Controlled surgery and L -homology
aa r X i v : . [ m a t h . G T ] A p r CONTROLLED SURGERY AND L -HOMOLOGY FRIEDRICH HEGENBARTH AND DUˇSAN REPOVˇS
Dedicated to the memory of Professor Andrew Ranicki (1948-2018)
Abstract.
This paper presents an alternative approach to controlled surgeryobstructions. The obstruction for a degree one normal map ( f, b ) : M n → X n with control map q : X n → B to complete controlled surgery is an element σ c ( f, b ) ∈ H n ( B, L ), where M n , X n are topological manifolds of dimension n ≥
5. Our proof uses essentially the geometrically defined L -spectrum as de-scribed by Nicas (going back to Quinn) and some well known homotopy theory.We also outline the construction of the algebraically defined obstruction, andwe explicitly describe the assembly map H n ( B, L ) → L n ( π ( B )) in terms offorms in the case n ≡ H n ( B, L ) → H n ( B, L ). Introduction
To solve a surgery problem one encounters an obstruction being an element of theWall group [20]. If one does controlled surgery with respect to a control map over B , the obstruction belongs to a controlled version of Wall groups. Both groupsare constructed in a purely algebraic way as equivalence classes of certain formsor formations. The principal result (cf. Theorem 3.3 in Section 3) of the presentpaper shows that controlled obstructions are elements of H n ( B, L ), where L is thegeometrically defined surgery spectrum as described by Nicas [13]. The basic ideaof our proof is that controlled surgeries are done in small regions of the manifoldwhen projecting it onto B (and this fits well with L -homology of B ). The proof isgiven in Section 3.In Section 1 we review the algebraic construction of controlled surgery obstruc-tions for the case n ≡ L -spectra. We follow theNicas description [13] (which goes back to Quinn [15]). The surgery spaces andspectra are defined semi-simplicially, i.e. by adic surgery problems. According tothe targets of the surgery problems, one obtains spectra denoted by L , resp. L P D .Here, the targets in L P D are adic Poincar´e duality complexes, whereas in L theyare adic manifolds. Date : April 30, 2019.2010
Mathematics Subject Classification.
Primary 57R67, 57P10, 57R65; Secondary 55N20,55M05.
Key words and phrases.
Generalized manifold, resolution obstruction, controlled surgery, con-trolled structure set, L q -surgery, Wall obstruction. Then we prove that the natural inclusion L → L P D is a homotopy equivalence(cf. Proposition 2.2). In particular, π n ( L ) ∼ = π n ( L P D ) , and as shown by Wall [20], π n ( L P D ) ∼ = L n ( { } ) , the Wall-group of the trivial group. We note that this problems was not addressedby Nicas [13].In Section 2.2 we describe elements of the L -homology group. The spectrum L is not connected, in fact, π ( L ) = L ∼ = Z . There is a fiber sequence of spectra L < > → L → K ( Z , L < > the connected covering of L , and K ( Z ,
0) the Eilenberg Mac-Lanespectrum. We study the induced map H n ( B, L ) → H n ( B, L )and give an explicit formula in Section 2.3 (cf. Corollary 2.6). It has particularsignificance when determining the resolution invariant of Quinn ([16, 17]).In Section 3 we treat H n ( B, L ) as the controlled Wall group and we present themain result of this paper - an alternative proof that H n ( B, L ) is the obstructiongroup for controlled surgery problems (cf. Theorem 3.3). Finally, in Epilogue wediscuss controlled Wall realizations of elements in H n +1 ( B, L ) on n -manifolds X .1. Controlled and uncontrolled surgery obstructions I. In this section we denote by B a finite connected polyhedron with fundamentalgroup π = π ( B ), giving rise to the group ring Λ = Z [ π ]. We shall restrict ourselvesonly to the oriented situation, i.e. when the usual orientation map π → {± } is 1.More precisely, we shall work in the category of oriented topological manifolds andtopological bundles. Normal degree one maps( f, b ) : M n → X n are defined as in Wall [20] (here, M in X are n -manifolds, possibly with nonemptyboundary ∂M and ∂X , respectively).We add to this a reference map q : X → B . In the controlled case it serves asthe control map, where B is equipped by a metric given by an embedding B ⊂ R m as a subcomplex, for a sufficiently large m . For controlled surgery we assume that q is a U V -map, i.e. for each contractible open set U ⊂ B , π ( q − ( U )) = 0 (cf.e.g., Ferry [4]).For dim X ≥
5, it was proved by Bestvina that q is homotopic to a U V -map(cf. [1, Theorem 4.4]). In the case when ∂X = 0, one must also assume that q | ∂X : ∂X → B is U V , so in this case one must have dim X ≥
6. Suppose that f restricts to a simplehomotopy equivalence on the boundary ∂X . The map f can be made highly con-nected. ONTROLLED SURGERY AND L -HOMOLOGY 3 In order to complete the surgery in the middle dimension, a surgery obstruction σ ( f, b ) , belonging to the Wall group L n ( π ) , must vanish. Here, we may assumewithout loss of generality that q ∗ : π ( X ) ∼ = −→ π ( B ) . Of course, this holds if q is U V . If σ ( f, b ) = 0, then we get a simple homotopyequivalence M ′ → X relative the boundary, if n ≥
5, which is normally cobordantto M → X .Controlled surgery is much more delicate (cf. [2]). One can define an obstruction σ c ( f, b ) , belonging to the controlled Wall group L n ( B, ε, δ ) (in the notations ofPedersen, Quinn and Ranicki [14]). Here, ε > ε > B and dim X , and δ > ε .When q is U V and n ≥
4, the following holds: If σ c ( f, b ) = 0 then ( f, b ) : M → X is normally cobodant to a δ -homotopy equivalence M ′ f ′ −→ X over B . The map f ′ : M ′ → X is unique up to ε -homotopy.This means that there exist a homotopy inverse g ′ : X → M and homotopies h t : f ′ ◦ g ′ ∼ Id X , g t : g ′ ◦ f ′ ∼ Id M ′ such that the tracks of the homotopies q ◦ h t , q ◦ f ′ ◦ g t are smaller than δ , measured in the metric of B . If ∂X = ∅ , one has to additionallyassume that f | ∂M is already a δ -homotopy equivalence, and f ′ is then a δ -homotopyequivalence relative the boundary.There is an obvious morphism L n ( B, ε, δ ) → L n ( π ) , forgetting the control, also considered as the assembly map. This is because con-trolled surgeries are done in small pieces which can be glued together to obtain theglobal result. We shall come back to this point in Section 3.Here, we point out how one can obtain the Wall obstruction σ ( f, b ) from thecontrolled obstruction σ c ( f, b ) (cf. Part IV below). We shall do this for n ≡ II.
Let now n = 2 k , where k is even. If f : M → X is highly connected then oneis left with the following exact sequence0 → K k ( f, Λ) → H k ( M, Λ) → H k ( X, Λ) → . By duality and the Hurewicz-Whitehead theorems, one has to kill K k ( f, Λ) ∼ = π k +1 ( X, M )by surgeries. Here, K k ( f, Λ) is a stably free based Λ-module, finitely generated,and carrying a Hermitian Λ-bilinear form λ : K k ( f, Λ) × K k ( f, Λ) → Λwhich is refined by a quadratic form µ , deduced from the bundle map b . In Wall [20,p. 47], this is called a special Hermitian form. Equivalence classes of such specialHermitian forms constitute the Wall group L k ( π ) (cf. Wall [20, Chapter 5] forprecise constructions). Hence σ ( f, b ) = [ K k ( f, Λ) , λ, µ ] ∈ L k ( π ) . F. HEGENBARTH AND D. REPOVˇS
III.
We are now going to describe the controlled surgery obstructions. It wasQuinn who explicitly constructed them (cf. Quinn [16, Section 2]). His aim was toprove the existence of resolutions of generalized manifolds. For this purpose it wasnot necessary to construct controlled Wall groups (cf. also Quinn [17]). A detailedconstruction can be found in Ferry [5]. To obtain controlled results one has to workwith the chain complex C ( X, M ) instead of homology. Here are the main steps:Step 1. ( f, b ) : M → X is normally cobordant to ( f , b ) : M → X so that C g ( X, M ) =0 for j ≤ k . This can be obtained for any surgery problem. To con-tinue, we recall that manifolds M satisfy the controlled Poincar´e duality,i.e. the cap product with a fundamental cycle is a δ -chain equivalence C ( M ) → C n − ( M ), and this implies a δ -chain equivalence C ( X, M ) → C n +1 − ( X, M )for arbitrary δ > δ -chain equivalence C ( X, M ) → C n +1 − ( X, M )and controlled cell trading, one proves that C ( X, M ) is δ -chain equivalentto a chain complex of the type0 → D k +1 → D k → . By doing surgery on small k -spheres in M , according to the basis of D k ,one obtains a chain complex of the type0 → A k +1 → . Let M ′ be the result of this surgery.Step 3. By Quinn [16, Proposition 2.4], the pair ( X, M ′ ) is δ -homotopy equivalentto a pair ( X ′ , M ′ ) such that C ( X ′ , M ′ ) = ( A k +1 k + 10 otherwise.Since the chain equivalence in Step 1 is a δ -equivalence for arbitrary small δ , wehave the same situation in Step 3. So the composition q ′ : X ′ ∼ −→ X g −→ B is a U V ( δ )-map. This will be sufficient for our purpose (cf. e.g., Ferry [5],Quinn [16], Yamasaki [21] for the concept of geometric algebra of chain complexes, U V ( δ ), and δ -chain equivalences).By Step 3, our original surgery problem M → X is replaced by a normal degreeone map ( f ′ , b ′ ) : M ′ → X ′ , where b ′ is a bundle map between the normal bundle ν M ′ of M ′ and the bundle ξ over X ′ , induced by the map X ′ → X from the normal bundle ν X of X .The result is a finitely generated geometric Z -module C k +1 ( X ′ , M ′ ), with obviousintersection number λ Z : C k +1 ( X ′ , M ′ ) × C k +1 ( X ′ , M ′ ) → Z , ONTROLLED SURGERY AND L -HOMOLOGY 5 refined by a quadratic form µ Z , determined by the normal data, such that the radiusof λ Z is δ -small: for basis elements a, b ∈ C k +1 ( X ′ , M ′ )one has λ Z ( a, b ) = 0 provided that d ( q ′ ( a ) , q ′ ( b )) > δ. The equivalence class of[ C k +1 ( X ′ , M ′ ) , λ Z , µ Z ] ∈ L n ( B, ε, δ )is the controlled surgery obstruction of the surgery problem( f ′ , b ′ ) : M ′ → X ′ . One notes that the Wall obstructions σ ( f, b ) and σ ( f ′ , b ′ ) in L n ( π ) coincide. IV.
The map L n ( B, ε, δ ) → L n ( π ) . We are given σ ( f, b ) ∈ L n ( π ) which we represent by the triple ( K k ( f ′ , Λ) , λ, µ ).One first notes that K k ( f ′ , Λ) = C k +1 ( X ′ , M ′ ) ⊗ Z Λ . Let a , . . . a r ∈ C k +1 ( X ′ , M ′ )be a Z -basis. Then e a i = a i ⊗ , i = 1 , . . . r is a Λ-basis of K k ( f ′ , Λ). To calculate λ Z ( a i , a j ), one observes that the a i ’s arerepresented by small maps ( D k +1 , S k ) → ( X ′ , M ′ ) , where ∂a i : S k → M ′ are framed immersions in general position. Let ∂a i ∩ ∂a j = { p , . . . , p m } . Then λ Z ( a i , a j ) = m X i =1 ε i , where ε i = ± p i .The elements e a , . . . , e a r ∈ K k ( f ′ , Λ) ∼ = C k +1 ( f X ′ , f M ′ )are considered as liftings of ∂a , . . . , ∂a r in the universal covering f M ′ of M ′ . Al-ternatively, e a , . . . , e a r are immersed spheres in M ′ together with connecting pathsto a base point of M ′ . We state our observation in the following Proposition 1.1.
With the above assumptions and notations we have λ ( e a i , e a j ) = λ Z ( a i , a j ) g ij ∈ Λ , where g ij ∈ π is determined by the paths connecting e a i , e a j to the base point.Proof. Since the radius of λ Z is as small as we want, and the immersed spheres aresmall, we may assume that their images in B are contained in a contractible subset.By the U V property we conclude that e a i ( S k ) ∪ e a j ( S k ) ⊂ U ⊂ M ′ with π ( U ) = { } . Calculating λ ( e a i , e a j ) as in the proof in Wall [20, Theorem 5.2], one obtains theclaim. (cid:3) F. HEGENBARTH AND D. REPOVˇS
The case when π is the fundamental group of the n -torus, this was first provedby Mio and Ranicki [12, Section 10.1]. Since any surgery problem ( f, b ) : M n → X n between n -manifolds without boundaries can be considered as a controlled problemover Id : X → X , we can get the following Corollary 1.2.
Let n ≡ . Then σ ( f, b ) ∈ L n ( π ( X )) has a representation ( G, λ, µ ) with G a free Λ -module with basis b , . . . , b r such that λ ( b i , b j ) = n ij g ij , n ij ∈ Z , and g ij ∈ π ( X ) . Remark 1.3. If ∂M, ∂X are nonempty, the restriction f | ∂M has to be a δ -controlledhomotopy equivalence. In the case of Id : X → X as the control map this impliesthat f | ∂M is a homeomorphism. However, if f | ∂M is a δ -homotopy equivalence forsome U V -map q : X → B, then the proof goes through. L -spectra and L -homology On the geometric construction of the L -spectrum. The geometric L -spectrum was introduced in Quinn [15] as a semi-simplicial Ω-spectrum. Detailscan also be found in Nicas [13] which we shall follow. We define surgery spaces L r ( B ), where B is a polyhedron. We are only interested in the case B = {∗} andwe shall write L r = L r {∗} .An s -simplex σ ∈ L r is a normal degree one map between ( r + s )-dimensionaloriented ( s + 3)-ads of manifolds( M, ∂ M, . . . , ∂ s M, ∂ s +1 M ) → ( X, ∂ X, . . . , ∂ s X, ∂ s +1 X )such that f restricted to ∂ s +1 M is a homotopy equivalence. To each σ belongs areference map of ( s + 3)-ads( X, ∂ X, . . . , ∂ s X, ∂ s +1 X ) → (∆ s , ∂ ∆ s , . . . , ∂ s ∆ s , ∆ s )to the standard s -simplex ∆ s . Note that the last face ∂ s +1 X maps to the interiorof ∆ s , and plays a special role in the constructions.Let L r ( s ) be the set of s -simplices. Then L r is a pointed semisimplicial complexwith base points the empty problem and there is a homotopy equivalence to thesimplicial loop space of L r − (cf. Nicas [13, Proposition 2.2.2]): L r → Ω L r − . The collection of surgery spaces { L r , r ∈ Z } defines a spectrum L + such that itshomotopy groups π n ( L + ) are the Wall groups L n (1). In the notation of [18], L + = L < > , whereas L denotes the periodic L -spectrum with the 0-term = Z × G / T OP . In order to do this we have to address two problems. The first one comes fromthe following easily proved (and well known) lemma.
Lemma 2.1.
The surgery space L defined above satisfies π ( L ) = { } .Proof. Recall, that we are working in the simplicial category. A typical element σ ∈ L (0) is a map of degree one of the type {± y , . . . , ± y k } → { x } . By the degreeone property one can reorder it as follows { y , + y , − y , . . . , + y l , − y l } → { x } . ONTROLLED SURGERY AND L -HOMOLOGY 7 The 1-simplex { I , . . . , I l } → J , with I j denoting the interval with ∂I j = { y j , − y j } ,shows that σ is equivalent to ( { y } → { x } ). Here we view J as a degenerate 1-simplex consisting of a single point. Moreover, ( { y } → { x } ) is equivalent to theempty set. Therefore π ( L ) = 0. (cid:3) The second problem arises from comparison with the Wall groups in Wall [20,Chapter 9] (cf. the proof of Nicas [13, Proposition 2.2.4]). The point is that inWall [20], Poincar´e duality spaces are used as targets, whereas in [13] manifolds areused. This point was not addressed in Nicas [13]. It might be not the same for ageneric polyhedron B , but it gives the same result when B = {∗} .To see this, we introduce the surgery spaces L P Dr in the same way as L r , butPoincar´e-ads as targets (this was used in Quinn [15]). One also proves that L P Dr ishomotopy equivalent to Ω L P Dr − . There is an obvious map L r → L P Dr , and π ( L r ) ∼ = π ( L P Dr ) = { } . We can define Ω-spectra L + and L P D using this.To match up with the usual notation, we write L + = { L − r , r ≥ } , L P D = { L P D − r , r ≥ } . Both spectra are connected and L + becomes L h i in the notations of Ranicki [18]. Proposition 2.2.
The map L + → L P D is a homotopy equivalence.Proof.
We shall show that the induced morphism π n ( L + ) → π n ( L P D )is an isomorphism for n ≥ . The assertion will then follow by the Whiteheadtheorem.Observe that π n ( L P D ) ∼ = π n + r ( L P D − r ) ∼ = π n ( L P D ) ∼ = π (Ω n L P D − n ) . However, the last one coincides with the group L n ( {∗} ) , considered by Wall [20,Chapter 9]. We begin with the higher dimensional case. Case I: n ≥ . Wall defines a restricted set L n ( {∗} ) ⊂ L n ( {∗} )consisting of simply-connected surgery problems (an adic version of this was con-sidered by Nicas [13, Chapter 2]). He shows that L n ( {∗} ) → L n ( {∗} )is bijective for n ≥ L n ( {∗} ) → L n (= Wall group of π = { } )is an isomorphism for n ≥ L n = π n ( L + ) → π n ( L P D ) ∼ = L n ( {∗} ) Θ −→ L n is the identity, this proves that we indeed have an isomorphism π n ( L + ) ∼ = −→ π n ( L P D )for all n ≥ F. HEGENBARTH AND D. REPOVˇS
Case II: n = 4 . The surgery obstruction map Θ is defined for n = 4 and thecomposition L = π ( L + ) → π ( L P D ) ∼ = L ( {∗} ) Θ −→ L is the identity. Therefore π ( L + ) → π ( L P D )is injective. Since L ( {∗} ) ∼ = −→ L ( {∗} ) , we can represent an element in π ( L P D ) by( f, b ) : M → X with π ( X ) = { } . Assume first that ∂X = ∅ . Then G = H ( X, Z ) is Z -free and the intersection form λ X : G × G → Z is unimodular. By Freedman [6, Theorem 1.5], there is a simply-connected 4-manifold M ′ realizing ( G, λ X ) . However, by Milnor [11], M ′ is homotopically equiv-alent to X , therefore ( f, b ) : M → X is equivalent to the surgery problem( f ′ , b ′ ) : M → M ′ arising from π ( L + ) . Now assume that ∂X = ∅ . Then f | ∂M : ∂M → ∂X is a homotopy equivalence. We obtain a closed surgery problem by glueing Id : M → M and f : M → X along the boundary N = M ∪ Id M Id ∪ f −−−→ M ∪ f | ∂M X = Y. By the van Kampen theorem, π ( Y ) = { } . It is now easy to see that the class of N → Y represents the same as the classes of( f, b ) : M → X and Id : M → M in L ( {∗} ) (cf. Supplement below). However, Id : M → M represents the trivialclass, so we are back in the closed case. Case III: n = 3 . (See also a short proof in Supplement below.) Let( f, b ) : M → X be given. As in the case n = 4, we may assume that ∂X = ∅ . There is a commutativediagram of well-known isomorphisms of Hurewicz maps between cobordism groupsΩ ( X ) Ω P D ( X ) H ( X, Z ) µ ONTROLLED SURGERY AND L -HOMOLOGY 9 It follows that µ is an isomorphism and since f is of degree one, M is P D -cobordantto X over X .Let q : Z → X be a P D -complex over X with q | X = Id and q | M = f. The Spivak fibration ν Z of Z restricts to the Spivak fibration ν X and ν M , and wehave the maps of the m -sphere into the Thom spaces( S m × I, S m × { } , S m × { } ) → ( T ν Z , T ν X , T ν M ) . Since M is a manifold, let us for simplicity write ν M also for the stable normalbundle of M ⊂ S m , i.e. b : ν M → ξ, where ξ is a certain topological reduction of ν X . Claim. If ν Z has a topological reduction ω which restricts to ξ on X , then ( f, b ) : M → X is equivalent to a normal degree one map ( f ′′ , b ′′ ) : M ′′ → M, where b ′′ : ν M ′′ → η and η = ω | M . This is obtained by taking the transverse inverse images of the composition of(
Z, X, M ):( S m × I, S m × { } , S m × { } ) → ( T ν Z , T ν X , T ν M ) h −→ ( T ω, T ξ, T η ) , where h comes from the reduction ω of ν Z .Now, the obstructions to existence of such ω belong to H r +1 ( Z, X, π r ( G / T OP ))hence there is only one in H ( Z, X, π ( G / T OP )) ∼ = H ( Z, X, Z ) . Since X ⊂ Z q −→ X is the identity, the homomorphism H r ( Z, Z ) → H r ( X, Z )is surjective, i.e. the short cohomology sequence0 → H ( Z, X, Z ) → H ( Z, Z ) → H ( X, Z ) → H ( Z, Z ) is 0 because ν z has topological re-duction (cf. Hambleton [7]). Therefore such ω exists which proves the surjectivityof { } = π ( L + ) → π ( L P D ) , i.e. π ( L P D ) = { } . Case IV: n = 1 , . These two cases are obvious since for n = 1 , P D -complexes are manifolds.This completes the proof of Proposition 2.2. (cid:3)
Supplement.
We add two remarks here. In the case n = 4 and ∂X = ∅ , a normal cobordism between N = M ∪ Id M → M ∪ f | ∂M X, M ( f,b ) −−−→ X, and Id : M → M can be constructed as follows: replace X by X ′ = X ∪ f | ∂M ∂M × I being homotopy equivalent to X with a collared boundary ∂M ⊂ X ′ . Then glue M × I · ∪ X ′ × I at M × { } ∪ X ′ × { } along the collar ∂M × [1 − ε, ⊂ M ∩ X ′ . This gives a
P D -complex V . A similar construction on M × M · ∪ M × I gives a 5-manifold W . An obvious degree one normal map can be constructedfrom Id M and ( f, b ). Note that ∂W = M · ∪ M · ∪ M ∪ Id M and ∂V = X · ∪ M · ∪ M ∪ f | ∂M X. In the case n = 3 it seems that one can replace the P D -complex Z by Z ′ with ∂Z ′ = ∂Z and π ( Z ′ ) = { } by Poincar´e surgeries. The obstruction to findinga reduction ω of ν Z ′ such that ω | X = ξ and ω | M = ν M belongs to H ( Z ′ , M · ∪ X, L ) ∼ = H ( Z ′ , L ) = 0 . Then we get a normal bordism between( f, b ) : M → X and Id : M → M, hence the class of ( f, b ) is trivial.2.2. Concerning the elements of H n ( B, L ) . We shall write as before L for theperiodic spectrum L h i , and L + = L h i for its connective covering spectrum. Recallthe fibration sequence (cf. Ranicki [18, Section 15]) L + → L → K ( L , , where K ( L ,
0) is the Eilenberg-MacLane spectrum. We shall study the homologyof this sequence in Subsection 2.3.Here, we want to describe elements x ∈ H n ( B, L ), where B ⊂ S m is a finitepolyhedron. We follow Ranicki [18, Section 12], to represent x by a cycle, using adual cell decomposition of S m . This is justified by Ranicki [18, Remark 12.5].If σ is a simplex of S m , let D ( σ, S m ) be its dual cell. It has a canonical ( m −| σ | + 3)-ad structure, where | σ | = dim σ and m − | σ | = dim D ( σ, S m ) . The element x is then represented by a simplicial map( S m , S m \ B ) → ( L n − m , ∅ ) ONTROLLED SURGERY AND L -HOMOLOGY 11 (one should merely replace S m \ B with the supplement of B , as done in Ran-icki [18]). Let us first consider the case when x : ( S m , S m \ B ) → ( L + n − m , ∅ )represents an element of H n ( B, L + ), i.e. x ( σ ) ∈ L + n − m ( m − | σ | ) . However, this is the surgery space described above, i.e. x ( σ ) is a degree one normalmap ( f σ , b σ ) : M n −| σ | σ → X n −| σ | σ between ( n − | σ | )-dimensional ( m − | σ | + 3)-ads with a reference map X n −| σ | σ → D ( σ, S m ). The cycle condition implies that they can be assembled (the colimit)to a degree one normal map ( f, b ) : M n → X n with boundaries ∂M, ∂X , so that f | ∂M is a homotopy equivalence, together with a reference map X → B . Note that x ( σ ) = ∅ if σ / ∈ B , and X → B is the colimit of all X n −| σ | σ → D ( σ, S m ) ⊂ S m with a retraction onto B (cf. Nicas [13, Theorem 3.3.2], or Laures and McClure [10,Proposition 6.6]). Moreover, the boundary map ∂M → ∂X is the colimit of thevarious homotopy equivalences ∂ m −| σ | +1 M n −| σ | σ → ∂ m −| σ | +1 X n −| σ | σ . To consider the general case x ∈ H n ( B, L ) we recall two properties:(a) (Periodicity): Suppose that dim B − ≤ r . Then there is a natural isomor-phism H r ( B, L ) → H r +4 ( B, L ) (cf. Ranicki [18, p. 289-290]);(b) If dim B < r , then H r ( B, L + ) ∼ = −→ H r ( B, L ) . Both properties also easily follow from the Atiyah-Hirzebruch spectral sequence H p ( B, π q ( L )) p + q = r −−−−→ H r ( B, L ) , and the periodicity of the L -spectrum: L r ∼ = L s if r − s ≡ . In order to represent x ∈ H n ( B, L ), we choose r sufficiently large with r − n ≡ x as an element of H r ( B, L ) ∼ = H r ( B, L + ) as above. Assembling(colimit) then gives a degree one normal map ( f, b ) : P r → Q r with the referencemap q : Q r → B , and f | ∂P a homotopy equivalence.A specific construction of the degree one normal map P r → Q r is given using theidentification H n ( B, L ) with the controlled Wall group L n ( B, ε, δ ), as establishedby Pedersen, Quinn and Ranicki [14]. Here are some details. Suppose that also n ≡ x corresponds to a triple { G, λ Z , µ Z } as described in Section 1. Itcan be considered as an element of L r ( B, ε, δ ) by the periodicity, r − n ≡ ∂N ofa regular neighbourhood N ⊂ R r of B ⊂ R r .We obtain P r which can be written as P r = N ∪ ∂N × I ∪ {∪ k D r × D r } . Here, k = rank G , and λ Z , µ Z are realized as framed immersions S r × I → ∂N × I. The handles D r × D r are attached to the top along the framed embeddings. Bythe controlled Hurewicz-Whitehead theorem and the α -approximation theorem onegets a degree one normal map P r → N of r -manifolds with boundary, such that ∂P r → ∂N is a homeomorphism. Then we can close this in the usual way to get P r = P r ∪ ∂ N → N ∪ ∂ N = Q r . It is more convenient to consider P r → N and we shall denote it by P r → N with ∂P r → ∂N a homeomorphism. Let q : N → B be the retraction. It can bemade transverse to the dual cell-decomposition, the map P r → N is in the naturalway a surgery mock bundle (cf. Nicas [13, Section 3.2]) Remark 2.3.
If conversely, we are given a degree one normal map ( f, b ) : P r → Q r with the reference map q : Q r → B , one can define an element x ∈ H r ( B, L + ) bysplitting ( f, b ) into pieces using transversality of q with respect to the dual cell-decomposition of B ⊂ S m . The homomorphism H n ( B, L ) → H n ( B, L ) . Without loss of generality wemay assume that dim B = n . Let B ( n − be the ( n − B . This impliesthat H n ( B, L ) ∼ = Z n ( B ) ⊗ L ֒ → C n ( B ) ⊗ L ∼ = H n ( B, B ( n − , L )is injective. Here, Z n ( B ) are the n -cycles of B and C n ( B ) are the n -chains. More-over, from the Atiyah-Hirzebruch spectral sequence one easily gets that H n ( B, B ( n − , L ) ∼ = −→ H n ( B, B ( n − , L ) . Lemma 2.4.
The natural map H n ( B, L ) → H n ( B, B ( n − , L ) factorizes as H n ( B, L ) → H n ( B, L ) ⊂ H n ( B, B ( n − , L ) ∼ = H n ( B, B ( n − , L ) . Proof.
This follows by the commutativity of the diagram: −−−−→ H n ( B, L ) −−−−→ H n (B , B (n-1) , L ) −−−−→ y y ∼ = −−−−→ H n ( B, L ) −−−−→ H n (B , B (n-1) , L ) −−−−→ induced by the map of spectra L → K ( L , (cid:3) To prepare the next lemma we must study the spectral sequence E pq ∼ = H p ( B, L q ) ==== ⇒ p + q = m H m ( B, L )in more detail. First, we note that E ∞ n,m − n ⊂ E n,m − n , since H p ( B, L q ) = 0 for p > n . Moreover, E ∞ n,m − n = F n,m − n / F n − ,m − n +1 , where F n,m − n = Im( H m ( B ( n ) , L ) → H m ( B, L )) ∼ = H m ( B, L ) . We consider the composite map α : H m ( B, L ) → E ∞ n,m − n ⊂ E n,m − n ∼ = H n ( B, L m − n ) ∼ = Z n ( B ) ⊗ L m − n . ONTROLLED SURGERY AND L -HOMOLOGY 13 Lemma 2.5.
Let B ⊂ S m , dim B = n , and m − n ≡ . Then H n ( B, L ) −−−−→ H n (B , L ) ∼ = Z n (B) ⊗ L y ∼ = ∼ = y β H m ( B, L ) −−−−→ α Z n (B) ⊗ L m-n commutes. Here, H n ( B, L ) ∼ = −→ H m ( B, L ) and β : Z n ( B ) ⊗ L ∼ = −→ Z n ( B ) ⊗ L m − n are isomorphisms induced by periodicity. The proof follows by the spectral sequences. (cid:3)
We now describe the image of x ∈ H n ( B, L ) in H n ( B, L ) ∼ = Z n ( B ) ⊗ L ⊂ C n ( B ) ⊗ L . It can be written as P k τ · τ , where τ ranges over the n -simplices of B .Step 1. Consider x ∈ H m ( B, L + ) ∼ = H m ( B, L ) ∼ = H n ( B, L ).Step 2. Represent x as the cycle x : ( S m , S m \ B ) → ( L , ∅ ).Step 3. Consider x ( τ ) : ( f τ , b τ ) : P m − nτ → Q m − nτ for τ < B , | τ | = n .One observes that ∂Q m − nτ = ∂P m − nτ = ∅ because its boundaries are composedof elements x ( ρ ), with | ρ | > n (because the boundary ∂D ( τ, S m ) is formed fromcells of type D ( ρ, S m ), | ρ | > n ). Now dim B = n , so ( f τ , b τ ) is a closed surgeryproblem.To summarize, we have obtained Corollary 2.6.
Let dim B = n , B ⊂ S m , with m − n ≡ . An element x ∈ H n ( B, L ) has the image in H n ( B, L ) ∼ = Z n ( B ) ⊗ L ∼ = Z n ( B ) ⊗ L m − n equal to X τ
The diagram in Lemma 2.5 can be rewritten as H n ( B, L ) −−−−→ H n (B , L ) y ∼ = y ∼ = H m ( B, L ) −−−−→ H n (B , L m-n )where the map H m ( B, L ) → H n ( B, L m − n )is the composition of H m ( B, L ) ∼ = H m ( B, L h m − n i )(cf. Ranicki [18, p. 156]) and H m ( B, L h m − n i ) → H n ( B, L m − n ) (cf. Ranicki [18, p. 289]). Note also the following commutativity L n ( B, ε, δ ) ∼ = H n (B , L ) y ∼ = y ∼ = L m ( B, ε, δ ) ∼ =H m (B , L ) . The above calculation resulting in Corollary 2.6 follows from the compositions H n ( B, L ) → H m ( B, L ) → H n ( B, L m − n )of the above diagrams.For the other composition one has to determine the map H n ( B, L ) → H n ( B, L ).This was done by Ranicki ([18]). In Prop. 15.3(II) therein an explicit formula isestablished using however the algebraic version of the L -spectrum. In fact, Propo-sition 15.3(II) is the formula for the case of the symmetric L -spectrum, but it issimilar for the quadratic L -spectrum.3. H n ( B, L ) as the controlled Wall group We mentioned in Section 1 the controlled Wall group L n ( B, ε, δ ). It can bedefined for any n ≥
0. As before, we assume that B is a finite polyhedron.Based on the work of Yamasaki [22], Quinn, Pedersen and Ranicki [14] provedthe following result. Theorem 3.1.
For finite dimensional ANR’s there is a morphism H n ( B, L ) → L n ( B, ε, δ ) which is an isomorphism for suitable ε > and δ > . Remark 3.2.
In the paper by Pedersen, Quinn and Ranicki [14] , L is the spec-trum of quadratic algebraic Poincar´e ads, and the morphism mentioned above is anassembling map. The proof of the theorem consists of showing that an element of L n ( B, ε, δ ) can be split into pieces giving an element of H n ( B, L ) . Now, the alge-braic L -spectrum is homotopy equivalent to the geometric one (cf. Ranicki [18] ), so H n ( B, L ) can be considered as the controlled Wall group. As in the classical surgery theory, the controlled version leads to the controlledsurgery sequence (cf. Ferry [5, Theorem 1.1.]). This involves the controlled struc-ture set for which one needs the ”stability properties” as proved in Ferry [5, Theo-rem 10.2].We shall now present the main result of this paper - an alternative proof that H n ( B, L ) is the obstruction group for controlled surgery problems. Theorem 3.3.
Let ( f, b ) : M n → X n be a degree one normal map between mani-folds, n ≥ , and π : X n → B a U V -map. Then an element σ c ( f, b ) ∈ H n ( B, L ) is defined so that σ c ( f, b ) = 0 if and only if ( f, b ) is normally cobordant to a δ -homotopy equivalence, uniquely up to ε -homotopy. Remark 3.4.
Note that the
U V -condition for π is no restriction when n ≥ .The theorem holds for n = 4 , if the U V -condition is satisfied.Proof. The map π : X → B can be assumed to be transverse to the dual cells of B (cf. Cohen [3]); i.e. π − ( D ( σ, B )) = X n −| σ | σ ONTROLLED SURGERY AND L -HOMOLOGY 15 is an ( n − | σ | )-dimensional submanifold. If we embed B ⊂ S m , for m sufficientlylarge, we have π − ( D ( σ, B )) = π − ( D ( σ, S m )) , and X n −| σ | σ has the corresponding ( m − | σ | + 3)-ad structure. By transversality wedefine M n −| σ | σ = f − ( X n −| σ | σ ) . The restrictions of b gives a family { ( f σ , b σ ) : M n −| σ | σ → X n −| σ | σ | σ ⊂ B } which obviously defines a cycle z : ( S m , S m \ B ) → ( L n − m , ∅ ) , i.e. an element [ z ] = σ c ( f, b ) ∈ H n ( B, L ) . We now suppose that [ z ] = 0, i.e. there is a simplicial map w : ( S m , S m \ B ) × ∆ → ( L n − m , ∅ )with w (0) = z, and w (1) = ∅ (cf. Ranicki [18, Section 12]). This means that thevarious ( m − | σ | + 3)-ads M n −| σ | σ → X n −| σ | σ normally bound. Since π is U V , wecan assume that these are simply-connected surgery problems. If f σ | ∂M σ : ∂M σ → ∂X σ is already a homotopy equivalence, it follows that ( f σ , b σ ) is normally cobordant toa homotopy equivalence. The proof now proceeds by induction on n − | σ | .Let X q = [ | σ |≥ q X n −| σ | σ and M q = [ | σ |≥ q M n −| σ | σ , hence X n ⊂ X n − ⊂ . . . ⊂ X ⊂ X = X, similarly for M . The induction hypothesis:
The restriction f to M q is a homotopy equivalence withthe inverse f : X q → M q such that the homotopies of f ◦ f ∼ Id X q and f ◦ f ∼ Id M q are controlled, i.e. when restricted onto X n −| σ | σ (resp. M n −| σ | σ ) they have tracksover D ( σ, B ) when projected down to B . More precisely, f | M σ : M n −| σ | σ → X n −| σ | σ is a homotopy equivalence with the inverse f (cid:12)(cid:12) X n −| σ | σ : X n −| σ | σ → M n −| σ | σ , and the homotopies above restrict to homotopies of f | M σ ◦ f (cid:12)(cid:12) X σ ∼ Id x σ and f (cid:12)(cid:12) X σ ◦ f | M σ ∼ Id M σ over D ( σ, B ). The inductive step:
Suppose we are given τ ⊂ B with | τ | = q −
1, i.e. dim X τ =dim M τ = n − q + 1, and ∂M τ = [ σ M σ , ∂X τ = [ σ X σ with | σ | = q , and σ a face of τ . By the inductive hypothesis, f | M σ is a homotopyequivalence. These can be glued together by the well known homotopy theory (cf.Hatcher [8], or Sullivan [19, Lemma H]) to give a homotopy equivalence f | ∂M τ : ∂M τ → ∂X τ . So let F τ : ( V τ , M τ , M ′ τ ) → ( X τ × I, X τ × , X τ × F τ | M τ = f τ , F τ | M ′ τ = f ′ τ arehomotopy equivalences, and because surgery was done in the interior of M τ , wehave that F τ | ∂V τ : ∂V τ = M τ ∪ ∂M τ × I ∪ M ′ τ → X τ × { } ∪ ∂X τ × I ∪ X τ × { } coincides with f τ ∪ ( f τ × I ) ∪ f ′ τ (note that f ′ τ | ∂M τ = f τ | ∂M τ ).We denote by f ′ τ : X τ → M τ a homotopy inverse of f ′ τ . In our construction weadd the cylinders ∂M τ × I and ∂X τ × I to M ′ τ and X τ ×
1, and again denote themby M ′ τ and X τ ×
1. Then f and f ′ τ can be glued to give a homotopy equivalence f ∪ f ′ τ : M q ∪ M ′ τ → X q ∪ X τ . This can be done for every τ ⊂ B with | τ | = q −
1. If M τ ∩ M τ ′ are nonempty,they intersect in a common face M σ , resp. X σ , where we have the map f . Gluedtogether they give a homotopy equivalence f ′ : M q − → X q − . Lemma 3.5.
There is a homotopy inverse f ′ : X q − → M q − such that f ′ (cid:12)(cid:12)(cid:12) X q = f ,and f ′ (cid:12)(cid:12)(cid:12) X τ is a homotopy inverse of f ′ τ for every τ ⊂ B with | τ | = q − .Proof. We fix τ ⊂ B , | τ | = q −
1. First note that f (cid:12)(cid:12) ∂X τ ∼ f ′ τ (cid:12)(cid:12)(cid:12) ∂X τ (where f ′ τ is theabove introduced inverse of f ′ τ ). This can be seen as follows: f ◦ f (cid:12)(cid:12) ∂X τ ∼ Id ∂X τ and f τ ◦ f ′ τ (cid:12)(cid:12)(cid:12) ∂X τ ∼ Id ∂X τ implies f τ ◦ f (cid:12)(cid:12) ∂X τ = f ◦ f (cid:12)(cid:12) ∂X τ ∼ f τ ◦ f ′ τ (cid:12)(cid:12)(cid:12) ∂X τ . However, f τ is a homotopy equivalence, hence f (cid:12)(cid:12) ∂X τ ∼ f ′ τ (cid:12)(cid:12)(cid:12) ∂X τ .Let H t : ∂X τ → ∂M ′ τ = ∂M τ be a homotopy such that H = f (cid:12)(cid:12) ∂X τ and H = f ′ τ (cid:12)(cid:12)(cid:12) ∂X τ . ONTROLLED SURGERY AND L -HOMOLOGY 17 By the Homotopy Extension Property we obtain a homotopy e H t : X τ × I → M ′ τ such that X τ × I [ ∂X τ × I ∪ X τ × { } H t ∪ f ′ τ ✲ ∂M ′ τ ∪ M ′ τ = M ′ τ e H t ✲ commutes. Hence e f τ = e H : X τ → M ′ τ is a homotopy equivalence such that e f τ (cid:12)(cid:12)(cid:12) ∂X τ = f (cid:12)(cid:12) ∂X τ . Hence f ∪ e f τ : X q ∪ X τ → M q ∪ M ′ τ is a homotopy inverse of f ′ and it has the desired property. Since at the intersection X τ ∩ X τ ′ the maps e f τ , e f τ ′ coincide with f , we can glue them together to get f ′ : X q − → M q − as claimed. (cid:3) In order to complete the proof of Theorem 3.3 it remains to prove that there arehomotopies of f ′ ◦ f ′ ∼ Id X q − , and f ′ ◦ f ′ ∼ Id M q − with small tracks. We shallconstruct such a homotopy for f ′ ◦ f ′ ∼ Id X q − . The other case is similar.We let H t : X q × I → X q be the homotopy of f ◦ f ∼ Id X q given by theinductive hypothesis, so h t = H t | ∂X τ is a homotopy of f ◦ f (cid:12)(cid:12) ∂X τ ∼ Id | ∂X τ . Recallthat X q ∩ X τ = ∂X τ , so f ′ τ ◦ f ′ τ coincides with h = f τ ◦ f τ on ∂X τ . We consider h t ∪ f ′ τ ◦ f ′ τ : ∂X τ × I ∪ X τ × { } → X τ and apply the Homotopy Extension Property to obtain h ′ t = X τ × I → X τ suchthat X τ × I [ ∂X τ × I ∪ X τ × { } ✲ X τ h ′ t ✲ The map h ′ : X τ → X τ is homotopic to Id X τ , since h ′ = f ′ τ ◦ f ′ τ and it satisfies h ′ | ∂X τ = h = Id ∂X τ .It follows from Hatcher [8, Proposition 0.19] that h ′ is homotopic relative ∂X τ to Id X τ by a homotopy h ′′ t (note that here Id X τ is a homotopy inverse of h ′ ). We cantherefore compose the homotopies h ′ t and h ′′ t in the usual way to get a homotopy( h ′ ∗ h ′′ ) t : X τ × I → X τ which coincides with H t on X q ∩ X τ , giving a homotopy H t ∪ ( h ′ ∗ h ′′ ) t : ( X q ∪ X τ ) × I → X q ∪ X τ between ( f ◦ f ) ∪ ( f ′ τ ◦ f ′ τ ) and Id . If τ, τ ′ ⊂ B are ( q + 1)-simplices such that X τ ∩ X τ ′ = ∅ , they intersect in acommon face σ, | σ | = q , so the above constructed homotopies coincide with H t , i.e.we can glue them together to get the desired controlled homotopies.One notes that the tracks can be arbitrary small (measured in B ) if we use anarbitrary small cell-decomposition of B . This proves the inductive step.We have in particular to consider the low-dimensional cases n , n −
1, and n −
3, because surgery does not apply (note that in dimension 4 one has to applyFreedman’s result).By the degree one property we can assume that M n = X n . For n − i, ≤ i ≤ f τ , b τ ) : M jτ → X jτ , ≤ j ≤ , are special. Namely, ∂X jτ is a ( j − π is U V . We can close ∂X jτ by a j -disk to get a closed simply-connected j -manifold, i.e. a j -sphere. By theinductive hypothesis, ∂M jτ must also be a ( j − M jτ can be closed.The closed problem M jτ → X jτ bounds a problem W j +1 τ → V j +1 τ (because σ c ( f, b ) = 0). Deleting the ( j + 1)-disks one obtains a normal cobordism between M jτ → X jτ and M ′ jτ = S j ∼ = −→ X jτ = S j . We can now choose a degree one map( V j +1 τ \ ˚ D j +1 , X jτ , S j ) → ( S j × I, S j × { } , S j × { } )and obtain a composition F τ : ( W j +1 τ \ ˚ D j +1 , M jτ , S j ) → ( X jτ × I, X jτ × { } , X jτ × { } ) . With this F τ , the proof proceeds as above and Theorem 3.3 is finally proved. (cid:3) Epilogue
We shall conclude this paper by a final remark on the controlled Wall realization.In our earlier paper [9], we showed that the controlled structure set of a manifold X with control map q : X → B is a subgroup of H n +1 ( B, X, L ). The controlledWall action of H n +1 ( B, L ) on it is then nothing but the canonical map H n +1 ( B, L ) → H n +1 ( B, X, L )of L -homology groups. Acknowledgements
This research was supported by the Slovenian Research Agency grants P1-0292,J1-7025, J1-8131, N1-0064, and N1-0083. We thank K. Zupanc for her technicalassistance with the preparation of the manuscript. We acknowledge the referee forcomments and suggestions.
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Dipartimento di Matematica ”Federigo Enriques”, Universit`a degli studi di Milano,20133 Milano, Italy
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