ M the chain map above is an isomorphism be-tween C ∗ ( X ; Z triv ) and C ∗ ( X ; Z ) , and induces one between C ∗ ( X ; Z ) and C ∗ ( X ; Z triv ) ,where C ∗ ( X ; Z ) and C ∗ ( X ; Z ) are the untwisted singular chain complexes.Proof. The isomorphism is given by lifting a simplex, which is always possible since asimplex is simply connected. If one has two different choices of lifts, then they differ byan element in π . But the action of Z [ π ] on Z is trivial and hence this indeterminacyvanishes. (cid:3) We will keep the notational difference between C ∗ ( X ; Z ) and C ∗ ( X ; Z triv ) to emphasisewhere our simplices live.As above let R be a ring and let A is an ( R, Z [ π ])-bimodule. Let K ⊂ M be a compactsubset of M . We define the (twisted) Cech cohomology groupsˇ H p ( K ; A ) := lim −→ K ⊂ U ⊂ M H p ( U ; A ) , where the direct limit runs over all open sets in M containing K . Since cohomology iscontravariant, we define the order on open sets in the reversed way i.e. U ≤ V if V ⊂ U .Now we assume that M is oriented. Being oriented gives us for any closed subset Z ⊂ M a preferred element θ Z ∈ H n ( M, M \ Z ; Z triv ) ∼ = H n ( M, M \ Z ; Z ), which restricts for all x ∈ Z to the generator in H n ( M, M \ { x } ; Z triv ).For any open set U ⊂ M containing K let ex U : H n ( M, M \ K ; Z triv ) → H n ( U, U \ K ; Z triv )be the inverse of the inclusion given by the excision isomorphism i.e. if j : U → M is theinclusion then j ∗ ◦ ex U = Id. We then obtain a map D U : H p ( U ; A ) −→ H n ( M, M \ K ; A ) φ (cid:55)−→ j ∗ ( φ X ex U ( θ K )) . Given another open set V ⊂ U denote by i : V → U the inclusion. Then one easilycalculates:PD V ( i ∗ φ ) = j ∗ i ∗ ( i ∗ φ X ex V ( θ K )) = j ∗ ( φ X i ∗ ex V ( θ K )) = j ∗ ( φ X ex U ( θ K )) = PD U ( φ ) . Or, with other words, the following diagram commutes: H p ( U ; A ) H n − p ( M, M \ K ; A ) H p ( V ; A ) i ∗ PD U PD V By the universal property of the direct limit we obtain the dualising map PD K : ˇ H p ( K ; A ) → H n − p ( M, M \ K ; A ).In the remainder of this section we will prove the following theorem. Theorem A.20 (Poincar´e duality) . The map PD K : ˇ H p ( K ; A ) → H n − p ( M, M \ K ; A ) isa left R -module isomorphism for all compact subsets K ⊂ M . Here, as above, A is an ( R, Z [ π ])-bimodule. In the subsequent section we will see thatthe Twisted Poincar´e Duality Theorem A.15 is a reasonably straightforward consequenceof Theorem A.20.The proof of Theorem A.20 will be an application of the following lemma. Lemma A.21 (Bootstrap lemma) . Let P M ( K ) be a statement about compact sets K in M . If P M ( · ) satisfies the following three conditions: (1) P M ( K ) holds true for all compact subsets K ⊂ M with the property that for all x ∈ K the inclusions { x } → K and M \ K → M \ { x } are deformation retracts, (2) If P M ( K ) , P M ( K ) and P M ( K ∩ K ) is true, then P M ( K ∪ K ) is true, (3) If · · · ⊂ K ⊂ K and P M ( K i ) is true for all i ∈ N , then P M ( (cid:84) i ∈ N K i ) is true.Then P M ( K ) is true for all K ⊂ M .Proof. See [Bre97, Chapter VI Lemma 7.9]. (cid:3)
The idea is to apply the bootstrap lemma to the statements that the conclusion ofTheorem A.20 holds for a given compact set K . It turns out that condition (3) is theeasiest to verify. It follows from formal properties about direct limits. For the verificationof condition (1) we have do to one explicit calculation. This is the content of the nextlemma. Lemma A.22.
Let x ∈ M be a point. The map PD { x } : ˇ H ( { x } ; A ) → H n ( M, M \ { x } ; A ) is an R -module isomorphism.Proof. Let p : (cid:102) M → M be the universal cover. Since x is a point in a manifold we cancalculate the dualising map PD { x } by taking the limit over open neighbourhoods U of x with the following two properties:(1) U is contractible,(2) for any connected component U ⊂ p − ( U ) the map p | U is a homeomorphism.This can be done, since any neighbourhood of x contains a neighbourhood with thesetwo properties. Let U be such a neighbourhood of x and U ⊂ p − ( U ) a fixed connectedcomponent. This choice of connected component gives us an isomorphism H ( U ; A ) ∼ = A asfollows. Let f ∈ H ( U ; A ) be arbitrary and x ∈ U be a point in our connected component,then we get an element in A by evaluating f ([ x ]). Conversely, given an element e ∈ A wecan construct a function in H ( U ; A ) by setting f ([ x ]) = e for all x ∈ U . Note that thereis a unique way to extend f equivariantly to C ( p − ( U ); Z ).We are now going to construct a representative of the orientation class θ K ∈ H n ( M, M \{ x } ; Z triv ) for which it is very simple to calculate the dualising map. Let x be the preimageof x in U . Now take a cycle (cid:80) di =1 k i σ i which generates H n ( U , U \ { x } ; Z ). By excision andLemma A.19 one easily sees that 1 ⊗ Z [ π ] (cid:80) di =1 k i σ i is a generator of H n ( M, M \ { x } ; Z triv ).Using the isomorphism H ( U ; A ) ∼ = A from above the dualising map becomes PD { x } : A → H n ( M, M \ { x } ; A ) , e (cid:55)→ e ⊗ Z [ π ] (cid:80) di =1 k i σ i . This is clearly an isomorphism, since on the HE FOUNDATIONS OF 4-MANIFOLD THEORY 91 chain level we have: C ∗ ( U, U \ { x } ; A ) = A ⊗ Z [ π ] (cid:77) γ ∈ π C ∗ ( γU , γU \ { γx } ; Z ) ∼ = A ⊗ Z C ∗ ( U , U \ { x } ; Z ) . (cid:3) In order to verify condition (2) of the bootstrap lemma we will need the following lemma(compare [Bre97, Lemma 8.2]).
Lemma A.23. If K and L are two compact subsets of M , then the diagram · · · ˇ H p ( K ∪ L ; A ) ˇ H p ( K ; A ) ⊕ ˇ H p ( L ; A ) ˇ H p ( K ∩ L ; A ) ˇ H p +1 ( K ∪ L ; A ) · · ·· · · H n − p ( M, M \ ( L ∪ K ); A ) H n − p ( M, M \ K ; A ) ⊕ H n − p ( M, M \ L ; A ) H n − p ( M, M \ ( L ∩ K ); A ) H n − p − ( M, M \ ( L ∪ K ); A ) · · · PD K ∪ L PD K ⊕ PD L PD K ∩ L PD K ∪ L has exact rows and it commutes up to a sign depending only on p .Proof. The rows are exact by Mayer-Vietoris and the fact that direct limit is an exactfunctor. The commutativity of the squares is clear except for the last one involving theboundary map. This will be a painful diagram chase. Let U ⊃ K and V ⊃ L be openneighbourhoods containg K resp. L . The sequence in the top row comes from the shortexact sequence ( U = { U, V } ):0 C ∗U ( U ∪ V ; A ) C ∗ ( U ; A ) ⊕ C ∗ ( V ; A ) C ∗ ( U ∩ V ; A ) 0 . An element φ ∈ ˇ H p ( K ∩ L ; A ) will already be represented by same element f ∈ C p ( U ∩ V ; A )for some U and V as above. We can extend f to an element f ∈ C p ( M ; A ) by f ( σ ) = (cid:40) f ( σ ) if Im σ ⊂ (cid:101) U ∩ (cid:101) V f ∈ C p ( M ; A ) since p − ( U ∩ V ) is an equivariant subspace and hence f isequivariant. If we consider f as an element in C p ( U ; A ) then the cohomology class δ ( φ ) isrepresented by the cochain h ∈ C p +1 ( U ∪ V ; A ) which is given by h ( σ ) = (cid:40) δ ( f )( σ ) if Im σ ⊂ (cid:101) U φ is a cocycle we have δ ( f )( σ ) = 0 for σ ∈ C ∗ ( U ∩ V ; A ). It follows in particularthat if σ is a simplex whose image is completely contained in (cid:101) V , then h ( σ ) = 0. We canrepresent our orientation class θ ∈ H n ( M, M \ ( K ∪ L )) by a cycle a = b + c + d + e with b ∈ C n ( U ∩ V ; Z triv ) c ∈ C n ( U \ ( U ∩ L ); Z triv ) d ∈ C n ( V \ ( V ∩ K ); Z triv ) , e ∈ C n ( M \ ( K ∪ L ); Z triv ) . Obviously e does not play a role since we kill it in the end. With these representatives onecomputes that δ ( φ )( θ ) is represented by h X ( b + c + d ) = δ ( f ) ∩ c + h X d + δ ( f ) ∩ b = δ ( f ) ∩ c. The pairing of h with b is zero since f was a cocycle in C ∗ ( U ∩ V ; A ) and the pairing of h with d is zero since d consist of simplices with image in (cid:101) V .The lower sequence comes from the short exact sequence:0 C ∗ ( M, M \ ( K ∪ L ); A ) C ∗ ( M, M \ K ; A ) ⊕ C ∗ ( M, M \ L ); A ) C ∗ ( M, M \ ( K ∩ L ); A ) 0 . Before we compute the other side ∂ ( φ X ex U ∩ V ( θ )) we want to recall that the cap productis natural on the chain complex level i.e. the following diagram commutes: C p ( U ; A ) × C n ( U, U \ K ; Z triv ) C ∗ ( U, U \ K ; A ) C p ( M ; A ) × C n ( M, M \ K ; Z triv ) C ∗ ( M, M \ K ; A ) . Therefore we use the representatives from above. To construct the boundary map ∂ , wetake as the preimage of f X a ∈ C ∗ ( M, M \ ( K ∩ L ); A ) the element ( f X a, ∈ C ∗ ( M, M \ K ; A ) ⊕ C ∗ ( M, M \ L ; A ). Then one computes in C ∗ ( M, M \ K ; A ) ∂ ( f X a ) = ( − p +1 · δ ( f ) X a ± f X ∂a (by Lemma A.13)= ( − p +1 · δ ( f ) X a (since f X ∂a ∈ C n − p − ( M \ ( K ∪ L ); A ))= ( − p +1 · δ ( f ) X b + c + d + e = ( − p +1 · δ ( f ) X ( c + d ) (same reason as above)= ( − p +1 · δ ( f ) X c (since d ∈ C n − p ( V \ ( K ∩ V ); A ))Therefore the element ∂ ( φ X ex U ∩ V ( θ )) is also represented by ( − p +1 · δ ( f ) X c ∈ C n − p − ( M, M \ ( K ∪ L ); A ). (cid:3) Proof of Theorem A.20.
Let P M ( K ) be the statement that the map PD K is an isomor-phism. Then it is sufficient to verify condition (1),(2) and (3) of the bootstrap lemma. Westart with verifying (1). In the case that K = { x } is just a point we have already seen inLemma A.22 that the statement holds true. For a general compact K with the property of(1) the statement follows from the following commutative diagram:ˇ H p ( K ; A ) H n − p ( M, M \ K ; A )ˇ H p ( { x } ; A ) H n − p ( M, M \ { x } ; A ) , (cid:39) (cid:39)(cid:39) HE FOUNDATIONS OF 4-MANIFOLD THEORY 93 where the vertical maps are isomorphisms by the homotopy invariance and the bottom rowby the observation above. Hence condition (1) is verified.Condition (2) follows immediately from the five-lemma and Lemma A.23.Let K i be a sequence of compact subsets such that P M ( K i ) holds for all i ∈ N . Weset K = (cid:84) i ∈ N K i . It is an exercise in pointset topology of manifolds that each K i has afundamental system U i,j of open neighbourhoods. Fundamental system means that U i,j ⊂ U i,k if j < k and that for each open set U containing K i there is a j such that U i,j ⊂ U .Another exercise in point set topology of manifolds shows, that one can construct thesesets such that U ,j ⊃ U ,j ⊃ U ,j ⊃ . . . for all j ∈ N . Then U i,j is a fundamental systemof open neighbourhoods of K with the order ( i, j ) ≤ ( k, l ) ⇔ i ≤ k ∧ j ≤ l . One has thenatural isomorphism [Bre97, Appendix D5]:lim −→ i ∈ N ˇ H p ( K i ; A ) = lim −→ i ∈ N lim −→ j ∈ N H p ( U i,j ; A ) (cid:39) −−→ lim −→ i,j ∈ N H p ( U i,j ; A ) ∼ = ˇ H p ( K ; A ) . And hence the theorem follows from the commutativity of the diagram:lim −→ i ∈ N ˇ H p ( K i ; A ) lim −→ i ∈ N H n − p ( M, M \ K i ; A )ˇ H p ( K ; A ) H n − p ( M, M \ K ; A ) . (cid:3) A.6.
Proof of the Twisted Poincar´e Duality Theorem A.15.
For the reader’s con-venience we recall the main theorem from the last section. Here, as above, R is a ring and A is an ( R, Z [ π ])-bimodule. Theorem A.20.
Let M be a compact, oriented, connected n -dimensional manifold. Themap PD K : ˇ H p ( K ; A ) → H n − p ( M, M \ K ; A ) is an isomorphism of left R -modules for allcompact subsets K ⊂ M . Furthermore, we also recall that we need to prove the following theorem.
Theorem A.15.
Let M an compact, oriented, connected n -dimensional manifold. Let S and T be codimension 0 compact submanifolds of ∂M such that ∂S = ∂T = S ∩ T and ∂M = S ∪ T . Let [ M ] ∈ H n ( M, ∂M ; Z ) be the fundamental class of M . The map − X [ M ] : H k ( M, S ; A ) → H n − k ( M, T ; A ) defined by Lemma A.13 is an isomorphism of left R -modules. In the remainder of this appendix we will explain how to deduce Theorem A.15 fromTheorem A.20. First note that if M is a closed manifold, then we can set K = M in Theorem A.20. Evidently we have ˇ H p ( M ; A ) = H ( M ; A ). Thus we obtain precisely thestatement of Theorem A.15 in the closed case.Next let M be a compact oriented manifold with non-empty boundary. First we considerthe case R = ∅ and S = ∂M . By the Collar neighbourhood theorem 2.5 there exists acollar ∂M × [0 , ⊂ M of the boundary such that ∂M = ∂M ×{ } . We obtain the followingchain of isomorphisms: H p ( M ; A ) ∼ = H p ( M \ ( ∂M × [0 , A ) (homotopy) ∼ = ˇ H p ( M \ ( ∂M × [0 , A ) (follows from considering the openneighborhoods M \ ( ∂M × [0 , − n ])) ∼ = H n − p ( M \ ∂M, ∂M × (0 , A ) (duality K = M \ ( ∂M × [0 , ∼ = H n − p ( M, ∂M × [0 , A ) (excision U = ∂M ) ∼ = H n − p ( M, ∂M ; A ) , It follows from the definition of the dualising map and naturality of cap product thatthese isomorphisms are given by capping with a generator [ M ] ∈ H n ( M, ∂M ; Z triv ) ∼ = H n ( M, ∂M ; Z ) as in the classical case.The proof of the general case of Theorem A.15 relies on the following lemma. Lemma A.24.
Let M a compact, oriented, connected n -dimensional manifold. Let R and S be compact codimension 0 submanifolds of ∂M such that ∂R = ∂S = R ∩ S and ∂M = R ∪ S . Then the following diagram commutes up to a sign: . . . H p ( M, R ; A ) H p ( M ; A ) H p +1 ( R ; A ) H p +1 ( M, R ; A ) . . .H n − p − ( R, ∂R ; A ) . . . H n − p ( M, S ; A ) H n − p ( M, ∂M ; A ) H n − p − ( R ∪ S, S ; A ) H n − p − ( M, S ; A ) . . . X [ M ] X [ M ] X [ R ] X [ M ]excision isom . Proof.
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