Extreme Nonuniqueness of End-Sum
aa r X i v : . [ m a t h . A T ] N ov EXTREME NONUNIQUENESS OF END-SUM
JACK S. CALCUT, CRAIG R. GUILBAULT, AND PATRICK V. HAGGERTY
Dedicated to the memory of Andrew Ranicki
Abstract.
We give explicit examples of pairs of one-ended, open 4-manifolds whose end-sumsyield uncountably many manifolds with distinct proper homotopy types. This answers stronglyin the affirmative a conjecture of Siebenmann regarding the nonuniqueness of end-sums. In ad-dition to the construction of these examples, we provide a detailed discussion of the tools usedto distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theoremis an understanding of this algebra for an end-sum in terms of the algebras of the summandstogether with ray-fundamental classes determined by the rays used to perform the end-sum. Dif-fering ray-fundamental classes allow us to distinguish the various examples, but only through thesubtle theory of infinitely generated abelian groups. An appendix is included which contains thenecessary background from that area. Introduction
Our primary goal is a proof of the following theorem, which emphatically affirms a conjecture ofSiebenmann [CKS12, p. 1805] addressed in an earlier article by the first and third authors of thepresent paper [CH14].
Main Theorem.
There exist one-ended, open -manifolds M and N such that the end-sum of M and N yields uncountably many manifolds with distinct proper homotopy types. In addition to definitions, background, and proofs, we carefully develop the tools needed to distin-guish between the aforementioned manifolds. Foremost among these is the end-cohomology algebraof an end-sum. We also discuss some intriguing open questions.End-sum is a technique for combining a pair of noncompact n -manifolds in a manner that pre-serves the essential properties of the summands. Sometimes called connected sum at infinity inthe literature, end-sum is the natural analogue of both the classical connected sum of a pair of n -manifolds and the boundary connected sum of a pair of n -manifolds with boundaries. The earli-est intentional use of the end-sum operation appears to have been by Gompf [Gom83] in his workon smooth structures on R . Other applications to the study of exotic R s can be found in Ben-nett [Ben16] and Calcut and Gompf [CG19]. End-sum has also been useful in studying contractible n -manifolds not homeomorphic to R n . This is due to the fact that, unlike with classical connectedsums, the end-sum of a pair of contractible manifolds is again contractible. For a sampling of suchapplications in dimension 3, see Myers [Mye99] and Tinsley and Wright [TW97]; in dimension 4,see Calcut and Gompf [CG19] and Sparks [Spa18]; and in dimensions n ≥
4, see Calcut, King, andSiebenmann [CKS12]. For “incidental” applications of end-sum to the study of contractible openmanifolds of dimension n ≥
4, see Curtis and Kwun [CW65] and Davis [Dav83]. These incidental(unintentional) applications are due to the fact that the interior of a boundary connected sum may
Date : November 19, 2020.2010
Mathematics Subject Classification.
Primary 57R19; Secondary 55P57.
Key words and phrases.
End-sum, connected sum at infinity, end-cohomology, proper homotopy, direct limit,infinitely generated abelian group. also be viewed as an end-sum of the corresponding interiors.Each variety of connected sum involves arbitrary choices that lead to questions of well-definedness.For example, to perform a classical connected sum in the smooth category , one begins with a pair ofsmooth, connected, oriented n -manifolds, then chooses smooth n -balls B ⊂ Int M and B ⊂ Int N and an orientation reversing diffeomorphism f : ∂B → ∂B . From there, one declares M N to bethe oriented manifold ( M − Int B ) ∪ f ( N − Int B ). Provided M and N are connected, standardbut nontrivial tools from differential topology can be used to verify that, up to diffeomorphism, M N does not depend upon the choices made. See Kosinski [Kos93, p. 90] for details. Note thatwell-definedness fails if one omits the connectedness hypothesis or ignores orientations.For smooth, oriented n -manifolds M and N with nonempty boundaries, a boundary connectedsum is performed by choosing smooth ( n − B ⊂ ∂M and B ⊂ ∂N , and an orientationreversing diffeomorphism f : B → B . Provided ∂M and ∂N are connected, an argument simi-lar to the one used for ordinary connected sums shows that the adjunction space M ∪ f N (suitablysmoothed and oriented) is well-defined up to diffeomorphism; it is sometimes denoted M ⋄ N . Again,see Kosinski [Kos93, p. 97] for details.An end-sum of a pair of smooth, oriented, noncompact n -manifolds M and N begins with thechoice of properly embedded rays r ⊂ Int M and s ⊂ Int N and regular neighborhoods νr and νs ofthose rays. The regular neighborhoods are diffeomorphic to closed upper half-space R n + , so each hasboundary diffeomorphic to R n − . Choose an orientation reversing diffeomorphism f : ∂νr → ∂νs to obtain the end-sum defined by ( M − Int νr ) ∪ f ( N − Int νs ); sometimes this end-sum is denotedinformally as M ♮N . By an argument resembling those used above, neither the choice of thicken-ings of r and s (that is, the regular neighborhoods νr and νs ) nor the diffeomorphism f affect thediffeomorphism type of M ♮N . However, the choices of rays r and s are another matter. For exam-ple, if M has multiple ends, then rays in M tending to different ends of M can yield inequivalentend-sums, even in the simple n = 2 case. For that reason, we focus on the most elusive case where M and N are one-ended. The existence of knotted rays in 3-manifolds poses problems unique tothat dimension. Indeed, Myers has exhibited an uncountable collection of topologically distinct end-sums where both summands are R . So, quickly we arrive at the appropriate question: For smooth,oriented, one-ended, open n -manifolds M and N where n ≥ , is end-sum well-defined up to dif-feomorphism? In many cases the answer is affirmative. For example, R ♮ R is always R [Gom85].More generally, the end-sum of n -manifolds with Mittag-Leffler ends and n ≥ n -manifolds will be their end-cohomology algebras—more specifically it is the ring structure of thatalgebra that holds the key. This is an essential point since every end-sum herein of a given pairof one-ended manifolds has homology and cohomology groups (absolute and “end”) in each dimen-sion that are isomorphic to those of any other end-sum of the same two manifolds. To allow fordifferences in these end-cohomology algebras, it will be necessary to work with manifolds that havesubstantial cohomology at infinity. That leads us naturally to the well-studied, but subtle, area ofinfinitely generated abelian groups. For the benefit of the reader with limited background in that Similar definitions, conventions, and arguments allow for analogous connected sum operations in the piecewiselinear and topological categories. For the sake of simplicity and focus, we will restrict our attention to the smoothcategory.
XTREME NONUNIQUENESS OF END-SUM 3 area, we have included an appendix with key definitions and proofs of the fundamental facts usedin this paper. Capturing this subtle algebra in the form of a manifold requires some care—mostsignificantly, a precise description of the end-cohomology algebra of an end-sum in terms of the end-cohomology algebras of the summands with input from so-called ray-fundamental classes determinedby the chosen rays. We provide a careful development of this topic, as suggested to us by Henry King.Given past applications of end-sum, the following open question deserves attention.
Question 1.1.
For contractible, open n -manifolds M and N of dimension n ≥
4, is
M ♮N well-defined up to diffeomorphism or up to homeomorphism?Note that, by Poincar´e duality “at the end” (see Geoghegan [Geo08, p. 361]), the end-cohomologyalgebra of a contractible, open n -manifold is isomorphic to the ordinary cohomology algebra of S n − .So, the methods used in the present paper appear to be of no use in attacking this problem.Using ladders based on exotic spheres, Calcut and Gompf [CG19, Ex. 3.4(a)] gave pairs of smooth,one-ended, open n -manifolds for some n ≥ n = 4 whose end-sumsare homeomorphic but not diffeomorphic. A key open question is the following (see [CH14, p. 3282]and [CG19, p. 1303]). Question 1.2.
Can the (oriented) end-sums of a smooth, oriented, one-ended, open 4-manifold M with a fixed oriented exotic R be distinct up to diffeomorphism?If such examples exist, then it appears that distinguishing them will be difficult [CG19, Prop. 5.3].The outline of this paper is as follows. Section 2 lays out some conventions, defines end-sum,and discusses end-cohomology. Section 3 defines some manifolds (stringers, surgered stringers, andladders) useful for our purposes and computes their end-cohomology algebras. Section 4 classifiesstringers, surgered stringers, and ladder manifolds based on closed surfaces. Section 5 defines ray-fundamental classes and presents a proof of an unpublished result of Henry King that computesthe end-cohomology algebra of a binary end-sum. Section 6 computes ray-fundamental classes insurgered stringers and ladders. Section 7 proves the Main Theorem. Appendix A presents somerelevant results from the theory of infinitely generated abelian groups. Acknowledgement
The authors are indebted to Henry King for valuable conversations and for allowing us to includean exposition of his unpublished Theorem 5.4. This research was supported in part by SimonsFoundation Grant 427244, CRG.2.
Conventions, End-sum, and End-cohomology
Conventions.
Throughout this paper, topological spaces are metrizable, separable, and locallycompact. In particular, each space has a compact exhaustion (see § ∂M of a manifold M is oriented by the outward normal first convention. Let Int M denote the manifold interior of M . A manifold without boundary is closed if it is compact and is open if it is noncompact. A map of spaces is proper provided the inverse image of each compactset is compact. A ray is a smooth proper embedding of the real half-line [0 , ∞ ). The submanifold[0 , ∞ ) ⊂ R is standardly oriented [GP74, Ch. 3]. By M ≈ N we indicate diffeomorphic manifolds(not necessarily preserving orientation). J. CALCUT, C. GUILBAULT, AND P. HAGGERTY
We will consider rays in manifold interiors as well as neatly embedded rays. Recall that a man-ifold A embedded in a manifold B is said to be neatly embedded provided A is a closed subspaceof B , ∂A = A ∩ ∂B , and A meets ∂B transversely (see Hirsch [Hir76, p. 30] and Kosinski [Kos93,pp. 27–31 & 62]). The closed subspace condition is automatically satisfied by any proper embed-ding. Now, let r ⊂ M be a neatly embedded ray. We let τ r ⊂ M denote a smooth closed tubularneighborhood of r in M as in Figure 2.1 (left). By definition, a closed tubular neighborhood is a Figure 2.1.
Neatly embedded ray r ⊂ M and a smooth closed tubular neighbor-hood τ r ⊂ M (left), and ray r ⊂ Int M and a smooth closed regular neighborhood νr ⊂ Int M (right).restriction of an open tubular neighborhood (see Hirsch [Hir76, pp. 109–118] and Kosinksi [Kos93,pp. 46–53]); we will always assume that closed tubular neighborhoods are restrictions of neat tubularneighborhoods. In particular, the disk bundle τ r over r meets ∂M in exactly the disk over the end-point 0. Closed tubular neighborhoods of r in M are unique up to ambient isotopy fixing r . Next,let r ⊂ Int M be a ray. We let νr ⊂ Int M denote a smooth closed regular neighborhood of r inInt M as in Figure 2.1 (right). Existence and ambient uniqueness of smooth closed tubular neighbor-hoods and collars imply the same results for smooth closed regular neighborhoods [CKS12, pp. 1815].2.2. End-sum.
We now define the end-sum of two noncompact manifolds. An end-sum pair ( M, r )consists of a smooth, oriented, connected, noncompact manifold M together with a ray r ⊂ Int M .We allow M to have arbitrarily many ends. Consider two end-sum pairs ( M, r ) and (
N, s ) where M and N have the same dimension m ≥
2. The end-sum of (
M, r ) and (
N, s ), which we denote by(
M, r ) ♮ ( N, s ), is defined as follows. Choose smooth closed regular neighborhoods νr ⊂ Int M and νs ⊂ Int N of r and s respectively. Delete the interiors of these regular neighborhoods and glue theresulting manifolds M − Int νr and N − Int νs along their boundaries ∂νr ≈ R m − and ∂νs ≈ R m − by an orientation reversing diffeomorphism as in Figure 2.2. Remark 2.1.
The manifold (
M, r ) ♮ ( N, s ) is smooth and oriented, and its diffeomorphism typeis independent of the choices of the regular neighborhoods and the glueing diffeomorphism [CH14, § End-cohomology.
Throughout, R denotes a commutative, unital ring. We use the singulartheory for ordinary (co)homology. We suppress the coefficient ring when that ring is Z .We will distinguish noncompact manifolds by the isomorphism types of their (graded) end-cohomology algebras. Just as cohomology is a homotopy invariant of spaces, end-cohomology is aproper homotopy invariant of spaces. We adopt the direct limit approach to end-cohomology. An al-ternative may be found in several places including Conner [Con57], Raymond [Ray60], Massey [Mas78,Ch. 10], and Geoghegan [Geo08, Ch. 12]. The alternative approach provides some advantages in XTREME NONUNIQUENESS OF END-SUM 5
Figure 2.2.
End-sum of two manifold/ray pairs.terms of establishing the foundations of end-cohomology and comparing it to other cohomologytheories. On the other hand, we find the direct limit approach invaluable for carrying out con-crete calculations. For the benefit of the reader—and since the arguments are straightforward andsatisfying—we take the time to develop the basics of end-cohomology straight from the direct limitdefinition .Fix a topological space X . Define the poset ( K , ≤ ) where K is the set of compact subsets of X and K ≤ K ′ means K ⊆ K ′ . We have a direct system of graded R -algebras H ∗ ( X − K ; R ) where K ∈ K .The morphisms of this direct system are restrictions induced by inclusions. Define H ∗ e ( X ; R ), the end-cohomology algebra , to be the direct limit of this direct system. For the relative version, let( X, A ) be a closed pair , namely a space X together with a closed subspace A ⊆ X . Regard X asthe closed pair ( X, ∅ ). Consider the direct system H ∗ ( X − K, A − K ; R ) where K ∈ K and the mor-phisms are restrictions. Define H ∗ e ( X, A ; R ) to be the direct limit of this direct system. Similarly,reduced end-cohomology e H ∗ e ( X, A ; R ) is the direct limit of the direct system e H ∗ ( X − K, A − K ; R ).We employ a standard explicit model of the direct limit [ES52, p. 222] where an element of H ∗ e ( X, A ; R ) is represented by an element of H ∗ ( X − K, A − K ; R ) for some compact K . Tworepresentatives α ∈ H ∗ ( X − K, A − K ; R ) and α ′ ∈ H ∗ ( X − K ′ , A − K ′ ; R ) are equivalent if theyhave the same restriction in some H ∗ ( X − K ′′ , A − K ′′ ; R ), where K, K ′ ⊆ K ′′ .Recall that a compact exhaustion of X is a nested sequence K ⊆ K ⊆ · · · of compact subsetsof X whose union equals X and where K j ⊆ K ◦ j +1 for each j . Here, K ◦ j +1 denotes the topologicalinterior of K j +1 as a subspace of X . By our hypotheses on spaces, each space has a compactexhaustion (see Hocking and Young [HY61, p. 75]). Let { K j } be any compact exhaustion of X .As { K j } is cofinal in K , we may compute H ∗ e ( X, A ; R ) using the direct system indexed by Z > .Namely, there is a canonical isomorphism (see [ES52, p. 224])(2.1) H ∗ e ( X, A ; R ) ∼ = lim −→ H ∗ ( X − K j , A − K j ; R )We claim that we may delete instead the topological interior K ◦ j of K j to obtain the canonicalisomorphism(2.2) H ∗ e ( X, A ; R ) ∼ = lim −→ H ∗ ( X − K ◦ j , A − K ◦ j ; R ) For background on proper homotopy, see Guilbault [Gui16, pp. 58–59] and Hughes and Ranicki [HR96, Ch. 3]. Forbackground on direct systems and direct limits, see Eilenberg and Steenrod [ES52, Ch. 8], Massey [Mas78, Appendix],and Rotman [Rot10, Ch. 6.9].
J. CALCUT, C. GUILBAULT, AND P. HAGGERTY
To prove the claim, we show that the right hand sides of (2.1) and (2.2) are canonically isomorphic.Let G j and G ′ j denote the j th terms in these direct systems. The inclusions K ◦ ⊆ K ⊆ K ◦ ⊆ K ⊆· · · induce the obvious maps between these direct systems and give the commutative diagram G / / ❆❆❆❆❆❆❆❆ G / / ❆❆❆❆❆❆❆❆ G / / ❆❆❆❆❆❆❆❆❆ · · · G ′ / / O O G ′ / / O O G ′ / / O O · · · (2.3)We get induced maps φ : lim −→ G j → lim −→ G ′ j and ψ : lim −→ G ′ j → lim −→ G j between the direct limits [ES52,p. 223]. It is a simple exercise to prove that ψ ◦ φ and φ ◦ ψ are the respective identity maps (useEilenberg and Steenrod [ES52, pp. 220–223]). This proves the claim. Passing to a subsequencein either (2.1) or (2.2) canonically preserves the isomorphism type of the direct limit since theseisomorphisms are independent of the choice of compact exhaustion (see also [ES52, p. 224]).A proper map of closed pairs is a map of closed pairs f : ( X, A ) → ( Y, B ) such that f : X → Y isproper; it follows that the restriction f | : A → B is proper. For example, if ( X, A ) is a closed pair,then the inclusions ( A, ∅ ) ֒ → ( X, ∅ ) and ( X, ∅ ) ֒ → ( X, A ) are proper maps of closed pairs. Each suchmap f induces a morphism f ∗ e : H ∗ e ( Y, B ; R ) → H ∗ e ( X, A ; R )Indeed, let { L j } be a compact exhaustion of Y . Observe that (cid:8) K j := f − ( L j ) (cid:9) is a compactexhaustion of X . In particular, K j ⊆ K ◦ j +1 . We have the commutative diagram H ∗ ( X − K , A − K ; R ) / / H ∗ ( X − K , A − K ; R ) / / · · · H ∗ ( Y − L , B − L ; R ) / / O O H ∗ ( Y − L , B − L ; R ) / / O O · · · (2.4)The rows are direct systems and the vertical maps are induced by the restrictions f | j : ( X − K j , A − K j ) → ( Y − L j , B − L j )These maps induce the morphism f ∗ e on the direct limits which are identified with the respectiveend-cohomology algebras by (2.1). The same argument applies to reduced cohomology. It is straight-forward to verify that id ∗ e = id and ( g ◦ f ) ∗ e = f ∗ e ◦ g ∗ e . Lemma 2.2.
Let f, g : (
X, A ) → ( Y, B ) be proper maps of closed pairs. If f and g are properlyhomotopic, then f ∗ e = g ∗ e .Proof. By hypothesis, there is a proper homotopy F : X × I → Y such that F = f , F = g , and F t ( A ) ⊆ B for all t ∈ I . Let pr : X × I → X be projection. Let { L j } be a compact exhaustion of Y . So, F − ( L j ) ⊆ X × I and K j := pr ( F − ( L j )) ⊆ X are compact. As projection maps are open, { K j } is a compact exhaustion of X . For each j , we have the restriction F | j : ( X − K j ) × I → Y − L j which is a homotopy between the restrictions f | j : X − K j → Y − L j g | j : X − K j → Y − L j Hence, f | ∗ j = g | ∗ j in (2.4). Therefore, the induced morphisms on direct limits are equal as desired. (cid:3) Corollary 2.3.
If the closed pairs ( X, A ) and ( Y, B ) are proper homotopy equivalent by the propermaps f : ( X, A ) → ( Y, B ) and g : ( Y, B ) → ( X, A ) , then f ∗ e and g ∗ e are graded R -algebra isomor-phisms. XTREME NONUNIQUENESS OF END-SUM 7
Proof.
By hypothesis, g ◦ f is proper homotopy equivalent to id X by a proper homotopy sending A into B at all times, and similarly for f ◦ g and id Y . By Lemma 2.2 and the preceding observations, f ∗ e ◦ g ∗ e = id and g ∗ e ◦ f ∗ e = id. (cid:3) Lemma 2.4.
For each closed pair ( X, A ) there is the induced long exact sequence · · · → H ke ( X, A ; R ) → H ke ( X ; R ) → H ke ( A ; R ) → H k +1 e ( X, A ; R ) → · · · Proof.
Let { K j } be a compact exhaustion of X . As A is closed in X , { A ∩ K j } is a compactexhaustion of A . Consider the biinfinite commutative diagram whose j th column is the long exactsequence for the pair ( X − K j , A − K j ). The rows in this diagram are the various direct systems H k ( A − K j ; R ), H k ( X − K j ; R ), and H k ( X − K j , A − K j ; R ). The maps in this diagram betweensuccessive rows induce maps of their direct limits. The resulting sequence of direct limits is exactsince the direct limit is an exact functor in the category of R -modules (see [ES52, p. 225] or [Mas78,p. 389]). (cid:3) A closed triple ( X, A, B ) is a space X together with subspaces B ⊆ A ⊆ X each closed in X .With the long exact sequences for the closed pairs ( A, B ), (
X, B ), and (
X, A ) in hand, a well-knowndiagram chase [ES52, p. 24] proves the following.
Corollary 2.5.
For each closed triple ( X, A, B ) there is the induced long exact sequence · · · → H ke ( X, A ; R ) → H ke ( X, B ; R ) → H ke ( A, B ; R ) → H k +1 e ( X, A ; R ) → · · · Remark 2.6.
It is crucial for end-cohomology that one consider closed pairs and triples. Otherwise,one does not obtain induced maps for the usual long exact sequences, and the direct system H ∗ ( A ∩ K j ; R ) (where { K j } is a compact exhaustion of X ) need not compute H ke ( A ; R ).An excisive triad ( X ; A, B ) is a space X together with two closed subspaces A ⊆ X and B ⊆ X such that X = A ◦ ∪ B ◦ where A ◦ and B ◦ are the topological interiors of A and B in X respectively . Lemma 2.7.
Let ( X ; A, B ) be an excisive triad and set C = A ∩ B . Then, the inclusion φ : ( A, C ) → ( X, B ) induces the excision isomorphism φ ∗ : H ∗ e ( X, B ; R ) → H ∗ e ( A, C ; R ) Proof.
Let { K j } be a compact exhaustion of X . So, { A ∩ K j } , { B ∩ K j } , and { C ∩ K j } are compactexhaustions of A , B , and C respectively. We have the two direct systems(2.5) H ∗ ( A − K j , C − K j ; R ) H ∗ ( X − K j , B − K j ; R )where the morphisms in both systems are induced by inclusions. For each j , we have the inclusion φ j : ( A − K j , C − K j ) → ( X − K j , B − K j )Observe that X − K j = ( A − K j ) ◦ ∪ ( B − K j ) ◦ where ( A − K j ) ◦ denotes the topological interior of A − K j as a subspace of X − K j and similarly for ( B − K j ) ◦ . Therefore, each φ ∗ j is an excision isomorphismon ordinary R -cohomology. By [ES52, p. 223], these isomorphisms induce an isomorphism betweenthe direct limits of the direct systems (2.5). Two applications of (2.1) now complete the proof. (cid:3) The following corollary is useful (compare [ES52, p. 32] and May [May99, pp. 145]).
Corollary 2.8.
Let ( X ; A, B ) be an excisive triad and set C = A ∩ B . Denote the inclusion mapsby i A : ( A, C ) ֒ → ( X, C ) and i B : ( B, C ) ֒ → ( X, C ) . Then, the map h : H ∗ e ( X, C ; R ) → H ∗ e ( A, C ; R ) ⊕ H ∗ e ( B, C ; R ) defined by h ( α ) = ( i ∗ A ( α ) , i ∗ B ( α )) is a graded R -algebra isomorphism. Recall that the product is coordinatewise in the direct sum of algebras. Note the subtle notational distinction between a triple and a triad.
J. CALCUT, C. GUILBAULT, AND P. HAGGERTY
Proof.
The commutative diagram of inclusions(
A, C ) (cid:15) (cid:15) i A $ $ ■■■■■■■■■ ( B, C ) (cid:15) (cid:15) i B z z ✉✉✉✉✉✉✉✉✉ ( X, C ) z z ✉✉✉✉✉✉✉✉✉ $ $ ■■■■■■■■■ ( X, B ) (
X, A )(2.6)induces the commutative diagram H ∗ e ( A, C ; R ) H ∗ e ( B, C ; R ) H ∗ e ( X, C ; R ) i ∗ A h h h h PPPPPPPPPPPP i ∗ B ♥♥♥♥♥♥♥♥♥♥♥♥ H ∗ e ( X, B ; R ) ∼ = exc. O O ) (cid:9) ♥♥♥♥♥♥♥♥♥♥♥♥ H ∗ e ( X, A ; R ) exc. ∼ = O O h h PPPPPPPPPPPP (2.7)The vertical maps are excision isomorphisms (Lemma 2.7). Hence, the two lower maps are injectiveand the two upper maps are surjective. The two diagonals are exact being portions of long exactsequences for triples (Corollary 2.5). These properties of (2.7) readily imply that h is both injectiveand surjective. (cid:3) Remark 2.9.
Let r ⊂ Int M be a ray and νr ⊂ Int M be a smooth closed regular neighborhoodof r . Define c M := M − Int νr . We claim that the inclusion φ : ( c M , ∂νr ) ֒ → ( M, νr ) induces anisomorphism φ ∗ e on end-cohomology. However, the corresponding triad ( M ; c M , νr ) is not excisivesince M is not the union of the topological interiors c M ◦ and νr ◦ of c M and νr in M respectively.This nuisance is easily fixed using a closed collar. Let Z ≈ ∂νr × [0 ,
1] be a closed collar on ∂νr in νr . Notice that φ equals the composition of the inclusions( c M , ∂νr ) i ֒ → ( c M ∪ Z, Z ) j ֒ → ( M, νr )Both induced morphisms i ∗ e and j ∗ e are isomorphisms. The former holds since i is properly homotopicto the identity map on ( c M , ∂νr ) using the obvious proper strong deformation retraction that collapsesthe closed collar Z to ∂νr . The latter holds since j ∗ e is the excision isomorphism from the excisivetriad ( M ; c M ∪ Z, νr ). Hence, φ ∗ e is an isomorphism and the claim is proved. Excision is used inSection 5 below and Corollary 2.8 is used in the proof of Theorem 5.4. In each of these places, weleave the standard collaring fix to the reader.For a general noncompact space or manifold, it appears to be difficult to compute the end-cohomology algebra in a comprehensible manner. So, we deliberately construct manifolds (stringers,surgered stringers, and ladders) with tractable algebras that fit into the following framework.Let M be a connected space with a compact exhaustion { K j } where j ∈ Z ≥ . Assume K = ∅ .Define M j := M − K ◦ j where K ◦ j is the topological interior of K j as a subspace of M . So, each M j is closed in M and M = M ⊇ M ⊇ M ⊇ · · · is a (closed) neighborhood system of infinity as in Figure 2.3. By (2.2), we have H ∗ e ( M ; R ) ∼ =lim −→ H ∗ ( M j ; R ). XTREME NONUNIQUENESS OF END-SUM 9 M M M K K Figure 2.3.
Manifold M with a compact exhaustion { K j } and a (closed) neigh-borhood system of infinity { M j } .For each j , let i j : M j +1 ֒ → M j be the inclusion. Suppose that for each j ∈ Z ≥ there is aretraction r j : M j → M j +1 (in Figure 2.3, the retraction r j folds up the bottom of M j ). Thecomposition r j ◦ i j equals the identity on M j +1 . So, i ∗ j ◦ r ∗ j equals the identity on H ∗ ( M j +1 ; R ) andeach i ∗ j is surjective. By [ES52, p. 222], each of the canonical morphisms q i : H ∗ ( M i ; R ) → H ∗ e ( M ; R )is surjective with kernel Q i equal to the submodule of elements that are eventually sent to 0 in thedirect system H ∗ ( M j ; R ). Here, q i ( α ) := J α K . Hence, for each i ∈ Z ≥ we have H ∗ ( M i , R ) /Q i ∼ = H ∗ e ( M ; R ). This discussion applies to relative and reduced end-cohomology as well.3. Stringers, Surgered Stringers, and Ladders
In this section, we define some manifolds and present their end-cohomology algebras. These willbe used in our proof of the Main Theorem.Let X be a closed, connected, oriented n -manifold with n ≥
2. The stringer based on X is[0 , ∞ ) × X with the product orientation [GP74, Ch. 3]. Let X t = { t } × X , so the oriented boundaryof the stringer is − X . The end-cohomology algebra of the stringer is e H ∗ e ([0 , ∞ ) × X ; R ) ∼ = e H ∗ ( X ; R )The surgered stringer S ( X ) based on X is obtained from the stringer on X by performing count-ably many oriented 0-surgeries as in Figure 3.1. We refer to the glued-in copies of D × S n as rungs .The space X ∨ J in Figure 3.1 is the wedge of X and J , where J is the wedge of a ray, n -spheres S j , and 1-spheres T j . It is a strong deformation retract of S ( X ) by an argument similar to the oneprovided in [CH14, Lemma 3.2].The surgered stringer S ( X ) is oriented using the orientation of the stringer [0 , ∞ ) × X . Let S [ j,k ] denote the points of S ( X ) with heights in the interval [ j, k ] as in Figure 3.2. We orient S [ j,k ] asa codimension-0 submanifold of S ( X ). We orient each n -sphere S j so that the oriented boundaryof the cobordism S [ j,j +1 / is X j +1 / − X j + S j . Thus, the oriented boundary of S [ j +1 / ,j +1] is X j +1 − X j +1 / − S j .Let s j denote the fundamental class [ S j ] of S j , and let t j denote the fundamental class [ T j ] of T j . So, the nonzero reduced integer homology groups of J are e H n ( J ) ∼ = Z [ s ] and e H ( J ) ∼ = Z [ t ]. X X X S S S X T S T S T S Figure 3.1.
Surgered stringer S ( X ) and a strong deformation retract X ∨ J of S ( X ). S j X j X j +1 S j X j X j + ½ S j X j +1 X j + ½ Figure 3.2.
Cobordisms S [ j,j +1] , S [ j,j +1 / , and S [ j +1 / ,j +1] in S ( X ).Define σ j and τ j to be the dual fundamental classes [ S j ] ∗ and [ T j ] ∗ so that the nonzero reducedcohomology groups of J are e H n ( J ; R ) ∼ = Hom Z ( Z [ s ] , R ) ∼ = R [[ σ ]] e H ( J ; R ) ∼ = Hom Z ( Z [ t ] , R ) ∼ = R [[ τ ]]All cup products in e H ∗ ( J ; R ) vanish.An argument similar, but simpler, to the one provided in [CH14, §
3] now shows that the end-cohomology algebra of S ( X ) is e H ke ( S ( X ) ; R ) ∼ = H n ( X ; R ) ⊕ R [[ σ ]] /R [ σ ] if k = n , H k ( X ; R ) ⊕ ≤ k ≤ n − H ( X ; R ) ⊕ R [[ τ ]] /R [ τ ] if k = 1,0 otherwiseThe cup product is coordinatewise in the direct sum; it is that of X in the first coordinate andvanishes in the second coordinate.Let X and Y be closed, connected, oriented n -manifolds with n ≥
2. The ladder manifold L ( X, Y ) based on X and Y is obtained from the stringers based on X and on Y by performingcountably many oriented 0-surgeries as in Figure 3.3 (Ladder manifolds were the primary objectsof study in [CH14]. See that paper for more details). Again, the glued-in copies of D × S n arecalled rungs . The ladder manifold is oriented using the orientations of the stringers based on X and Y . The oriented boundary of L ( X, Y ) is − X − Y . Let L [ j,k ] denote the points of L ( X, Y )with heights in the interval [ j, k ]. We orient L [ j,k ] as a codimension-0 submanifold of L ( X, Y ). The
XTREME NONUNIQUENESS OF END-SUM 11 X X X Y Y Y S S S X T S T Y S S Figure 3.3.
Ladder manifold L ( X, Y ) and a strong deformation retract X ∨ J ∨ Y .cobordism L [ j,j +1] is the union of two connected cobordisms with shared boundary component S j .We orient each S j so that the oriented boundaries of these cobordisms are X j +1 − X j + S j and Y j +1 − Y j − S j . The ladder manifold L ( X, Y ) also contains 1-spheres T j as shown in Figure 3.3, andit strong deformation retracts to the wedge X ∨ J ∨ Y as explained in [CH14, p. 3287].Let s j denote the fundamental class [ S j ] of S j , and let t j denote the fundamental class [ T j ] of T j . Again, the nonzero reduced integer homology groups of J are e H n ( J ) ∼ = Z [ s ] and e H ( J ) ∼ = Z [ t ].Define σ j and τ j to be the dual fundamental classes [ S j ] ∗ and [ T j ] ∗ so that the nonzero reducedcohomology groups of J are e H n ( J ; R ) ∼ = Hom Z ( Z [ s ] , R ) ∼ = R [[ σ ]] e H ( J ; R ) ∼ = Hom Z ( Z [ t ] , R ) ∼ = R [[ τ ]]All cup products in e H ∗ ( J ; R ) vanish. By [CH14, § L ( X, Y ) is e H ke ( L ( X, Y ) ; R ) ∼ = ( H n ( X ; R ) ⊕ R [[ σ ]] ⊕ H n ( Y ; R )) /K if k = n , H k ( X ; R ) ⊕ ⊕ H k ( Y ; R ) if 2 ≤ k ≤ n − H ( X ; R ) ⊕ R [[ τ ]] /R [ τ ] ⊕ H ( Y ; R ) if k = 1,0 otherwisewhere K := (cid:8) ( P β i , β, − P β i ) | β = P β i σ i ∈ R [ σ ] (cid:9) ∼ = R [ σ ]. The cup product is coordinatewise inthe direct sum; it is that of X in the first coordinate, that of Y in the third coordinate, and vanishesin the second coordinate. Remark 3.1. As X and Y are closed, connected, and oriented n -manifolds, we have that e H ne ( L ( X, Y ) ; R ) ∼ = ( R ⊕ R [[ σ ]] ⊕ R ) /K When R = Z , we show in Appendix A.2 below that the dual module of this Z -module is isomorphicto Z . On the other hand, for any ring R the canonical R -module homomorphism R ⊕ R → ( R ⊕ R [[ σ ]] ⊕ R ) /K defined by ( r, s ) J ( r, , s ) K is injective and, hence, an R -module isomorphism onto its image( R ⊕ ⊕ R ) /K . When R is a field, ( R ⊕ ⊕ R ) /K is a two dimensional R -vector space. When R = Z , ( R ⊕ ⊕ R ) /K is a rank two free Z -module. For any ring R , each cup product with valueof degree n must lie in ( R ⊕ ⊕ R ) /K . For many base manifolds, surgered stringers and ladder manifolds have nonisomorphic end-cohomology algebras. The proof of Theorem 4.2 below shows various techniques for distinguishingthese algebras. However, in some exceptional cases these manifolds have diffeomorphic ends.
Proposition 3.2.
Let X be a closed, connected, oriented n -manifold where n ≥ . Let M = L ( X, S n ) ∪ ∂ D n +1 be the ladder manifold with the S n boundary component capped by an ( n + 1) -disk. Then, M is diffeomorphic to S ( X ) . In particular, L ( X, S n ) and S ( X ) have diffeomorphicends and, hence, isomorphic end-cohomology algebras.Proof. Let N be the (classical) connected sum of the stringer [0 , ∞ ) × X and countably many ( n +1)-spheres as in Figure 3.4. Note that N ≈ [0 , ∞ ) × X . Performing countably many oriented 0-surgeries X X X Figure 3.4.
Classical connected sum N of the stringer based on X with countablymany ( n + 1)-spheres.on N yields M , and performing them on [0 , ∞ ) × X yields S ( X ). (cid:3) Remark 3.3.
Given a manifold Y (not necessarily connected) and two proper disjoint rays in Y , ladder surgery is the operation where one performs countably many oriented 0-surgeries on Y usingthe 0-spheres given by the corresponding integer points on the rays. Properly homotopic rays yielddiffeomorphic manifolds (see Definition 3.1 and Corollary 4.13 in [CG19]). Corollary 3.4.
For any ring R , there is an R -module isomorphism f : R ⊕ R [[ x ]] /R [ x ] −→ ( R ⊕ R [[ x ]] ⊕ R ) /K (cid:0) r, q P ∞ i =0 c i x i y (cid:1) q (cid:0) r − c , P ∞ i =1 ( c i − c i − ) x i , c (cid:1) y Proof.
The topological proof of Proposition 3.2 determines the function f . With f in hand, it isstraightforward to verify (purely algebraically) that f is a well-defined R -module isomorphism. Theinverse function f − is given by q (cid:0) r, P ∞ i =0 a i x i , s (cid:1) y (cid:16) r + s, q P ∞ i =0 ( s + P ij =0 a j ) x i y (cid:17) In particular, f maps (1 , J K ) J (1 , , K and (cid:16) , q − x y (cid:17) J (0 , , K . (cid:3) Consider the Z -module G = Z ⊕ Z [[ x ]] / Z [ x ]. The submodule 0 ⊕ Z [[ x ]] / Z [ x ] is determined alge-braically in an isomorphism invariant manner as the elements of G sent to 0 by every element ofthe dual Z -module of G (see Corollary A.2 in Appendix A below). However, the submodule Z ⊕ Corollary 3.5.
Consider the Z -module G = Z ⊕ Z [[ x ]] / Z [ x ] . The elements (1 , J K ) and (cid:16) , q − x y (cid:17) generate a rank two free Z -submodule of G . Further, there is a Z -module automorphism of G thatinterchanges these two elements. In particular, ⊕ Z [[ x ]] / Z [ x ] has unequal complements in G . XTREME NONUNIQUENESS OF END-SUM 13
We emphasize that G does not split off Z ⊕ Z as a direct summand by Corollary A.2. Proof.
Let f : G → ( Z ⊕ Z [[ x ]] ⊕ Z ) /K be the isomorphism from Corollary 3.4. The first conclusionfollows by Remark 3.1. Consider an involution of the ladder manifold L ( S n , S n ) (for example, aproduct of two reflections) that interchanges the stringers, reverses the orientation of each sphere S j , and induces the involution ρ of ( Z ⊕ Z [[ x ]] ⊕ Z ) /K given by J ( r, γ, s ) K J ( s, − γ, r ) K . Theautomorphism ψ : G → G given by ψ = f − ◦ ρ ◦ f interchanges the desired elements. (cid:3) In our proof of the Main Theorem, we will need to algebraically detect the submodule Z ⊕ G . The previous corollary shows that this will require more of the end-cohomology algebra thanjust the top degree module. We will use base manifolds X with nontrivial cup products in order toalgebraically detect this submodule.4. Stringers, Surgered Stringers, and Ladders Based on Surfaces
This section classifies all stringers, surgered stringers, and ladder manifolds based on closed sur-faces. It demonstrates various methods for distinguishing end-cohomology algebras up to isomor-phism. In interesting cases, the ring structure plays the deciding role. This classification of laddersbased on surfaces answers a question raised by Calcut and Haggerty [CH14, p. 3295]. In addition, itsproof is good preparation for the more complicated situations that arise in subsequent sections. LetΣ g denote the closed, connected, and oriented surface of genus g ∈ Z ≥ . Throughout this section,we use integer coefficients.The end-cohomology algebra of the stringer [0 , ∞ ) × Σ g is e H ke ([0 , ∞ ) × Σ g ) ∼ = e H k (Σ g ) ∼ = Z if k = 2, Z g if k = 1,0 otherwiseThe cup product pairing H (Σ g ) × H (Σ g ) → Z is nonsingular and is given by L g (cid:20) − (cid:21) .The end-cohomology algebra of the surgered stringer S (Σ g ) is e H ke ( S (Σ g )) ∼ = Z ⊕ Z [[ σ ]] / Z [ σ ] if k = 2, Z g ⊕ Z [[ τ ]] / Z [ τ ] if k = 1,0 otherwiseThe cup product is coordinatewise in the direct sum, vanishes in the second coordinate, and is thatof the cohomology ring of Σ g in the first coordinate.Given g , g ∈ Z ≥ , the end-cohomology algebra of the ladder manifold L (Σ g , Σ g ) is e H ke ( L (Σ g , Σ g )) ∼ = ( Z ⊕ Z [[ σ ]] ⊕ Z ) /K if k = 2, Z g ⊕ Z [[ τ ]] / Z [ τ ] ⊕ Z g if k = 1,0 otherwisewhere K = { ( P β i , β, − P β i ) | β = P β i σ i ∈ Z [ σ ] } ∼ = Z [ σ ]. The cup product is coordinatewise inthe direct sum and vanishes in the middle coordinate. Define the matrices C = (cid:20) J (1 , , KJ − (1 , , K (cid:21) D = (cid:20) J (0 , , KJ − (0 , , K (cid:21) where J α K is the class of α in ( Z ⊕ Z [[ σ ]] ⊕ Z ) /K . For degree one elements, the cup product in thefirst coordinate is given by L g C , and in the third coordinate by L g D . Of course, all of these manifolds may be capped with compact 3-manifolds (handlebodies, forexample) to eliminate boundary and obtain open, one-ended 3-manifolds. However, compact capswill not alter the isomorphism types of their graded end-cohomology algebras (which is our focus).So, we choose to work with the non-capped manifolds. We will use the following basic fact.
Lemma 4.1.
Let F be a free Z -module of finite rank. Let G and H be submodules of F . Then, rank ( G ∩ H ) ≥ rank ( G ) + rank ( H ) − rank ( F ) .Proof. The hypotheses imply that G , H , and G + H are free Z -modules of rank at most rank ( F ) [DF04,p. 460]. We have the exact sequence of free Z -modules0 → G ∩ H → G ⊕ H → G + H → g ( g, − g ) and the third map is ( g, h ) g + h . Recall two facts: (i) if E is a free Z -module, then rank ( E ) = dim Q ( E ⊗ Z Q ) [DF04, pp. 373 & 471], and (ii) tensoring with Q is an exact functor (since Q is a flat Z -module [DF04, p. 401]). It follows thatrank ( G ) + rank ( H ) = rank ( G ∩ H ) + rank ( G + H ) ≤ rank ( G ∩ H ) + rank ( F )as desired. (cid:3) Now, we will classify up to isomorphism the algebras listed for the three types of manifolds:stringers, surgered stringers, and ladder manifolds based on surfaces. The classification of thesemanifolds up to various types of equivalence will then readily follow. Plainly, L ( X, Y ) ≈ L ( Y, X )for any manifolds X and Y . Theorem 4.2.
Two of the algebras listed are isomorphic if and only if their corresponding manifoldshave the same type and are based on surfaces of equal genus, with the exception: for each g ∈ Z ≥ the algebras for S (Σ g ) , L (Σ g , Σ ) , and L (Σ , Σ g ) are isomorphic. In particular, the algebras for L (Σ g , Σ g ) and L (Σ g , Σ g ) are isomorphic if and only if { g , g } = { g , g } .Proof. For stringers based on surfaces with unequal genus, the algebras are distinguished by theranks of e H e . The algebras for a stringer and a surgered stringer or a ladder manifold are distin-guished by the cardinalities of e H e . Corollary A.2 implies that the algebras for surgered stringersbased on surfaces with unequal genus are distinguished by the ranks of the duals of e H e .For each g ∈ Z ≥ , the manifolds S (Σ g ) and L (Σ g , Σ ) ≈ L (Σ , Σ g ) have diffeomorphic ends byProposition 3.2. So, their algebras are isomorphic. In all other cases, the algebras for S (Σ g ) and L (Σ g , Σ g ) are not isomorphic. If g = 0 and g = g (or g = 0 and g = g ), then use the ranks ofthe duals of e H e . If g = 0 and g = 0, then use the ranks of the (degree two) subgroups generatedby all cup products of degree one elements. For S (Σ g ) this rank is zero or one, and for L (Σ g , Σ g )it is two (see Remark 3.1).It remains to classify the algebras for ladder manifolds based on surfaces. Suppose the followingis an isomorphism f : e H ∗ e ( L (Σ g , Σ g )) → e H ∗ e ( L (Σ g , Σ g ))Corollary A.2 implies that the ranks of the duals of e H e are 2 g + 2 g and 2 g + 2 g respectively. So, g + g = g + g . Suppose, by way of contradiction, that { g , g } 6 = { g , g } . Then, g + g = g + g implies that some g i is strictly greater than the other three. Without loss of generality, we have g > g ≥ g > g ≥ Z [[ τ ]] / Z [ τ ] sum-mands in an isomorphism invariant manner. Let J denote the set of elements in e H e ( L (Σ g , Σ g ))that are sent to 0 by every element in the dual of e H e ( L (Σ g , Σ g )). Note that J is a subgroup of e H e ( L (Σ g , Σ g )) and, in fact, is an ideal in e H ∗ e ( L (Σ g , Σ g )). Similarly, we define the ideal J ′ in XTREME NONUNIQUENESS OF END-SUM 15 e H ∗ e ( L (Σ g , Σ g )). Evidently, f ( J ) = J ′ and so we obtain an induced isomorphism of the quotient al-gebras where we mod out by J and J ′ respectively. Corollary A.2 implies that J = 0 ⊕ Z [[ τ ]] / Z [ τ ] ⊕ J ′ ). Therefore, we have an isomorphism f : A → B of the algebras A = ( Z ⊕ Z [[ σ ]] ⊕ Z ) /K if k = 2, Z g ⊕ ⊕ Z g if k = 1,0 otherwise B = ( Z ⊕ Z [[ σ ]] ⊕ Z ) /K if k = 2, Z g ⊕ ⊕ Z g if k = 1,0 otherwiseLet V = Z g ⊕ ⊕
0, a rank 2 g and degree one submodule of A . Recalling Remark 3.1, cupproducts of elements of V generate ( Z ⊕ ⊕
0) + K , a rank one and degree two submodule of A . Wewill show that cup products of elements of f ( V ) generate a rank two and degree two submodule of B . This contradiction will complete the proof.Note the following facts. For each element 0 = α ∈ V , there exists α ′ ∈ V such that α ∪ α ′ = 0(since the degree one cup product pairing for Σ g is nonsingular). As f is an isomorphism, the previ-ous fact holds for f ( V ) as well. If γ ∈ Z g ⊕ ⊕ δ has degree one, then γ ∪ δ ∈ ( Z ⊕ ⊕
0) + K .Similarly, if γ ∈ ⊕ ⊕ Z g and δ has degree one, then γ ∪ δ ∈ (0 ⊕ ⊕ Z ) + K . The last two factshold since the cup product is coordinatewise.Recalling that g + g = g + g and g > g ≥ g > g ≥
0, Lemma 4.1 implies that there existelements 0 = α ∈ f ( V ) ∩ ( Z g ⊕ ⊕ = β ∈ f ( V ) ∩ (0 ⊕ ⊕ ∩ Z g )By the previous paragraph, there exist α ′ , β ′ ∈ f ( V ) such that0 = α ∪ α ′ ∈ ( Z ⊕ ⊕
0) + K = β ∪ β ′ ∈ (0 ⊕ ⊕ Z ) + K By Remark 3.1, ( Z ⊕ ⊕ Z ) + K is free of rank two. So, these two nonzero cup products generate arank two submodule of degree two. This contradiction completes the proof. (cid:3) Corollary 4.3.
Ladder manifolds L (Σ g , Σ g ) and L (Σ g , Σ g ) based on surfaces of genera g , g , g , g ∈ Z ≥ are proper homotopy equivalent if and only if { g , g } = { g , g } . Hence, the same classificationholds up to homeomorphism and up to diffeomorphism. End-Cohomology Algebra of Binary End-Sum
We present a proof of an unpublished result of Henry King. It computes the end-cohomologyalgebra of a binary end-sum in terms of the algebras of the two summands together with certainray-fundamental classes determined by the rays used to perform the end-sum.First, recall the analogue for classical connected sum. Consider two closed, connected, oriented n -manifolds X and Y . The reduced cohomology ring e H ∗ ( X Y ) is isomorphic to the quotient ofthe sum e H ∗ ( X ) ⊕ e H ∗ ( Y ) by the principal ideal generated by (cid:0) [ X ] ∗ , − [ Y ] ∗ (cid:1) where [ X ] ∗ ∈ H n ( X )and [ Y ] ∗ ∈ H n ( Y ) are the cohomology fundamental classes dual to the respective homology (orien-tation) fundamental classes. The cup product is coordinatewise in the sum. For the unreduced ring H ∗ ( X Y ), let P be the subring of H ∗ ( X ) ⊕ H ∗ ( Y ) consisting of all elements of positive degree andonly those of degree zero of the form ( r, r ) for r ∈ Z . The desired ring is the quotient of P by theprincipal ideal generated by (cid:0) [ X ] ∗ , − [ Y ] ∗ (cid:1) . One may prove these well-known facts by an argumentstructurally the same as our proof of Theorem 5.4 below. For end-sum and end-cohomology, thecohomology fundamental classes will be replaced by ray-fundamental classes that we now define. Let M be a smooth, connected, oriented, noncompact manifold of dimension n + 1 ≥ r ⊂ Int M be a ray, and let νr ⊂ Int M be a smoothclosed regular neighborhood of r in Int M oriented as a codimension-0 submanifold of M . Orientthe hyperplane ∂νr ≈ R n as the boundary of νr . We will define nonzero cohomology classes[ M, r ] ∗ e ∈ H ne ( M, νr ; R )[ r ] ∗ e ∈ e H ne ( M ; R )called (respectively) the relative and absolute ray-fundamental classes determined by the ray r . Ournotation is chosen since, as will emerge, these elements are intimately related to classical fundamen-tal classes of compact manifolds.Recall that a Morse function h : M → R is exhaustive provided h is proper and the image of h isbounded below. Lemma 5.1.
There exists an exhaustive Morse function h : M → R such that: (i) h | r is projection,(ii) h | νr has just one critical point, namely a global minimum in ∂νr , (iii) h − ([ t, ∞ )) ∩ ( νr, ∂νr ) ≈ [ t, ∞ ) × ( D n , S n − ) for each t ≥ , and (iv) each j ∈ Z ≥ is a regular value of h .Proof. By Whitney’s embedding theorem, we may assume M ⊂ R n +3 is a submanifold that isembedded as a closed subspace. As 2 n + 3 >
3, we may assume, by an ambient isotopy of R n +3 ,that r is straight in R n +3 . Next, ambiently untwist νr while fixing r . Define h ( x ) := k x − p k + c for an appropriate point p ∈ R n +3 and c ∈ R [Mil69, p. 36]. (cid:3) Let h : M → R be a Morse function given by Lemma 5.1. For each j ∈ Z ≥ , define M j := h − ([ j, ∞ )) and K j := h − (( −∞ , j ]), both oriented as codimension-0 submanifolds of M (see Fig-ure 5.1). The K j provide a compact exhaustion of M . For all sufficiently large j , the boundary B j ∂ν rZ j K j M j h ¯ ( j ) Figure 5.1.
Manifold M n +1 where the Morse function h is depicted as height. The( n − ∂ b Z j = ∂B j is depicted as two dots.of M (compact by hypothesis) is contained in the interior of K j ; without loss of generality, weassume this holds for all j ∈ Z ≥ (shrink r towards infinity if necessary). So, for all j ∈ Z ≥ , h − ( j ) = K j ∩ M j is a finite disjoint union of closed, connected n -manifolds. Let Z j be the compo-nent of h − ( j ) that meets νr , and let b Z j := Z j − Int νr , both oriented as codimension-0 submanifoldsof ∂K j . The ( n − ∂ b Z j is given the boundary orientation. Define B j := ∂νr ∩ K j ≈ D n oriented as a codimension-0 submanifold of ∂νr . Observe that ∂ b Z j = ∂B j as oriented ( n − XTREME NONUNIQUENESS OF END-SUM 17
For each j ∈ Z ≥ , we define the following (see Figure 5.2). c M j := M j − Int νrF j := νr ∩ M j ≈ [ j, ∞ ) × D n b F j := ∂νr ∩ M j ≈ [ j, ∞ ) × S n − ∆ j := νr ∩ Z j ≈ D n The fundamental class h ∂ b Z j i is our preferred generator of H n − (cid:16) b F j (cid:17) . By Universal Coefficients, ∂ M j Z j ɵɵ F j ɵ M j Z j F j Z j ɵ ∆ j Figure 5.2.
Manifold M j and some relevant submanifolds.its dual h ∂ b Z j i ∗ is our preferred generator ofHom Z (cid:16) H n − (cid:16) b F j (cid:17) , R (cid:17) ∼ = e H n − (cid:16) b F j ; R (cid:17) ∼ = R where the latter isomorphism sends our preferred generator to 1 ∈ R . In the direct system e H n − (cid:16) b F j ; R (cid:17) , j ∈ Z ≥ , each morphism is an isomorphism carrying one preferred generator toanother. Therefore, the direct limit e H n − e ( ∂νr ; R ) ∼ = lim −→ e H n − (cid:16) b F j ; R (cid:17) ∼ = R has a preferred generator h ∂ b Z j i ∗ e that is represented by each h ∂ b Z j i ∗ . In the proof of Theorem 5.4below, we use γ M to denote h ∂ b Z j i ∗ e .The inclusion ι j : (cid:16) b Z j , ∂ b Z j (cid:17) → (cid:16) c M j , b F j (cid:17) induces the following diagram, where the rows are thelong exact sequences for pairs. / / e H n − (cid:16) b F j ; R (cid:17) (cid:31) (cid:127) δ j / / ι ∗ j ∼ = (cid:15) (cid:15) H n (cid:16) c M j , b F j ; R (cid:17) ι ∗ j (cid:15) (cid:15) / / / / e H n − (cid:16) ∂ b Z j ; R (cid:17) δ ′ j ∼ = / / H n (cid:16) b Z j , ∂ b Z j ; R (cid:17) / / (5.1) Notational mnemonic: intuitively b X is a “nicely punctured” copy of X . The diagram is commutative by naturality of the coboundary map. As δ ′ j and the left ι ∗ j areisomorphisms, δ j is injective. We have the diagram H n − (cid:16) b F j ; R (cid:17) (cid:31) (cid:127) δ j / / H n (cid:16) c M j , b F j ; R (cid:17) H n ( M j , F j ; R ) φ j ∼ = o o ψ j ∼ = / / H n ( M j ; R ) h ∂ c Z j i ∗ ✤ / / δ j (cid:16)h ∂ b Z j i ∗ (cid:17) [ M, r ] ∗ j ✤ o o ✤ / / [ r ] ∗ j (5.2)where φ j is the excision isomorphism, ψ j is the isomorphism from the long exact sequence for thepair, and [ M, r ] ∗ j and [ r ] ∗ j are defined by the diagram. Consider the commutative diagram D whose j th row, j ∈ Z ≥ , equals (5.2). The four vertical maps in D from row j to row j + 1 are inclusioninduced. Passing to the direct limit in D yields e H n − e ( ∂νr ; R ) (cid:31) (cid:127) δ M / / H ne (cid:16) c M , ∂νr ; R (cid:17) H ne ( M, νr ; R ) φ M ∼ = o o ψ M ∼ = / / e H ne ( M ; R ) h ∂ b Z j i ∗ e ✤ / / δ M (cid:16)h ∂ b Z j i ∗ e (cid:17) [ M, r ] ∗ e ✤ o o ✤ / / [ r ] ∗ e (5.3)where δ M is injective, c M := M − Int νr , φ M is the excision isomorphism, and ψ M is the isomor-phism from the long exact sequence for the pair. The relative and absolute ray-fundamental classes[ M, r ] ∗ e ∈ H ne ( M, νr ; R ) and [ r ] ∗ e ∈ e H ne ( M ; R ) are defined by (5.3). Remarks 5.2. (1) Let D e be the diagram D augmented by the direct limit row (5.3) together with the canonicalmaps in each column from the terms in the direct system to the direct limit. The diagram D e iscommutative and shows immediately that each [ M, r ] ∗ j and [ r ] ∗ j represent (respectively) [ M, r ] ∗ e and [ r ] ∗ e . This observation holds without any additional assumptions on D (such as surjectivityof the vertical maps in the last column).(2) The ray-fundamental classes are well-defined, up to isomorphism, independent of the choice ofregular neighborhood νr by uniqueness of such neighborhoods. They are also independent of theMorse function h satisfying Lemma 5.1. To see this, let h ′ be another such Morse function anddistinguish corresponding submanifolds of M by primes. As our Morse functions are exhaustive,each M j contains M ′ k for all sufficiently large k , and conversely. It follows that H ∗ e ( M ; R ) ∼ = lim −→ H ∗ ( M j ; R ) ∼ = lim −→ H ∗ ( M ′ j ; R )and the latter of these isomorphisms carries the absolute ray-fundamental classes to one another.A similar argument applies to the relative case.(3) If r ⊂ M is neatly embedded, then we define the ray-fundamental classes [ M, r ] ∗ e and [ r ] ∗ e asfollows. As in Figure 5.3 (left), let τ r ⊂ M be a smooth closed tubular neighborhood of r in M (see Section 2 for our conventions on tubular neighborhoods). Let C be the boundary Figure 5.3.
Manifold M containing a neatly embedded ray r and a smooth closedtubular neighborhood τ r (left), and M ′ = M ∪ (closed collar) (right).component of M containing ∂r . Let M ′ equal M union a closed collar on C . The closed collar XTREME NONUNIQUENESS OF END-SUM 19 contains an ( n + 1)-disk B such that B ∪ τ r is a smooth closed regular neighborhood νr of r contained in the interior of M ′ as in Figure 5.3 (right). Evidently H ∗ e ( M ′ , νr ; R ) ∼ = H ∗ e ( M, τ r ; R ) H ∗ e ( M ′ ; R ) ∼ = H ∗ e ( M ; R )We define the ray-fundamental classes [ M, r ] ∗ e and [ r ] ∗ e for r ⊂ M to be the images under theseisomorphisms of the ray-fundamental classes for r ⊂ M ′ .(4) The existence of the nonzero absolute ray-fundamental class [ r ] ∗ e implies that if M is a smooth,oriented, connected, noncompact manifold of dimension n + 1 ≥ e H ∗ e ( M ; R ) is nonzero. In particular, R injects into H ne ( M ; R ). For such a manifold M , H ne ( M ; R ) may indeed be the only nonzero cohomology group in e H ∗ e ( M ; R ). Consider thebasic example of euclidean space. e H ∗ e (cid:0) R n +1 ; R (cid:1) ∼ = e H ∗ ( S n ; R ) ∼ = H n ( S n ; R ) ∼ = R If M has noncompact boundary, then e H ∗ e ( M ; R ) may vanish. Consider the basic example ofclosed upper half-space R n +1+ which is proper homotopy equivalent to a ray. e H ∗ e (cid:0) R n +1+ ; R (cid:1) ∼ = e H ∗ e ([0 , ∞ ); R ) ∼ = 0 Example 5.3.
We will compute the absolute ray-fundamental class determined by a neat straightray in a stringer. Fix a smooth, closed, connected, oriented n -manifold Z where n ≥
1. Let ∆ ⊂ Z be a smoothly embedded n -disk, and let z ∈ Int ∆. So, r = [0 , ∞ ) ×{ z } is a neat straight ray in thestringer [0 , ∞ ) × Z with smooth closed tubular neighborhood F = [0 , ∞ ) × ∆. We let M j = [ j, ∞ ) × Z and reuse the notation from Figure 5.2 and thereafter.We have the following diagram in integer homology. H n − (cid:16) ∂ b Z (cid:17) H n (cid:16) b Z, ∂ b Z (cid:17) ∂ ∗ ∼ = o o exc. ∼ = / / H n ( Z, ∆) H n ( Z ) l.e. ∼ = o o h ∂ b Z i h b Z, ∂ b Z i ✤ o o ✤ / / [ Z, ∆] [ Z ] ✤ o o (5.4)Each of these groups is a copy of Z . We claim that the preferred generators map as shown. It iswell-known that ∂ ∗ is an isomorphism here. Seemingly less well-known is the more explicit fact that ∂ ∗ (cid:16)h b Z, ∂ b Z i(cid:17) = h ∂ b Z i for fundamental classes and the outward normal first orientation convention;a proof appears in Kreck [Kre13, Thm. 8.1]. A moment of reflection reveals that the second andthird isomorphisms in (5.4) send the preferred generators to the same generator, denoted [ Z, ∆], of H n ( Z, ∆) as claimed.The Universal Coefficients Theorem now yields the following since all relevant Ext groups vanish. H n − (cid:16) ∂ b Z (cid:17) δ ∼ = / / H n (cid:16) b Z, ∂ b Z (cid:17) H n ( Z, ∆) exc. ∼ = o o l.e. ∼ = / / H n ( Z ) h ∂ b Z i ∗ ✤ / / h b Z, ∂ b Z i ∗ [ Z, ∆] ∗ ✤ o o ✤ / / [ Z ] ∗ (5.5)Diagram (5.5) is canonically isomorphic to row j = 0 in (5.2) by the obvious strong deformationretractions. The latter diagram is canonically isomorphic to the direct limit diagram (5.3) sinceevery vertical map in D is an isomorphism. Making the canonical identifications H ne ([0 , ∞ ) × Z ; R ) ∼ = H n ([0 , ∞ ) × Z ; R ) ∼ = H n ( Z ; R ) ∼ = R with the last given by [ Z ] ∗
1, we have that [ r ] ∗ e = 1 ∈ R . This completes our example. Let (
M, r ) and (
N, s ) be end-sum pairs (see § M and N have the same dimension n + 1 ≥ νr ⊂ Int M and νs ⊂ Int N besmooth closed regular neighborhoods of r and s respectively. Let H ⊂ ( M, r ) ♮ ( N, s ) denote theimage of ∂νr (which also equals the image of ∂νs ). Let u ⊂ H be an unknotted ray, and let νu bea smooth closed regular neighborhood of u in the interior of S := ( M, r ) ♮ ( N, s ). Theorem 5.4 (H. King) . There are isomorphisms of graded R -algebras H ∗ e ( S, νu ; R ) ∼ = ( H ∗ e ( M, νr ; R ) ⊕ H ∗ e ( N, νs ; R )) / (cid:10)(cid:0) [ M, r ] ∗ e , − [ N, s ] ∗ e (cid:1)(cid:11)e H ∗ e ( S ; R ) ∼ = ( e H ∗ e ( M ; R ) ⊕ e H ∗ e ( N ; R )) / (cid:10)(cid:0) [ r ] ∗ e , − [ s ] ∗ e (cid:1)(cid:11) where (cid:10)(cid:0) [ M, r ] ∗ e , − [ N, s ] ∗ e (cid:1)(cid:11) and (cid:10)(cid:0) [ r ] ∗ e , − [ s ] ∗ e (cid:1)(cid:11) are homogeneous principal ideals of degree n .Proof. Recall that S is obtained from the disjoint union of c M := M − Int νr and b N := N − Int νs by identifying ∂νr and ∂νs using an orientation reversing diffeomorphism. We have inclusions i M : (cid:16) c M , ∂νr (cid:17) ֒ → ( S, H ) i N : (cid:16) b N , ∂νs (cid:17) ֒ → ( S, H )We orient H so that i M | : ∂νr → H is an orientation preserving diffeomorphism; it follows that i N | : ∂νs → H is an orientation reversing diffeomorphism. Let ω ∈ H n − e ( H ; R ) ∼ = R be thepreferred generator for this orientation. Hence, the following hold for our preferred generators. e H n − e ( H ; R ) i M | ∗ ∼ = / / e H n − e ( ∂νr ; R ) e H n − e ( H ; R ) i N | ∗ ∼ = / / e H n − e ( ∂νs ; R ) ω ✤ / / γ M ω ✤ / / − γ N (5.6)The long exact sequences for the pairs give isomorphisms ψ M : H ∗ e ( M, νr ; R ) ∼ = −→ e H ∗ e ( M ; R ) ψ N : H ∗ e ( N, νs ; R ) ∼ = −→ e H ∗ e ( N ; R ) ψ S : H ∗ e ( S, νu ; R ) ∼ = −→ e H ∗ e ( S ; R )Equation (5.3) shows that ψ M (cid:0) [ M, r ] ∗ e (cid:1) = [ r ] ∗ e and ψ N (cid:0) [ N, s ] ∗ e (cid:1) = [ s ] ∗ e . So, the reduced cohomologyresult will follow immediately from the relative cohomology result. Further, the isomorphism ψ S shows that it suffices to prove the following e H ∗ e ( S ; R ) ∼ = ( H ∗ e ( M, νr ; R ) ⊕ H ∗ e ( N, νs ; R )) / (cid:10)(cid:0) [ M, r ] ∗ e , − [ N, s ] ∗ e (cid:1)(cid:11) Consider the long exact sequence for the pair(5.7) −→ e H k − e ( H ; R ) δ −→ H ke ( S, H ; R ) j ∗ −→ e H ke ( S ; R ) −→ e H ke ( H ; R ) −→ We claim that j ∗ is an isomorphism unless k = n , in which case j ∗ is surjective. As e H ke ( H ; R ) = 0for k = n −
1, the claim is clear except for surjectivity of j ∗ for k = n −
1. By exactness, it sufficesto prove that δ S : e H n − e ( H ; R ) → H ne ( S, H ; R )is injective. The inclusions i M and i N together with naturality of the coboundary map imply thefollowing(5.8) i ∗ M ◦ δ S = δ M ◦ i M | ∗ i ∗ N ◦ δ S = δ N ◦ i N | ∗ Either of these equations imply that δ S is injective since both δ M and δ N are injective (see (5.3))and both i M | ∗ and i N | ∗ are isomorphims. The claim is proved. XTREME NONUNIQUENESS OF END-SUM 21
The claim implies that e H ∗ e ( S ; R ) is isomorphic to the quotient of H ∗ e ( S, H ; R ) by the kernel of j ∗ . By exactness of (5.7), this kernel is generated by δ S ( ω ).By Corollary 2.8, the inclusions i M and i N induce the isomorphism h : H ∗ e ( S, H ; R ) ∼ = −→ H ∗ e (cid:16) c M , ∂νr ; R (cid:17) ⊕ H ∗ e (cid:16) b N, ∂νs ; R (cid:17) where h ( α ) = ( i ∗ M ( α ) , i ∗ N ( α )). We also have the excision isomorphisms φ M : H ∗ e ( M, νr ; R ) ∼ = −→ H ∗ e (cid:16) c M , ∂νr ; R (cid:17) φ N : H ∗ e ( N, νs ; R ) ∼ = −→ H ∗ e (cid:16) b N, ∂νs ; R (cid:17) Therefore, the theorem will follow provided we show that the image of δ S ( ω ) under the isomorphism h equals the image of (cid:0) [ M, r ] ∗ e , − [ N, s ] ∗ e (cid:1) under the isomorphism φ M ⊕ φ N . We have h ( δ S ( ω )) = ( i ∗ M ( δ S ( ω )) , i ∗ N ( δ S ( ω )))= ( δ M ◦ i M | ∗ ( ω ) , δ N ◦ i N | ∗ ( ω ))= ( δ M ( γ M ) , δ N ( − γ N ))= (cid:0) φ M (cid:0) [ M, r ] ∗ e (cid:1) , φ N (cid:0) − [ N, s ] ∗ e (cid:1)(cid:1) where we used (5.8), (5.6), and (5.3). This completes our proof of the theorem. (cid:3) Remarks 5.5. (1) Recall that the number of ends of a space Y equals the rank of H e ( Y ; R ) where R is a principalideal domain [Geo08, Prop. 13.4.11]. Thus, the reduced end-cohomology result in Theorem 5.4implies that the number of ends (finite or infinite) of the binary end-sum S equals the sum ofthe numbers of ends of M and N minus one. In particular, if M and N are one-ended, then sois S .(2) The results in this section likely hold in the piecewise-linear and topological categories and alsofor nonorientable manifolds with R = Z . In this paper, we will not need these generalizations.6. Ray-Fundamental Classes
Ray-Fundamental Classes in Ladders.
Fix X and Y to be closed, connected, oriented n -manifolds where n ≥
2. Let L := L ( X, Y ) be the ladder manifold based on X and Y as definedin Section 3. Let r be a ray in L emanating from x ∈ X and intersecting each S j transverselyas in Figure 6.1. Let F be a smooth closed tubular neighborhood of r with a parameterization X X X Y Y Y S S S x x x Figure 6.1.
Ray r in ladder manifold L = L ( X, Y ). τ : [0 , ∞ ) × D n → F such that r = τ ([0 , ∞ ) ×
0) and, for each j , F ∩ S j = τ ( P j × D n ), where P j is the (finite) set of preimages of points where r intersects S j . If p ∈ P j , then let D p denote τ ( p × D n ),and let D = τ (0 × D n ) = F ∩ X . Viewing r as a properly embedded oriented submanifold of L we may consider the Z -intersection numbers ε Z ( r, S j ) (see [RS72, p. 68] or [GP74, p. 112]). Underthis convention, p ∈ P j contributes +1 to ε Z ( r, S j ) if r passes from the X -side of L to the Y -sideon a small neighborhood of p and it contributes − D the orientation induced by X and slide that orientation along the product structure of F to orienteach D p . Then p ∈ P j contributes +1 to ε Z ( r, S j ) if D p ֒ → S j is orientation preserving and − D p ֒ → S j is orientation reversing.Let b L = L − F ◦ and b F = F − F ◦ = τ ([0 , ∞ ) × S n − ), where F ◦ denotes the topological interior of F as a subspace of L . Our first goal is to understand the coboundary map δ : H n − (cid:16) b F (cid:17) → H n (cid:16)b L , b F (cid:17) .To accomplish this, we use the familiar diagram(6.1) H n − (cid:16) b F (cid:17) H n (cid:16)b L , b F (cid:17) ∂ ∗ o o exc. ∼ = / / H n ( L , F ) H n ( L ) l.e. ∼ = o o and examine the boundary map ∂ ∗ .By calculations in Section 3, the fundamental classes [ X ], [ Y ], and [ S j ], j ∈ Z ≥ , form a freebasis for H n ( L ). By the long exact sequence for ( L , F ) and excision, H n (cid:16)b L , b F (cid:17) has a free basisconsisting of the relative fundamental classes h b X , ∂ b X i of b X := X − Int D and h b S j , ∂ b S j i ofthe b S j := S j − ∪ p ∈ P j Int D p together with the fundamental class [ Y ] of Y . The ( n − ∂ b X is given the boundary orientation; the fundamental class h ∂ b X i is our preferred generatorof H n − (cid:16) b F (cid:17) ∼ = Z . (This agrees with our orientation conventions in Section 5 where ∂ b Z playedthe role of ∂ b X .) The orientation conventions established earlier in the current section imply that[ ∂D p ] = − h ∂ b X i in H n − (cid:16) b F (cid:17) .Now, ∂ ∗ : H n (cid:16)b L , b F (cid:17) → H n − (cid:16) b F (cid:17) is determined by its action on this basis. We have ∂ ∗ ([ Y ]) = 0and ∂ ∗ (cid:16)h b X , ∂ b X i(cid:17) = h ∂ b X i (see Example 5.3 above). For each j ∈ Z ≥ , we have ∂ ∗ (cid:16)h b S j , ∂ b S j i(cid:17) = h ∂ b S j i = X p ∈ P j − [ ∂D p ] = ε Z ( r, S j ) · h ∂ b X i We now return to the pertinent coboundary map δ where we will employ the following diagram. H n − (cid:16) b F (cid:17) δ / / h ′ ∼ = (cid:15) (cid:15) H n (cid:16)b L , b F (cid:17) h ∼ = (cid:15) (cid:15) Hom Z (cid:16) H n − (cid:16) b F (cid:17) , Z (cid:17) ∂ ∗ / / Hom Z (cid:16) H n (cid:16)b L , b F (cid:17) , Z (cid:17) (6.2)Here h ′ and h are the surjective homomorphisms provided by Universal Coefficients, and commuta-tivity is verified in [Hat02, p. 200]. Injectivity of h ′ and h requires some specifics of the situationat hand, but both are immediate by Universal Coefficients when H n − (cid:16) b F (cid:17) and H n − (cid:16)b L , b F (cid:17) aretorsion free. That is clearly the case for H n − (cid:16) b F (cid:17) . Next, excision and the long exact sequence for( L , F ) imply that H n − (cid:16)b L , b F (cid:17) ∼ = H n − ( L ). By calculations in Section 3, the latter is isomorphicto H n − ( X ) ⊕ H n − ( Y ). By Poincar´e duality, H n − ( X ) ∼ = H ( X ) and similarly for Y . By XTREME NONUNIQUENESS OF END-SUM 23
Universal Coefficients, degree one Z -cohomology is always torsion-free and our assertion follows.The Universal Coefficients Theorem gives the following diagram dual to (6.1) since all relevantExt groups vanish.(6.3) H n − (cid:16) b F (cid:17) δ / / H n (cid:16)b L , b F (cid:17) H n ( L , F ) exc. ∼ = o o l.e. ∼ = / / H n ( L )As in Section 3, we identify H n ( L ) with Z ⊕ Z [[ x ]] ⊕ Z where the dual fundamental class [ X ] ∗ corresponds to the positive generator of the first summand, [ S j ] ∗ corresponds to the monomial x j ,and [ Y ] ∗ corresponds to the positive generator in the third summand. We also identify H n − (cid:16) b F (cid:17) with Z by h ∂ b X i ∗ ↔
1. Thus, the composite map H n − (cid:16) b F (cid:17) → H n ( L ) may be written as µ : Z → Z ⊕ Z [[ x ]] ⊕ Z Define ε i = ε Z ( r, S i ). With these conventions, diagram (6.2) and our description of ∂ ∗ imply that µ (1) = (cid:0) , P ∞ i =0 ε i x i , (cid:1) .By the end of Section 2.3, we have the canonical surjection q : H n ( L ) ։ H ne ( L ) ∼ = ( Z ⊕ Z [[ x ]] ⊕ Z ) /K By Remarks 5.2(1), the following is now immediate.
Proposition 6.1.
Let r be a ray in L emanating from x ∈ X and intersecting each S i transversely,and let ε i = ε Z ( r, S i ) . Then, the absolute ray-fundamental class determined by r is [ r ] ∗ e = q (cid:0) , P ∞ i =0 ε i x i , (cid:1) y ∈ ( Z ⊕ Z [[ x ]] ⊕ Z ) /K ∼ = H ne ( L )Next, we prove a simple realization theorem whose proof is reminiscent of a Mazur-Eilenberginfinite swindle. Proposition 6.2. If α = P ∞ i =0 a i x i ∈ Z [[ x ]] , then there exists a ray r in L emanating from x ∈ X such that [ r ] ∗ e = J (1 , α, K .Proof. Recall the definition of L [ j,k ] ⊆ L in Section 3. Let x = (0 , x ) ∈ X be our usual basepoint,and for each i ∈ Z > choose x i = ( i + 1 / , x ) ∈ L [ i,i +1] as in Figure 6.1. Let r : [0 , → L [0 , be asmooth oriented path beginning at x , ending at x , and circling through the rungs of L [0 , so as torealize intersection numbers ε Z ( r , S ) = a and ε Z ( r , S ) = − a . With respect to Figure 6.1, thispath will circle counterclockwise if a > a <
0; if a = 0, then it is a vertical arc.Similarly, let r : [1 , → L [1 , be a path beginning at x , ending at x , and circling through therungs of L [1 , so as to realize intersection numbers ε Z ( r , S ) = a + a and ε Z ( r , S ) = − ( a + a ).Notice that ε Z ( r ∪ r , S ) = a and ε Z ( r ∪ r , S ) = − a + ( a + a ) = a .In general, choose r k : [ k, k + 1] → L [ k,k +2] beginning at x k , ending at x k +1 , and realizingintersection numbers ε Z ( r k , S k ) = P ki =0 a i and ε Z ( r k , S k +1 ) = − P ki =0 a i . Then, let r : [0 , ∞ ) → L be the union of these paths, adjusted, if necessary, to make it a smooth embedding. Choosing a nicesmooth closed tubular neighborhood of r and applying the proof of Proposition 6.1 complete theproof. (cid:3) Ray-Fundamental Classes in Surgered Stringers.
The above propositions for laddershave simpler analogues for surgered stringers. Fix a closed oriented n -manifold X where n ≥
2. Let S := S ( X ) be the surgered stringer based on X as defined in Section 3. Let r be a ray in S emanatingfrom x ∈ X and intersecting each S j transversely. Recall the definition of S [ j,k ] from Section 3.Working as we did in ladder manifolds, we consider the Z -intersection numbers ε Z ( r, S j ). A point of r ∩ S j at which r exits S [ j,j +1 / contributes +1, and a point where r enters S [ j,j +1 / contributes − F be a smooth closed tubular neighborhood of r chosen so there exists a parame-terization τ : [0 , ∞ ) × D n → F with r = τ ([0 , ∞ ) ×
0) and, for each j , F ∩ S j = τ ( P j × D n ), where P j is the set of preimages of r ∩ S j . Let D = F ∩ X and for each p ∈ P j let D p = τ ( p × D n ) ⊆ S j .Let b S = S − F ◦ ; b F = F − F ◦ ; b X := X − Int D ; and b S j := S j − ∪ p ∈ P j Int D p . Using calculationsfrom Section 3, the long exact sequence for ( S , F ), excision, and notation as above for ladders, therelative fundamental classes h b X , ∂ b X i and h b S j , ∂ b S j i , j ∈ Z ≥ , form a free basis for H n (cid:16)b S , b F (cid:17) .Our preferred generator of H n − (cid:16) b F (cid:17) ∼ = Z is h ∂ b X i , and [ ∂D p ] = − h ∂ b X i in H n − (cid:16) b F (cid:17) .The map ∂ ∗ : H n (cid:16)b S , b F (cid:17) → H n − (cid:16) b F (cid:17) is given by h b X , ∂ b X i h ∂ b X i and h b S j , ∂ b S j i ε Z ( r, S j ) · h ∂ b X i . The Universal Coefficients Theorem gives the following diagram.(6.4) H n − (cid:16) b F (cid:17) δ / / H n (cid:16)b S , b F (cid:17) H n ( S , F ) exc. ∼ = o o l.e. ∼ = / / H n ( S )Identify H n ( S ) with Z ⊕ Z [[ x ]] by [ X ] ∗ ↔ (1 ,
0) and [ S j ] ∗ ↔ (0 , x j ). Identify H n − (cid:16) b F (cid:17) with Z by h ∂ b X i ∗ ↔
1. The composite map H n − (cid:16) b F (cid:17) → H n ( S ) is now written as µ : Z → Z ⊕ Z [[ x ]]. Define ε i = ε Z ( r, S i ). Then, µ (1) = (cid:0) , P ∞ i =0 ε i x i (cid:1) . We have the canonical surjection q : H n ( S ) ։ H ne ( S ) ∼ = Z ⊕ Z [[ x ]] / Z [ x ]Our work yields the following. Proposition 6.3.
Let r be a ray in S emanating from x ∈ X and intersecting each S i transversely,and let ε i = ε Z ( r, S i ) . Then, the absolute ray-fundamental class determined by r is [ r ] ∗ e = (cid:0) , q P ∞ i =0 ε i x i y (cid:1) ∈ Z ⊕ Z [[ x ]] / Z [ x ] ∼ = H ne ( S ) Furthermore, if α = P ∞ i =0 a i x i ∈ Z [[ x ]] , then there exists a ray r in S emanating from x ∈ X suchthat [ r ] ∗ e = (1 , J α K ) . The proof of the realization result in this proposition is simpler than that for ladder manifolds.No “swindle” is needed here. 7.
Proof of the Main Theorem
We first prove the Main Theorem using specific one-ended, open 4-manifolds. Then, we describevarious ways of adapting the proof to other one-ended, open manifolds. Let T k = × k S be the k -torus. Define M = S (cid:0) T (cid:1) ∪ ∂ ( T × D ) N = ([0 , ∞ ) × ( S × S )) ∪ ∂ ( S × D )So, M is the surgered stringer based on T capped with T × D , and N is the stringer based on S × S capped with S × D as in Figure 7.1.Let α = P ∞ i =0 a i x i ∈ Z [[ x ]]. By Proposition 6.3, there is a ray r ⊂ Int M such that[ r ] ∗ e = (1 , J α K ) ∈ Z ⊕ Z [[ x ]] / Z [ x ] ∼ = e H e ( M )As N is one-ended, collared at infinity, and has dimension at least four, it contains a unique ray upto ambient isotopy. So, let s ⊂ Int N be a straight ray as in Figure 7.1. By Example 5.3, we have[ s ] ∗ e = 1 ∈ Z ∼ = e H e ( N ) XTREME NONUNIQUENESS OF END-SUM 25
XM YN sr
Figure 7.1.
Open manifolds M and N with rays r ⊂ Int M and s ⊂ Int N .Let S = ( M, r ) ♮ ( N, s ). By Section 3 and Theorem 5.4, we have e H ∗ e ( S ) ∼ = A where e H ke ( S ) ∼ = A k := (( Z ⊕ Z [[ x ]] / Z [ x ]) ⊕ Z ) /I if k = 3, Z ⊕ ⊕ Z if k = 2, Z ⊕ Z [[ τ ]] / Z [ τ ] ⊕ Z if k = 1,0 ⊕ ⊕ I is the homogeneous ideal of degree 3 generated by ((1 , J α K ) , − C be an arbitrary graded Z -algebra (possibly non-unital). We assign a sequence to C bythe following procedure:(1) Let J ≤ C be the subgroup of elements in C that are sent to 0 by every element of thedual Hom Z ( C , Z ) of C . If J is not a two-sided ideal of C , then return the empty sequence() and end the procedure. Otherwise, J is a two-sided homogeneous ideal of C .(2) Let D = C /J , which is a graded Z -algebra.(3) In D , let U ≤ D be the subgroup generated by all products of three elements of degree one.(4) In D , let V ≤ D be the subgroup generated by elements that are a product of a degree oneelement and a degree two element but are not a product of three degree one elements.(5) If V is not infinite cyclic, then return the empty sequence () and end the procedure. Other-wise, let v be either generator of V .(6) Let π : D → D /U be the canonical homomorphism.(7) Return the height of π ( v ) in D /U and end the procedure.The notion of the height of an element in an abelian group is reviewed below in Appendix A.3. Let h ( C ) denote the sequence (empty or infinite in length) determined by the procedure. Proposition 7.1. If C and C ′ are isomorphic as graded Z -algebras, then h ( C ) = h ( C ′ ) . Appliedto the specific case of the end-cohomology algebra of the end-sum S described above, this yields h (cid:16) e H ∗ e ( S ) (cid:17) = h ( A ) which equals the height of J α K ∈ Z [[ x ]] / Z [ x ] .Proof. Let f : C → C ′ be a graded Z -algebra isomorphism. Note that f respects gradings, products,sums, and the Z -module structure. In particular, f restricts to an isomorphism f | : C → C ′ , and thelatter induces the isomorphism of dual modules Hom Z ( C ′ , Z ) → Hom Z ( C , Z ) given by ψ ψ ◦ f | .It follows that f ( J ) = J ′ . If J is not a two-sided ideal of C , then J ′ is not a two-sided ideal of C ′ and we have h ( C ) = () = h ( C ′ ). Otherwise, both J and J ′ are two-sided homogeneous idealsand f induces the graded Z -algebra isomorphism F : D → D ′ given by F ( x + J ) = f ( x ) + J ′ . As F respects gradings, products, sums, and the Z -module structure, we get that F restricts toisomorphisms D → D ′ , U → U ′ , and V → V ′ . So, if V is not infinite cyclic, then neither is V ′ and we have h ( C ) = () = h ( C ′ ). Otherwise, V and V ′ are both infinite cyclic and F ( v ) = ± v ′ . Theisomorphism F induces the isomorphism G : D /U → D ′ /U ′ given by G ( x + U ) = F ( x ) + U ′ . So,the following diagram commutes D F |∼ = / / π (cid:15) (cid:15) D ′ π ′ (cid:15) (cid:15) D /U G ∼ = / / D ′ /U ′ (7.1)In particular, G sends π ( v )
7→ ± π ′ ( v ′ ). The sequence h ( C ) is the height of π ( v ) ∈ D /U , and thesequence h ( C ′ ) is the height of π ′ ( v ′ ) ∈ D ′ /U ′ . These sequences are equal since height is invariantunder isomorphism (see Corollary A.19 in the appendix below) and sign change. This proves thefirst claim in the proposition and implies that h (cid:16) e H ∗ e ( S ) (cid:17) = h ( A ). It remains to show that h ( A )equals the height of J α K ∈ Z [[ x ]] / Z [ x ].Applying the procedure to A , Corollary A.2 implies that J = 0 ⊕ Z [[ τ ]] / Z [ τ ] ⊕ ≤ A Recall the product structure of A given in Section 3. It implies that the product (in either order)of any element of J with any element of A vanishes. So, J is a two-sided homogeneous ideal of A .Taking the quotient of A by J , we obtain the algebra D where D k := (( Z ⊕ Z [[ x ]] / Z [ x ]) ⊕ Z ) /I if k = 3, Z ⊕ ⊕ Z if k = 2, Z ⊕ ⊕ Z if k = 1,0 ⊕ ⊕ U ≤ D and V ≤ D as in the procedure. Using the well-known Z -cohomology rings of T and S × S (see Hatcher [Hat02, p. 216]), we have U = (( Z ⊕ ⊕ /I ∼ = Z and V = ((0 ⊕ ⊕ Z ) /I ∼ = Z which are both infinite cyclic. Let v be either generator of V . Let π : D → D /U be the canonicalhomomorphism. Then, h ( A ) equals the height of π ( v ) ∈ D /U .Conceptually, A = D is obtained from Z ⊕ Z [[ x ]] / Z [ x ] by summing with Z and then identify-ing the new Z with an infinite cyclic subgroup of Z ⊕ Z [[ x ]] / Z [ x ]; essentially, this doesn’t changethe group. More precisely, consider the homomorphism η : D → Z ⊕ Z [[ x ]] / Z [ x ] defined by J (( i, J β K ) , j ) K ( i + j, J β K + j J α K ). Noting that J (( i, J β K ) , j ) K = J (( i + j, J β K + j J α K ) , K , we seethat η is an isomorphism of groups. Observe that η ( U ) = Z ⊕
0, and η ( V ) is the infinite cyclicsubgroup generated by (1 , J α K ). We have the commutative diagram D η ∼ = / / π (cid:15) (cid:15) Z ⊕ Z [[ x ]] / Z [ x ] (cid:15) (cid:15) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ D /U E ∼ = / / ( Z ⊕ Z [[ x ]] / Z [ x ]) / ( Z ⊕ ∼ = / / Z [[ x ]] / Z [ x ](7.2) XTREME NONUNIQUENESS OF END-SUM 27 where E is induced by η and the two downward homomorphisms in the triangle are the canonicalprojections which induce the horizontal isomorphism. At the level of elements, we have v ✤ / / ❴ (cid:15) (cid:15) ± (1 , J α K ) ❴ (cid:15) (cid:15) ☞ % % ▲▲▲▲▲▲▲▲▲▲ π ( v ) ✤ / / J ± (1 , J α K ) K ✤ / / ± J α K (7.3)As height is invariant under isomorphism and sign change, the height of π ( v ) ∈ D /U equals theheight of J α K ∈ Z [[ x ]] / Z [ x ] proving the proposition. (cid:3) As there exist uncountably many heights of elements in Z [[ x ]] / Z [ x ] (see Lemma A.17), the MainTheorem is proved. Crucial to our proof was the detection of the specific subgroups U and V inan isomorphically invariant manner. To enable this, we chose manifolds with useful cup productstructures. In the absence of such cup products, results at the end of Section 3 above show thatthese subgroups cannot be so detected.We close this section with a sample of variations of our proof of the Main Theorem. Always, weconsider a pair of one-ended, open m -manifolds M and N .(1) To prove the Main Theorem for each dimension m ≥
5, consider the manifolds M = S (cid:0) S × S m − (cid:1) ∪ ∂ ( S × D m − ) N = ([0 , ∞ ) × ( S × S m − )) ∪ ∂ ( S × D m − )The proof is the same, except we consider the subgroups U and V of D m − ∼ = (( Z ⊕ Z [[ x ]] / Z [ x ]) ⊕ Z ) /I where U = (( Z ⊕ ⊕ /I ∼ = Z is the subgroup generated by elements that are a product of adegree two element and a degree m − m = 5, then this means the product of twodegree two elements), and V = ((0 ⊕ ⊕ Z ) /I ∼ = Z is the subgroup generated by elements thatare a product of a degree one element and a degree m − m = 3, consider closed, oriented surfaces Σ g and Σ g of distinct positive genera g > g (the case g < g may be handled similarly). Consider the manifolds M = S (Σ g ) ∪ ∂ H g N = ([0 , ∞ ) × Σ g ) ∪ ∂ H g where boundaries are capped with handlebodies. As in the main proof, we let S = ( M, r ) ♮ ( N, s )and, by Section 4 and Theorem 5.4, we have D k := (( Z ⊕ Z [[ x ]] / Z [ x ]) ⊕ Z ) /I if k = 2, Z g ⊕ ⊕ Z g if k = 1,0 ⊕ ⊕ J α K = 0. So, products of degree one elements form a rank two subgroupof D ; products in the first factor generate U := (( Z ⊕ ⊕ /I , and products in the third factorgenerate V which is the Z -span of J ((1 , J α K ) , − K .Consider subgroups C ≤ D such that: (1) products of elements of C generate a rank onesubgroup D ≤ D , and (2) the product of each element of C with each element of D lies in D .Note that Z g ⊕ ⊕ C ′ ofmaximal rank. Suppose that C ′ is not contained in Z g ⊕ ⊕
0. Then, using Lemma 4.1, we seethat C ′ meets both Z g ⊕ ⊕ ⊕ ⊕ Z g nontrivially. By Poincar´e duality, elements thatare the product of an element of C ′ and an element of D generate a rank two subgroup of D .This contradicts the defining properties of C ′ . Therefore, C ′ is contained in Z g ⊕ ⊕
0. Among all such subgroups of maximal rank, Z g ⊕ ⊕ Z g ⊕ ⊕ U . The rest of the proof is unchanged.(3) Similarly, one may prove the Main Theorem in each dimension m ≥ m , one mayuse any compact caps to eliminate boundary and the proof is unchanged. Appendix A. Infinitely Generated Abelian Group Theory
We present some relevant results from the theory of infinitely generated abelian groups. Thistheory is subtle, beautiful, and (in our experience) not widely known among topologists. For thatreason, we provide proofs, as elementary as possible, for a few foundational results. These resultsare then applied to prove a few propositions tailored specifically to our needs in this paper. We closethis appendix with a discussion of height in an abelian group.A.1.
Classical Results.
The additive abelian group Z [[ x ]] ∼ = Z × Z × · · · is called the Baer-Specker group . Famously, it is not a free Z -module; we include a proof of this fact below (see alsoSchr¨oer [Sch08]). The additive abelian group Z [ x ] ∼ = Z ⊕ Z ⊕ · · · is, of course, a free Z -module withbasis (cid:8) , x, x , . . . (cid:9) . Throughout this and the next appendix, all maps are Z -module homomorphisms.The dual module of a Z -module M is the Z -module M ∗ := Hom Z ( M, Z ) Fact A.1. If f : Z [[ x ]] → Z vanishes on Z [ x ], then f = 0. Proof.
Fix an integer p >
1. Consider the element α = a p x + a p x + a p x + · · · ∈ Z [[ x ]]where each a i ∈ Z . As f vanishes on Z [ x ], we get f ( α ) = f ( a k p k x k + a k +1 p k +1 x k +1 + · · · ) = p k f ( a k p x k + a k +1 p x k +1 + · · · )and so p k divides f ( α ) for each integer k >
0. Hence, f ( α ) = 0.Next, fix coprime integers p, q >
1. Let γ = P c i x i be an arbitrary element of Z [[ x ]]. We wishto write γ = α + β where α = P a i p i x i and β = P b i q i x i . For each i ≥
0, we seek integers a i and b i such that c i = a i p i + b i q i , which is always possible since p i and q i are coprime. Now f ( γ ) = f ( α ) + f ( β ) = 0 + 0 = 0. (cid:3) Corollary A.2. ( Z [[ x ]] / Z [ x ]) ∗ = { } .Proof. Let f : Z [[ x ]] / Z [ x ] → Z , and let π : Z [[ x ]] ։ Z [[ x ]] / Z [ x ] be the canonical surjection. So, f ◦ π vanishes on Z [ x ]. By Fact A.1, f ◦ π = 0. As π is surjective, f = 0. (cid:3) Recall that Hom Z ( − , Z ) distributes over finite direct sums. Corollary A.3.
The Z -module Z [[ x ]] / Z [ x ] is uncountable and torsion-free, but does not split off Z as a direct summand and is not a free Z -module.Proof. The first two claims are simple exercises. For the last two claims, otherwise ( Z [[ x ]] / Z [ x ]) ∗ would be nonzero, contradicting Corollary A.2. (cid:3) Intuitively, Z [[ x ]] / Z [ x ] is flexible and large regarding injective maps of free Z -modules into it, butis rigid regarding maps to free Z -modules. Corollary A.4. ( Z ⊕ Z [[ x ]] / Z [ x ] ⊕ Z ) ∗ ∼ = Z .Proof. Immediate by Corollary A.2. (cid:3)
XTREME NONUNIQUENESS OF END-SUM 29
Fact A.1 also implies that two maps Z [[ x ]] → Z that agree on Z [ x ] must be equal (consider theirdifference). Combining this observation with projections, we see that two maps Z [[ x ]] → Z [[ x ]] thatagree on Z [ x ] must be equal (see also Fuchs [Fuc73, Lemma 94.1]). We mention that Z [ x ] ∗ ∼ = Z [[ x ]]since Z [ x ] is free with Z -basis (cid:8) , x, x , . . . (cid:9) . The isomorphism is f P i ≥ f ( x i ) x i . For complete-ness, we prove the “dual” fact that Z [[ x ]] ∗ ∼ = Z [ x ].The following fact is long known. Fact A.5. If f : Z [[ x ]] → Z , then f ( x k ) = 0 for cofinitely many k . Proof.
Consider an element α = a ! + a ! x + a ! x + · · · ∈ Z [[ x ]]for integers 0 < a < a < a < · · · to be determined. For each k , we have a tail of α denoted a k ! β k = a k ! x k + a k +1 ! x k +1 + · · · ∈ Z [[ x ]]where β k = a k ! a k ! x k + a k +1 ! a k ! x k +1 + · · · ∈ Z [[ x ]]Notice that β k lies in Z [[ x ]] since the a j ’s are increasing. We have f ( α ) = a ! f (1) + a ! f ( x ) + · · · + a k ! f ( x k ) + a k +1 ! f ( β k +1 ) ∈ Z Hence a k +1 ! f ( β k +1 ) = f ( α ) − a ! f (1) − a ! f ( x ) − · · · − a k ! f ( x k )By the triangle inequality a k +1 ! | f ( β k +1 ) | ≤ | f ( α ) | + a ! | f (1) | + a ! | f ( x ) | + · · · + a k ! (cid:12)(cid:12) f ( x k ) (cid:12)(cid:12) Therefore | f ( β k +1 ) | ≤ | f ( α ) | a k +1 ! + a ! | f (1) | + a ! | f ( x ) | + · · · + a k ! (cid:12)(cid:12) f ( x k ) (cid:12)(cid:12) a k +1 !The first term on the right side tends to zero as k → ∞ , and we may inductively choose the positiveintegers a j so that the second term is less than 1 / ( k + 1). Hence, the nonnegative integers | f ( β k +1 ) | tend to 0 as k → ∞ . So, f ( β k +1 ) = 0 for cofinitely many k .Now, a k ! β k − a k +1 ! β k +1 = a k ! x k and so f ( x k ) = 0 for cofinitely many k , as desired. (cid:3) Corollary A.6. Z [[ x ]] ∗ ∼ = Z [ x ] .Proof. Let f : Z [[ x ]] → Z . Fact A.5 implies that f ( x k ) = 0 for cofinitely many k . Thus, g : Z [[ x ]] → Z defined by g X i ≥ c i x i = X i ≥ c i f ( x i ) x i is a well-defined Z -module homomorphism. As f and g agree on Z [ x ], we see that f = g . Note thatthe isomorphism Z [[ x ]] ∗ → Z [ x ] is f P i ≥ f ( x i ) x i . (cid:3) Corollary A.7. Z [[ x ]] is not a free Z -module.Proof. Otherwise, there is a Z -basis for Z [[ x ]] which is necessarily uncountable and so Z [[ x ]] ∗ isuncountable. This contradicts Corollary A.6. (cid:3) A.2.
Applications of Duals to End-Cohomology of Ladder Manifolds.
We now move to Z -modules and algebras arising from ladder manifolds. Let L ( X, Y ) be a ladder manifold based onclosed, connected, and oriented n -manifolds X and Y where n ≥
2. Recall from Section 3 that thedegree n subgroup of the end-cohomology algebra of L ( X, Y ) is e H ne ( L ( X, Y )) ∼ = ( Z ⊕ Z [[ σ ]] ⊕ Z ) /K where K = (cid:8) ( P β i , β, − P β i ) | β = P β i σ i ∈ Z [ σ ] (cid:9) ∼ = Z [ σ ]. Sometimes, we write x instead of σ in K . We begin by computing the dual module of ( Z ⊕ Z [[ x ]] ⊕ Z ) /K . Lemma A.8. If h : Z [[ x ]] → Z vanishes on L := n β = X β i x i ∈ Z [ x ] (cid:12)(cid:12)(cid:12) X β i = 0 o then h = 0 .Proof. By Fact A.5, there exists k ≥ h ( x k ) = 0. Let α = P a i x i be an arbitrary elementof Z [ x ] and let n = P a i . Then, α − nx k ∈ L and 0 = h ( α − nx k ) = h ( α ). So, h vanishes on Z [ x ].By Fact A.1, h = 0. (cid:3) Corollary A.9. If f : Z ⊕ Z [[ x ]] ⊕ Z → Z vanishes on K , then f = 0 on ⊕ Z [[ x ]] ⊕ and f ( r, γ, s ) = j ( r + s ) for some fixed integer j .Proof. Consider the inclusion i : Z [[ x ]] ֒ → Z ⊕ Z [[ x ]] ⊕ Z given by i ( γ ) = (0 , γ, f ◦ i : Z [[ x ]] → Z vanishes on L . By Lemma A.8, f ◦ i = 0. Thus, f vanishes on 0 ⊕ Z [[ x ]] ⊕
0. As(1 , , − ∈ K , we get0 = f (1 , , −
1) = f (1 , ,
0) + f (0 , , − f (0 , ,
1) = f (1 , , − f (0 , , f (1 , ,
0) = f (0 , , j := f (1 , , f ( r, γ, s ) = j ( r + s ). (cid:3) Corollary A.10. (( Z ⊕ Z [[ x ]] ⊕ Z ) /K ) ∗ ∼ = Z .Proof. Let f : ( Z ⊕ Z [[ x ]] ⊕ Z ) /K → Z , and let π : Z ⊕ Z [[ x ]] ⊕ Z ։ ( Z ⊕ Z [[ x ]] ⊕ Z ) /K be the canonicalsurjection. The composition f ◦ π vanishes on K . By the previous corollary, f ◦ π ( r, γ, s ) = j ( r + s )for some fixed integer j . As π is surjective, f ( J ( r, γ, s ) K ) = j ( r + s ). In particular, j = 1 gives agenerator for the dual module in question. (cid:3) Corollary A.11.
The uncountable, torsion-free Z -modules Z ⊕ Z [[ x ]] / Z [ x ] ⊕ Z and ( Z ⊕ Z [[ x ]] ⊕ Z ) /K are not isomorphic.Proof. They have nonisomorphic duals by Corollaries A.4 and A.10. (cid:3)
As an application of Corollary A.11, consider the space in Figure A.1. The wedge space W ( X, Y ) S S S Y Y Y X X X Figure A.1.
Wedge space W ( X, Y ). XTREME NONUNIQUENESS OF END-SUM 31 based on X and Y (a nonmanifold) is obtained from the disjoint union of the stringers on X and Y by simply wedging on the rungs as shown. The end-cohomology algebra of W ( X, Y ) is e H ke ( W ( X, Y ) ; R ) ∼ = H n ( X ; R ) ⊕ R [[ σ ]] /R [ σ ] ⊕ H n ( Y ; R ) if k = n , H k ( X ; R ) ⊕ ⊕ H k ( Y ; R ) if 2 ≤ k ≤ n − H ( X ; R ) ⊕ R [[ τ ]] /R [ τ ] ⊕ H ( Y ; R ) if k = 1,0 otherwisewhere the cup product is coordinatewise in the direct sum and vanishes in the middle coordinate.While the end-cohomology algebra of the wedge space W ( X, Y ) bears a striking resemblance to thatof the ladder manifold L ( X, Y ), they are not isomorphic.
Corollary A.12.
The end-cohomology algebras of the ladder manifold L ( X, Y ) and the wedge space W ( X, Y ) are not isomorphic. In particular, these two spaces are not proper homotopy equivalent.Proof. The degree n subgroups of these two algebras are nonisomorphic by Corollary A.11. (cid:3) A.3.
Height in Abelian Groups.
The notion of the height of an element plays an important rolein the study of infinite abelian groups. Let G be an additive abelian group, g ∈ G , and p > G for integers k ≥ † ) p k x = g The height of g ∈ G at p is H p ( g ) := k where k ∈ Z ≥ is maximal such that ( † ) has a solution x ∈ G . If ( † ) has a solution for every k ∈ Z ≥ ,then we write H p ( g ) = ∞ . Let 2 = p < p < · · · be the primes. The height of g ∈ G is the sequence( ‡ ) H ( g ) := ( H p ( g ) , H p ( g ) , . . . ) ∈ { , , , . . . , ∞} N For example, H (0) = ( ∞ , ∞ , ∞ , . . . ) in every abelian group G . If G is a field of characteristiczero, then H ( g ) = ( ∞ , ∞ , ∞ , . . . ) for all g ∈ G . If G is a field of characteristic p and 0 = g ∈ G ,then H p ( g ) = 0 and H q ( g ) = ∞ for each prime q = p . In general, the height of g depends on G , since the solutions x of ( † ) are required to lie in G . For the sake of intuition, it is useful tolook at some examples. In what follows, the partial ordering (cid:22) on { , , , . . . , ∞} N is defined by:( m i ) i ∈ N (cid:22) ( n i ) i ∈ N if and only if m i ≤ n i for all i ∈ N . Example A.13.
Let G = Z and g = 1400. The prime factorization 1400 = 2 H ( g ) = 3, H ( g ) = 2, H ( g ) = 1, and H p ( g ) = 0 for all other primes p . Inserting this data into ( ‡ )give us H (1400) = (3 , , , , , , , . . . )More generally, if 0 = g = Q ∞ i =1 p k i i , then H ( g ) = ( k , k , k , . . . )where all k i < ∞ and all but finitely many k i are zero. Example A.14.
Let G = Z [ x ] and α = P ni =0 a i x i ∈ Z [ x ] where a n = 0. Recall that the content of α is c ( α ) = gcd ( a , a , . . . , a n ) ∈ Z > It is straightforward to verify that the height of α ∈ Z [ x ] equals the height of c ( α ) ∈ Z . Thus, as inthe previous example, all heights of nonzero elements contain finitely many nonzero entries each ofwhich is finite. Example A.15.
Let G = Z [[ x ]] and 0 = α = P ∞ i =0 a i x i ∈ Z [[ x ]]. We define the content of α to be c ( α ) = gcd ( a , a , a , . . . ) ∈ Z > This is well-defined since some a n = 0 and ∞ > gcd( a , a , . . . , a n ) ≥ gcd( a , a , . . . , a n +1 ) ≥ gcd( a , a , . . . , a n +2 ) ≥ · · · Only finitely many of these inequalities can be strict by the well-ordering principle. In particular, c ( α ) = gcd ( a , a , . . . , a N ) for some nonnegative integer N = N ( α ). Again, it is straightforward toverify that the height of α ∈ Z [[ x ]] equals the height of c ( α ) ∈ Z , which yields exactly the samecollection of heights as in the previous two examples. Therefore, even though Z [[ x ]] is uncountable,its elements realize only countably many heights. Example A.16. In G = Z [[ x ]] / Z [ x ] height becomes more interesting. Consider, for example H (cid:0) q x + 2 x + · · · y (cid:1) = (1 , , , . . . ) H (cid:0) q + 2 x + 2 x + · · · y (cid:1) = (3 , , ∞ , , , . . . ) H (cid:0) q x + 2 x + 2 x + · · · y (cid:1) = ( ∞ , ∞ , ∞ , . . . )The key here is that, for any power series P ∞ i =0 a i x i ∈ Z [[ x ]] representing an element of Z [[ x ]] / Z [ x ],we may ignore any finite initial sum P ji =0 a i x i . This leads to the following. Lemma A.17.
Let G = Z [[ x ]] / Z [ x ] and let h = ( h , h , . . . ) ∈ { , , , . . . , ∞} N be a height. Then,there exists g ∈ G such that H ( g ) = h .Proof. The idea of the proof is contained in Example A.16. Let 2 = p < p < · · · be the rationalprimes. For each integer i ≥
1, define a i := p e (1 ,i )1 p e (2 ,i )2 · · · p e ( i,i ) i ∈ Z > where e ( n, i ) := ( i if h n ≥ ih n if h n < i Let g = q P ∞ i =1 a i x i y ∈ G . Then, the height of g ∈ G is H ( g ) = h . (cid:3) For a general discussion of height, see Fuchs [Fuc58, Ch. VII]. For our purposes, the crucial factsare Lemma A.17 and the following.
Lemma A.18.
Let φ : G → G ′ be a homomorphism of abelian groups. Then, for all g ∈ G we have H ( g ) (cid:22) H ( φ ( g )) .Proof. Let p > g = p k x in G . Then φ ( g ) = φ ( p k x ) = p k φ ( x ) in G ′ . Therefore, H p ( g ) ≤ H p ( φ ( g )). (cid:3) Corollary A.19.
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