Compactifications of manifolds with boundary
aa r X i v : . [ m a t h . G T ] N ov COMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY
SHIJIE GU AND CRAIG R. GUILBAULT
Abstract.
This paper is concerned with compactifications of high-dimensionalmanifolds. Siebenmann’s iconic 1965 dissertation [Sie65] provided necessary andsufficient conditions for an open manifold M m ( m ≥
6) to be compactifiable byaddition of a manifold boundary. His theorem extends easily to cases where M m is noncompact with compact boundary; however when ∂M m is noncompact, thesituation is more complicated. The goal becomes a “completion” of M m , ie, acompact manifold c M m containing a compactum A ⊆ ∂M m such that c M m \ A ≈ M m . Siebenmann did some initial work on this topic, and O’Brien [O’B83] extendedthat work to an important special case. But, until now, a complete characterizationhad yet to emerge. Here we provide such a characterization.Our second main theorem involves Z -compactifications. An important openquestion asks whether a well-known set of conditions laid out by Chapman andSiebenmann [CS76] guarantee Z -compactifiability for a manifold M m . We cannotanswer that question, but we do show that those conditions are satisfied if and only if M m × [0 ,
1] is Z -compactifiable. A key ingredient in our proof is the above ManifoldCompletion Theorem—an application that partly explains our current interest inthat topic, and also illustrates the utility of the π -condition found in that theorem. Introduction
This paper is about “nice” compactifications of high-dimensional manifolds. Thesimplest of these compactification is the addition of a boundary to an open manifold.That was the topic of Siebenmann’s famous 1965 dissertation [Sie65], the main re-sult of which can easily be extended to include noncompact manifolds with compactboundaries. When M m has noncompact boundary, one may ask for a compactifi-cation c M m that “completes” ∂M m . That is a more delicate problem. Siebenmannaddressed a very special case in his dissertation, before O’Brien [O’B83] characterizedcompletable n -manifolds in the case where M m and ∂M m are both 1-ended. Sincecompletable manifolds can have infinitely many (non-isolated) ends, O’Brien’s the-orem does not imply a full characterization of completable n -manifolds. We obtainsuch a characterization here, thereby completing an unfinished chapter in the studyof noncompact manifolds. Date : November 3, 2018.
Key words and phrases. manifold, end, inward tame, completion, Z-compacification, Wall finite-ness obstruction, Whitehead torsion.This research was supported in part by Simons Foundation Grants 207264 and 427244, CRG.
A second type of compactification considered here is the Z -compactification. Theseare similar to the compactifications discussed above—in fact, those are special cases—but Z -compactifications are more flexible. For example, a Z -boundary for an openmanifold need not be a manifold, and a manifold that admits no completion can ad-mit a Z -compactification. These compactifications have proven to be useful in bothgeometric group theory and manifold topology, for example, in attacks on the Boreland Novikov Conjectures. A major open problem (in our minds) is a characterizationof Z -compactifiable manifolds. A set of necessary conditions was identified by Chap-man and Siebenmann [CS76], and it is hoped that those conditions are sufficient.We prove what might be viewed the next best thing: If M m satisfies the Chapman-Siebenmann conditions (and m = 4), then M m × [0 ,
1] is Z -compactifiable. We dothis by proving that M m × [0 ,
1] is completable—an application that partly explainsthe renewed interest in manifold completions, and also illustrates the usefulness ofthe conditions found in the Manifold Completion Theorem.1.1.
The Manifold Completion Theorem. An m -manifold M m with (possiblyempty) boundary is completable if there exists a compact manifold c M m and a com-pactum C ⊆ ∂ c M m such that c M m \ C is homeomorphic to M m . In this case c M m iscalled a (manifold) completion of M m . A primary goal of this paper is the followingcharacterization theorem for m ≥
6. Definitions will be provided subsequently.
Theorem 1.1 (Manifold Completion Theorem) . An m -manifold M m ( m ≥ ) iscompletable if and only if(a) M m is inward tame,(b) M m is peripherally π -stable at infinity,(c) σ ∞ ( M m ) ∈ lim ←− n e K ( π ( N )) | N a clean neighborhood of infinity o is zero, and(d) τ ∞ ( M m ) ∈ lim ←− { Wh( π ( N )) | N a clean neighborhood of infinity } is zero. Together, Conditions (a) and (c) ensure that (nice) neighborhoods of infinity havefinite homotopy type, while Condition (d) allows one to upgrade certain, naturallyarising, homotopy equivalences to simple homotopy equivalences. These conditionshave arisen in other contexts, such as [Sie65] and [CS76].Condition (b) can be thought of as “ π -stability rel boundary”; it seems unique tothe situation at hand. In the special case where M m is 1-ended and N ⊇ N ⊇ · · · is a cofinal sequence of (nice) connected neighborhoods of infinity, it demands thateach sequence π ( ∂ M N i ∪ N i +1 ) ← π ( ∂ M N i ∪ N i +2 ) ← π ( ∂ M N i ∪ N i +3 ) ← · · · be stable where ∂ M N i denotes ∂M m ∩ N i . This reduces to ordinary π -stability when ∂M m is compact. A complete discussion of this condition can be found in § Remark 1.
Several comments are in order:(1) Dimensions ≤ §
2; our main focus is m ≥ ∂M m is compact and M m is inward tame then M m has finitely many ends(see § ∂M m . In that case Theorem OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 3 M m and ∂M m are 1-ended, was proved by O’Brien [O’B83]; that is where “peripheral π -stability” was first defined. But since candidates for completion can be infinite-ended (e.g., let C ⊆ S m − be a Cantor set and M m = B m \ C ), the generaltheorem is not a corollary. In the process of generalizing [O’B83], we simplifythe proof presented there and correct an error in the formulation of Condition(c). We also exhibit some interesting examples which answer a question posedby O’Brien about a possible weakening Condition (b).(4) If Condition (b) is removed from Theorem 1.1, one arrives at Chapman andSiebenmann’s conditions for characterizing Z -compactifiable Hilbert cube man-ifolds [CS76]. A Z -compactification theorem for finite-dimensional manifoldsis the subject of the second main result of this paper. We will describe thattheorem and the necessary definitions now.1.2. The Stable Z -compactification Theorem for Manifolds. To extend theidea of a completion to Hilbert cube manifolds Chapman and Siebenmann introducedthe notion of a “ Z -compactification”. A compactification b X = X ⊔ Z of a space X isa Z -compactification if there is a homotopy H : b X × [0 , → b X such that H = id b X and H t (cid:16) b X (cid:17) ⊆ X for all t >
0. Subsequently, this notion has been fruitfully appliedto more general spaces—notably, finite-dimensional manifolds and complexes; see,for example, [BM91],[CP95],[FW95],[AG99], and [FL05]. A completion of of a finite-dimensional manifold is a Z -compactification, but a Z -compactification need not bea completion. In fact, a manifold that allows no completion can still admit a Z -compactification; the exotic universal covers constructed by Mike Davis are someof the most striking examples (just apply [ADG97]). Such manifolds must satisfyConditions (a), (c) and (d), but the converse remains open. Question.
Does every finite-dimensional manifold that satisfies Conditions (a), (c)and (d) of Theorem 1.1 admit a Z -compactification? This question was posed more generally in [CS76] for locally compact ANRs, but in[Gui01] a 2-dimensional polyhedral counterexample was constructed. The manifoldversion remains open. In this paper, we prove a best possible “stabilization theorem”for manifolds.
Theorem 1.2 (Stable Z -compactification Theorem for Manifolds) . An m -manifold M m ( m ≥ ) satisfies Conditions (a), (c) and (d) of Theorem 1.1, if an only if M m × [0 , admits a Z -compactification. In fact, M m × [0 , is completable if andonly if M m satisfies those conditions. Remark 2.
In [Fer00], Ferry showed that if a locally finite k -dimensional polyhe-dron X satisfies Conditions (a), (c) and (d), then X × [0 , k +5 is Z -compactifiable. SHIJIE GU AND CRAIG R. GUILBAULT
Theorem 1.1 can be viewed as a sharpening of Ferry’s theorem in cases where X is amanifold.1.3. Outline of this paper.
The remainder of this paper is organized as follows.In § <
6. In § § § §
11 we prove Theorem1.2. In §
12 we provide a counterexample to a question posed in [O’B83] about apossible relaxation of Condition (b), and in §
13 we provide the proof of a technicallemma that was postponed until the end of the paper.2.
Manifold completions in dimensions < ≤
3, but much simplerversions are possible in those dimensions. For example, Tucker [Tuc74] showed that a3-manifold can be completed if and only if each component of each clean neighborhoodof infinity has finitely generated fundamental group—a condition that is implied byinward tameness alone.Since we have been unable to find the optimal 2-dimensional completion theorem inthe literature, we take this opportunity to provide such a theorem. If M has finitelygenerated first homology (e.g., if M is inward tame), then by classical work (see[Ker23] and [Ric63]) int( M ) ≈ Σ − P , where Σ is a closed surface and P is a finiteset of points. Therefore, M contains a compact codimension 1 submanifold C suchthat each of the the components { N i } ki =1 of M \ C is a noncompact manifold whosefrontier is a circle onto which it deformation retracts. Complete the N i individuallyas follows:i) If N i contains no portion of ∂M , add a circle at infinity; andii) If N i contains components of ∂M , perform the Ker´ekj´art´o-Freudenthal end-point compactification to N i .Classification 9.26 of [CKS12], applied to each N i of type ii), ensures that the resultis a manifold completion of M . As a consequence, we have the following: Theorem 2.1.
A connected -manifold M is completable if and only if H ( M ) isfinitely generated; in particular, Theorem 1.1 is valid when n = 2 . In dimension 5 our proof of Theorem 1.1 goes through verbatim, provided it isalways possible to work in neighborhoods of infinity with boundaries in which Freed-man’s 4-dimensional Disk Embedding Theorem holds. That issue is discussed in[Qui82] and [FQ90, § π groups that are “good”. A caveat is that,whenever [Fre82] is applied, conclusions are topological, rather than PL or smooth. OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 5
Remarkably, Siebenmann’s thesis fails in dimension 4 (see [Wei87] and [KS88]).Counterexamples to his theorem are, of course, counterexamples to Theorem 1.1 aswell.As for low-dimensional versions of Theorem 1.2: if m ≤ M m satisfies Con-dition (a) then M m is completable (hence Z -compactifiable), so M m × [0 ,
1] is com-pletable and Z -compactifiable. If m = 4, then M × [0 ,
1] is a 5-manifold, which (see §
11) satisfies the conditions of Theorem 1.1. Whether that leads to a completion de-pends on 4-dimensional issues, in particular the “goodness” of the (stable) peripheralfundamental groups of the ends of M × [0 , M . If desired, a precisegroup-theoretic condition can be formulated from Proposition 11.1 and [Gui07].3. Conventions, notation, and terminology
For convenience, all manifolds are assumed to be piecewise-linear (PL). That as-sumption is particularly useful for the topic at hand, since numerous instances of“smoothing corners” would be required in the smooth category (an issue that is cov-ered nicely in [O’B83]). With proper attention to such details, analogous theoremscan be obtained in the smooth or topological category. Unless stated otherwise, an m -manifold M m is permitted to have a boundary, denoted ∂M m . We denote the man-ifold interior by int M m . For A ⊆ M m , the point-set interior will be denoted Int M m A and the frontier by Fr M m A (or for conciseness, Int M A and the frontier by Fr M A ).A closed manifold is a compact boundaryless manifold, while an open manifold is anon-compact boundaryless manifold.For q < m , a q -dimensional submanifold Q q ⊆ M m is properly embedded if it is aclosed subset of M m and Q q ∩ ∂M m = ∂Q q ; it is locally flat if each p ∈ int Q q has aneighborhood pair homeomorphic to ( R m , R q ) and each p ∈ ∂Q q has a neighborhoodpair homeomorphic to (cid:0) R m + , R q + (cid:1) . By this definition, the only properly embeddedcodimension 0 submanifolds of M m are unions of its connected components; a moreuseful type of codimension 0 submanifold is the following: a codimension 0 subman-ifold Q m ⊆ M m is clean if it is a closed subset of M m and Fr M Q m is a properlyembedded locally flat (hence, bicollared) ( m − M m . In that case, M m \ Q m is also clean, and Fr M Q m is a clean codimension 0 submanifold of both ∂Q m and ∂ ( M m \ Q m ).When the dimension of a manifold or submanifold is clear, we sometimes omit thesuperscript; for example, denoting a clean codimension 0 submanifold by Q . Similarly,when the ambient space is clear, we denote (point-set) interiors and frontiers by Int A and Fr A For any codimension 0 clean submanifold Q ⊆ M m , let ∂ M Q denote Q ∩ ∂M m ;alternatively ∂ M Q = ∂Q \ int(Fr Q ). Similarly, we will let int M Q denote Q ∩ int M m ;alternatively int M Q = Q \ ∂M m . SHIJIE GU AND CRAIG R. GUILBAULT Ends, pro - π , and the peripheral π -stability condition Neighborhoods of infinity, partial neighborhoods of infinity, and ends.
Let M m be a connected manifold. A clean neighborhood of infinity in M m is a cleancodimension 0 submanifold N ⊆ M m for which M m \ N is compact. Equivalently,a clean neighborhood of infinity is a set of the form M m \ C where C is a compactclean codimension 0 submanifold of M m . A clean compact exhaustion of M m isa sequence { C i } ∞ i =1 of clean compact connected codimension 0 submanifolds with C i ⊆ Int M C i +1 and ∪ C i = M m . By letting N i = M m \ C i we obtain the corresponding cofinal sequence of clean neighborhoods of infinity . Each such N i has finitely manycomponents (cid:8) N ji (cid:9) k i j =1 . By enlarging C i to include all of the compact componentsof N i , we can arrange that each N ji is noncompact; then, by drilling out regularneighborhoods of arcs connecting the various components of each Fr M N ji (furtherenlarging C i ), we can also arrange that each Fr M N ji is connected. A clean N i withthese latter two properties is called a 0 -neighborhood of infinity . Most constructionsin this paper will begin with a clean compact exhaustion of M m with a correspondingcofinal sequence of clean 0-neighborhoods of infinity.Assuming the above arrangement, an end ε of M m is determined by a nestedsequence (cid:0) N k i i (cid:1) ∞ i =1 of components of the N i ; each component is called a neighborhoodof ε . More generally, any subset of M m that contains one of the N k i i is a neighborhoodof ε , and any nested sequence ( W j ) ∞ j =1 of connected neighborhoods of ε , for which ∩ W j = ∅ , also determines the end ε . A more thorough discussion of ends can befound in [Gui16]. Here we will abuse notation slightly by writing ε = (cid:0) N k i i (cid:1) ∞ i =1 ,keeping in mind that a sequence representing ε is not unique.At times we will have need to discuss components { N j } of a neighborhood of infinity N without reference to a specific end of M m . In that situation, we will refer to the N j as a partial neighborhoods of infinity for M m ( partial -neighborhoods if N is a0-neighborhood of infinity). Clearly every noncompact clean connected codimension0 submanifold of M m with compact frontier is a partial neighborhood of infinity withrespect to an appropriately chosen compact C ; if its frontier is connected it is a partial0-neighborhood of infinity.4.2. The fundamental group of an end.
For each end ε of M m , we will define the fundamental group at ε by using inverse sequences. Two inverse sequences of groups A α ←− A α ←− A α ←− · · · and B β ←− B β ←− B β ←− · · · are pro-isomorphic if theycontain subsequences that fit into a commutative diagram of the form(4.1) G i < λ i +1 ,i G i < λ i +1 ,i G i < λ i +1 ,i G i · · · H j < µ j +1 ,j < < H j < µ j +1 ,j < < H j < µ j +1 ,j < < · · · OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 7 where the connecting homomorphisms in the subsequences are (as always) compo-sitions of the original maps. An inverse sequence is stable if it is pro-isomorphicto a constant sequence C id ←− C id ←− C id ←− · · · . Clearly, an inverse sequence ispro-isomorphic to each of its subsequences; it is stable if and only if it contains asubsequence for which the images stabilize in the following manner(4.2) G < λ G < λ G < λ G · · · Im ( λ ) < ∼ = < < Im ( λ ) < ∼ = < < Im ( λ ) < ∼ = < < · · · where all unlabeled homomorphisms are restrictions or inclusions. (Here we have sim-plified notation by relabelling the entries in the subsequence with integer subscripts.)Given an end ε = (cid:0) N k i i (cid:1) ∞ i =1 , choose a ray r : [1 , ∞ ) → M m such that r ([ i, ∞ )) ⊆ N k i i for each integer i > π (cid:0) N k , r (1) (cid:1) λ ←− π (cid:0) N k , r (2) (cid:1) λ ←− π (cid:0) N k , r (3) (cid:1) λ ←− · · · where each λ i is an inclusion induced homomorphism composed with the change-of-basepoint isomorphism induced by the path r | [ i − ,i ] . We refer to r as the base ray and the sequence (4.3) as a representative of the “fundamental group at ε based at r ” —denoted pro- π ( ε, r ). Any similarly obtained representation (e.g., by choosinga different sequence of neighborhoods of ε ) using the same base ray can be seen tobe pro-isomorphic. We say the fundamental group at ε is stable if (4.3) is a stablesequence. A key observation from the theory of ends is that stability of pro- π ( ε, r )depends on neither the choice of neighborhoods nor that of the base ray. See [Gui16]or [Geo08].4.3. Relative connectedness, relative π -stability, and the peripheral π -stability condition. Let Q be a manifold and A ⊆ ∂Q . We say that Q is A - connected at infinity if Q contains arbitrarily small neighborhoods of infinity V forwhich A ∪ V is connected. Example 1. If P is a compact manifold with connected boundary, X ⊆ ∂P is aclosed set, and Q = P \ X , then Q has one end for each component of X but Q is ∂Q -connected at infinity. More generally, if B is a clean connected codimension 0manifold neighborhood of X in ∂P and A = B \ X , then Q is A -connected at infinity.The following lemma is straightforward. Lemma 4.1.
Let Q be a noncompact manifold and A a clean codimension subman-ifold of ∂Q . Then Q is A -connected at infinity if and only if Q \ A is -ended. If A ⊆ ∂Q and Q is A -connected at infinity: let { V i } be a cofinal sequence ofclean neighborhoods of infinity for which each A ∪ V i is connected; choose a ray SHIJIE GU AND CRAIG R. GUILBAULT r : [1 , ∞ ) → int Q such that r ([ i, ∞ )) ⊆ V i for each i >
0; and form the inversesequence(4.4) π ( A ∪ V , r (1)) µ ←− π ( A ∪ V , r (2)) µ ←− π ( A ∪ V , r (3)) µ ←− · · · where bonding homomorphisms are obtained as in (4.3). We say Q is A - π -stable atinfinity if (4.4) is stable. Independence of this property from the choices of { V i } and r follows from the traditional theory of ends by applying Lemmas 4.1 and 4.2. Lemma 4.2.
Let Q be a noncompact manifold and A a clean codimension subman-ifold of ∂Q for which Q is A -connected at infinity. Then, for any cofinal sequence ofclean neighborhoods of infinity { V i } and ray r : [1 , ∞ ) → Q as described above, thesequence (4.4) is pro-isomorphic to any sequence representing pro - π ( Q \ A, r ) .Proof. It suffices to find a single cofinal sequence of connected neighborhoods of in-finity { N i } in Q \ A for which the corresponding representation of pro- π ( Q \ A, r ) ispro-isomorpic to (4.4). Toward that end, for each i let C ⊇ C ⊇ · · · be a nested se-quence of relative regular neighborhoods of A in Q such that ∩ C i = A . By “cleanness”of the V i , each C i can be chosen so that C i ∪ V i is a clean codimension 0 submanifoldof Q which deformation retracts onto A ∪ V i . Then N i = ( C i ∪ V i ) \ A is a cleanneighborhood of infinity in Q \ A and N i ֒ → C i ∪ V i is a homotopy equivalence. Foreach i there is a canonical isomorphism α i : π ( A ∪ V i , r ( i )) → π ( N i , r ( i )) which isthe composition π ( A ∪ V i , r ( i )) ∼ = −→ π ( C i ∪ V i , r ( i )) ∼ = ←− π ( N i , r ( i ))These isomorphisms fit into a commuting diagram π ( A ∪ V , r (1)) µ ←− π ( A ∪ V , r (2)) µ ←− π ( A ∪ V , r (3)) µ ←− · · · α ↓∼ = α ↓∼ = α ↓∼ = π ( N , r (1)) λ ←− π ( N , r (2)) λ ←− π ( N , r (3)) λ ←− · · · completing the proof. (cid:3) Remark 3.
In the above discussion, we allow for the possibility that A = ∅ . Inthat case, A -connectedness at infinity reduces to 1-endedness and A - π -stability toordinary π -stability at that end. Definition 4.3.
Let M m be a manifold and ε an end of M m .(1) M m is peripherally locally connected at infinity if it contains arbitrarily small0-neighborhoods of infinity N with the property that each component N j is ∂ M N j -connected at infinity.(2) M m is peripherally locally connected at ε if ε has arbitrarily small 0-neighbor-hoods P that are ∂ M P -connected at infinity.An N with the property described in condition (1) will be called a strong -neigh-borhood of infinity for M m , and a P with the property described in condition (2)will be called a strong -neighborhood of ε . More generally, any connected partial0-neighborhood of infinity Q that is ∂ M Q -connected at infinity will be called a strongpartial -neighborhood of infinity. OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 9
Lemma 4.4. M m is peripherally locally connected at infinity iff M m is peripherallylocally connected at each of its ends.Proof. Clearly the initial condition implies the latter. For the converse, let N ′ be anarbitrary neighborhood of infinity in M m and for each end ε , let P ε be a 0-neighbor-hoods of ε , contained in N ′ , which is ∂ M P ε -connected at infinity. By compactnessof the Freudenthal boundary of M m , there is a finite subcollection { P ε k } nk =1 thatcovers the end of M m ; in other words, C = M m − ∪ nk =1 P ε k is compact. If the P ε k arepairwise disjoint, we are finished; just let N = ∪ nk =1 P ε k . If not, adjust the P ε k within N ′ so they are in general position with respect to one another, then let { Q j } sj =1 bethe set of components of ∪ nk =1 P ε k and note that each Q j is a ∂ M Q j -connected partial0-neighborhood of infinity. (cid:3) Remark 4.
In the next section, we show that every inward tame manifold M m isperipherally locally connected at infinity. As a consequence, that condition plays lessprominent role than the next definition. Definition 4.5.
Let M m be a manifold and ε an end of M m .(1) M m is peripherally π -stable at infinity if contains arbitrarily small strong0-neighborhoods of infinity N with the property that each component N j is ∂ M N j - π -stable at infinity.(2) M m is peripherally π -stable at ε if ε has arbitrarily small strong 0-neighbor-hoods P that are ∂ M P - π -stable at infinity.It is easy to see that peripheral π -stability at infinity implies peripheral π -stabilityat each end; and when M m is finite-ended, peripheral π -stability at each end impliesperipheral π -stability at infinity. A argument could be made for defining peripheral π -stability at infinity to mean “peripherally π -stability at each end”. For us, thatpoint is moot; in the presence of inward tameness the two alternatives are equivalent. Lemma 4.6.
An inward tame manifold M m is peripherally π -stable at infinity ifand only if it is peripherally π -stable at each of its ends. Proof of this lemma is technical, and not central to the main argument. For thatreason, we save the proof for later (see § Finite domination and inward tameness
A topological space P is finitely dominated if there exists a finite polyhedron K and maps u : P → K and d : K → P such that d ◦ u ≃ id P . If choices can be madeso both d ◦ u ≃ id P and u ◦ d ≃ id K , i.e., P ≃ K , we say P has finite homotopytype . For simplicity, we will restrict our attention to cases where P is a locally finitepolyhedron—a class that contains the (PL) manifolds, submanifolds, and subspacesconsidered here. Lemma 5.1.
Let M m be a manifold and A ⊆ ∂M . Then M m is finitely dominated[resp., has finite homotopy type] if and only if M m \ A is finitely dominated [resp., hasfinite homotopy type].Proof. M m \ A ֒ → M m is a homotopy equivalence, and these properties are homotopyinvariants. (cid:3) Lemma 5.2.
A locally finite polyhedron P is finitely dominated if and only if thereexists a homotopy H : P × [0 , → P such that H = id P and H ( P ) is compact.Proof. Assuming a finite domination, as described above, the homotopy between id P and d ◦ u has the desired property. For the converse, let K be a compact polyhedralneighborhood of H ( P ), u : K ֒ → P , and d = H : P → K . (cid:3) A locally finite polyhedron P is inward tame if it contains arbitrarily small polyhe-dral neighborhoods of infinity that are finitely dominated. Equivalently, P contains acofinal sequence { N i } of closed polyhedral neighborhoods of infinity each admitting a“taming homotopy” H : N i × [0 , → N i that pulls N i into a compact subset of itself.By an application of the Homotopy Extension Property (similar to [GM18, Lemma3.4]) we can require taming homotopies to be fixed on Fr N i . From there, it is easy tosee that, in an inward tame polyhedron, every closed neighborhood of infinity admitsa taming homotopy. Lemma 5.3.
Let M m be a manifold and A a clean codimension submanifold of ∂M m . If M m is inward tame then so is M m \ A .Proof. For an arbitrarily small clean neighborhood of infinity N in M m , let H be ataming homotopy that fixes Fr N . Then H extends via the identity to a homotopy thatpulls A ∪ N into a compact subset of itself, so A ∪ N is finitely dominated. Arguing asin Lemma 4.2, M m \ A has arbitrarily small clean neighborhoods of infinity homotopyequivalent to such an A ∪ N . (cid:3) Remark 5.
Important cases of Lemma 5.3 are when A = ∂M m and when V is aclean neighborhood of infinity (or a component of one) and A = ∂ M V . Notice thatLemma 5.3 is valid when M m is compact and H is the “empty map”.A finitely dominated space has finitely generated homology, from which it can beshown that an inward tame manifold with compact boundary is finite-ended (see[GT03, Prop.3.1]). That conclusion fails for manifolds with noncompact boundary;see item (3) of Remark 1. The following variation is crucial to this paper. Proposition 5.4.
If a noncompact connected manifold M m and its boundary eachhave finitely generated homology, then M m has finitely many ends. More specifically,the number of ends of M m is bounded above by dim H m − ( M m , ∂M m ; Z ) + 1 .Proof. Let C be a clean connected compact codimension 0 submanifold of M m , withthe property that N = M m \ C is a 0-neighborhood of infinity, and let { N j } kj =11 For a discussion of “tameness” terminology and its variants, see [Gui16, § OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 11 be the collection of connected components of N n . It suffices to show that k ≤ dim H m − ( M m , ∂M m ; Z ) + 1. For the remainder of this proof (and only this proof),all homology is with Z -coefficients.Note that ∂C is the union of clean codimension 0 submanifolds ∂ M C and Fr C ,which intersect in their common boundary ∂ (Fr C ). So by a generalized version ofPoincar´e duality [Hat02, Th.3.43] and the Universal Coefficients Theorem, for all i ,we have(5.1) H i ( C, ∂ M C ) ∼ = H m − i ( C, Fr C ) . Claim 1. dim H m − ( C, ∂ M C ) ≥ k − C, Fr C ), we have · · · → H ( C, Fr C ) ։ e H (Fr C ) → e H ( C ) q q ( Z ) k − Claim 2. rank H m − ( N, ∂ M N ) ≥ k This claim follows from the long exact sequence for the triple (
N, ∂N, ∂ M N ) → H m ( N, ∂N ) → H m − ( ∂N, ∂ M N ) H m − ( N, ∂ M N ) → q q Z ) k where triviality of H m ( N, ∂N ) is due to the noncompactness of all components of N ,and the middle equality is from excision.The relative Mayer-Vietoris Theorem for pairs [Hat02, § M m , ∂M m )expressed as ( C ∪ N, ∂ M C ∪ ∂ M N ), contains(5.2) H m − (Fr C, ∂ Fr C ) → H m − ( C, ∂ M C ) ⊕ H m − ( N, ∂ M N ) → H m − ( M m , ∂M m )from which we can deducedim ( H m − ( C, ∂ M C ) ⊕ H m − ( N, ∂ M N )) ≤ dim H m − (Fr C, ∂ Fr C ) + dim H m − ( M m , ∂M m )Since H m − (Fr C, ∂ Fr C ) ∼ = ( Z ) k (from excision), then by Claims 1 and 2 we have( k −
1) + k ≤ k + dim H m − ( M m , ∂M m ) . So k ≤ dim H m − ( M m , ∂M m ) + 1 . (cid:3) Corollary 5.5. If M m is inward tame, then M m is peripherally locally connected atinfinity. Proof.
By Lemma 4.1, it suffices to show that each compact codimension 0 cleansubmanifold D ⊆ M m is contained in a compact codimension 0 clean submanifold C ⊆ M m so that if N = M m \ C , then each component N j of N has the property that N j \ ∂M m is 1-ended.Since M m is inward tame, each of its clean neighborhoods of infinity is finitely dom-inated, so M m \ D has finitely many components, each of which is finitely dominated.Let P l be one of those components. Then, Fr P l is a compact clean codimension 0submanifold of ∂D , whose interior is the boundary of P l \ ∂M m . Since int (cid:0) Fr P l (cid:1) and P l \ ∂M m each have finitely generated homology ( P l \ ∂M m is finitely dominated),then by Proposition 5.4, P l \ ∂M m has finitely many ends. Choose a compact cleancodimension 0 submanifold K l of P l \ ∂M m that intersects int(Fr P l ) nontrivially andhas exactly one (unbounded) complementary component in P l \ ∂M m for each of thoseends. After doing this for each of the component P l of M m \ D , let C = D ∪ ( ∪ K l ). (cid:3) Finite homotopy type and the σ ∞ -obstruction Finitely generated projective left Λ-modules S and T are stably equivalent if thereexist finitely generated free Λ-modules F and F such that S ⊕ F ∼ = T ⊕ F . Under theoperation of direct sum, the stable equivalence classes of finitely generated projectivemodules form a group e K (Λ), the reduced projective class group of Λ. In [Wal65],Wall associated to each path connected finitely dominated space P a well-defined σ ( P ) ∈ e K ( Z [ π ( P )]) which is trivial if and only if P has finite homotopy type. (Here Z [ π ( P )] denotes the integral group ring corresponding to π ( P ). In the literature, e K ( Z [ G ]) is sometimes abbreviated to e K ( G ).) As one of the necessary and sufficientconditions for completability of a 1-ended inward tame open manifold M m ( m > π , Siebenmann defined the end obstruction σ ∞ ( M m ), to be (up tosign) the finiteness obstruction σ ( N ) of an arbitrary clean neighborhood of infinity N whose fundamental group “matches” the stable pro- π ( ε ( M m )). In cases where M m is multi-ended or has non-stable pro- π (or both), a more generaldefinition of σ ∞ ( M m ), introduced in [CS76], is required. Its definition employs severalideas from [Sie65, § e K from groups toabelian groups taking G to e K ( Z [ G ]), which may be composed with the π -functor toget a functor from path connected spaces to abelian groups; here we use an observationby Siebenmann allowing base points to be ignored. Next extend the functor and thefiniteness obstruction to non-path-connected P (abusing notation slightly) by letting e K ( Z [ π ( P )]) = L e K ( Z (cid:2) π (cid:0) P j (cid:1)(cid:3) ) The main theorem of [O’B83] incorrectly uses σ ( M m ) —the finiteness obstruction of the entiremanifold M m — in place of σ ∞ ( M m ). The mistake is an erroneous application of Siebenmann’sSum Theorem to conclude that triviality of σ ( M m ) implies triviality of σ ( N ) for each clean neigh-borhood of infinity N . Siebenmann [Sie65] (correctly) used the Sum Theorem to show that, in thecase of stable pro- π , it is enough to check the obstruction once—for a well-chosen clean neigh-borhood of infinity. He denoted that obstruction σ ( ε ). In our situation (and O’Brien’s) such asimplification is not possible. We use the subscripted “ ∞ ” to help distinguish the general situationfrom Siebenmann’s special case. OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 13 where { P j } is the set of path components of P , and letting σ ( P ) = (cid:0) σ ( P ) , · · · , σ (cid:0) P k (cid:1)(cid:1) recalling that P is finitely dominated and, hence, has finitely many components—eachfinitely dominated.Now, for an inward tame locally finite polyhedron P (or more generally locally com-pact ANR), let { N j } be a nested cofinal sequence of closed polyhedral neighborhoodsof infinity and define σ ∞ ( P ) = ( σ ( N ) , σ ( N ) , σ ( N ) , · · · ) ∈ lim ←− n e K [ Z [ π ( N i )] o The bonding maps of the target inverse sequence e K [ Z [ π ( N )] ← e K [ Z [ π ( N )] ← e K [ Z [ π ( N )] ← · · · are induced by inclusion, with the Sum Theorem for finiteness obstructions [Sie65,Th.6.5] assuring consistency. Clearly, σ ∞ ( P ) vanishes if and only if each N i has finitehomotopy type; by another application of the Sum Theorem, this happens if and onlyif every closed polyhedral neighborhood of infinity has finite homotopy type. Remark 6.
Alternatively, we could define σ ∞ ( P ) to lie in the inverse limit of the inverse system corresponding to all closed polyhedral neighborhoods of infinity, par-tially ordered by inclusion. These inverse limits are isomorphic, and in either case,the combination of Conditions (a) and (c) of Theorem 1.1 is equivalent to the require-ment that all clean neighborhoods of infinity have finite homotopy type—a propertyreferred to as absolute inward tameness in [Gui16].We close this section with an observation that builds upon Lemma 5.3. Both playkey roles in the proof of Theorem 1.1. Lemma 6.1.
Let M m be a manifold and A a clean codimension submanifold of ∂M m . If M m is inward tame and σ ∞ ( M m ) vanishes, then M m \ A is inward tameand σ ∞ ( M m \ A ) also vanishes.Proof. Lemma 5.3 assures us that if M m is inward tame, then so too is M m \ A .The latter ensures that σ ∞ ( M m \ A ) is defined. Arguing as we did in the proof ofLemma 5.3, M m \ A contains arbitrarily small neighborhoods of infinity which arehomotopy equivalent to A ∪ N , where N is a clean neighborhood of infinity in M m . If σ ∞ ( M m ) = 0, then N has finite homotopy type; and since A ∪ N = A \ N ∪ N , where A \ N is a compact ( m − A ∪ N has finite homotopy type (by a directargument or easy application of the Sum Theorem for the finiteness obstruction). Thevanishing of σ ∞ ( M m \ A ) then follows from the above discussion. (cid:3) The τ ∞ -obstruction The τ ∞ obstruction in Condition (d) of Theorem 1.1 was first defined in [CS76]and applied to Hilbert cube manifolds; the role it plays here is similar. It lies inthe derived limit of an inverse sequence of Whitehead groups. For a more detaileddiscussion, the reader should see [CS76]. C W W C C Figure 1.
Decomposition of M m into { W i } ∞ i =1 .The derived limit of an inverse sequence G λ ←− G λ ←− G λ ←− · · · of abelian groups is the quotient group:lim ←− { G i , λ i } = ∞ Y i =0 G i ! / { ( g − λ g , g − λ g , g − λ g , · · · ) | g i ∈ G i } It is a standard fact that pro-isomorphic inverse sequences of abelian groups haveisomorphic derived limits.Suppose a manifold M m contains a cofinal sequence { N i } of clean neighborhoods ofinfinity with the property that each inclusion Fr N i ֒ → N i is a homotopy equivalence .Let W i = N i \ N i +1 and note that Fr N i ֒ → W i is a homotopy equivalence. See Figure1. Since Fr N i and W i are finite polyhedra, the inclusion determines a Whiteheadtorsion τ ( W i , Fr N i ) ∈ Wh( π (Fr N i )) (see [Coh73]). As in the previous section, wemust allow for non-connected Fr N i so we defineWh( π (Fr N i )) = L Wh( π (Fr N ji ))where (cid:8) Fr N ji (cid:9) is the (finite) set of components of Fr N i and τ ( W i , Fr N i ) = (cid:0) τ (cid:0) W i , Fr N i (cid:1) , · · · , τ (cid:0) W ki , Fr N ki (cid:1)(cid:1) . A manifold admitting such sequence of neighborhoods of infinity is called pseudo-collarable . See[Gui00], [GT03] and [GT06] for discussion of that topic.
OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 15
These groups fit into and inverse sequence of abelian groupsWh( π ( N )) ← Wh( π ( N )) ← Wh( π ( N )) ← · · · where the bonding homomorphisms are induced by inclusions. (To match [CS76],we have substituted π ( N i ) for the canonically equivalent π (Fr N i ).) Let τ i = τ ( W i , Fr N i ) ∈ Wh( π ( N i )). Then τ ∞ ( M m ) = [( τ , τ , τ , · · · )] ∈ lim ←− { Wh( π ( N i )) } where [( τ , τ , τ , · · · )] is the coset containing ( τ , τ , τ , · · · ).If τ ∞ ( M m ) is trivial, it is possible to adjust the choices of the N i so that eachinclusion Fr N i ֒ → W i has trivial torsion, and hence is a simple homotopy equivalence.Roughly speaking, the adjustment involves “lending and borrowing torsion to andfrom immediate neighbors of the W i ”. The procedure is as described in [CS76, § §
4] is flawed; we recommend [CS76].8.
Geometric characterization of completable manifolds and areview of h- and s-cobordisms
The following geometric characterization of completable manifolds, which has analogsin [Tuc74] and [O’B83], paves the way for the proof of Theorem 1.1. It leads naturallyto the consideration of h- and s-cobordisms, which we will briefly review for later use.
Lemma 8.1 (Geometric characterization of completable manifolds) . A non-compactmanifold with boundary M m is completable iff M m = ∪ ∞ i =1 C i where, for all i :(i) C i is a compact clean codimension submanifold of M m ,(ii) C i ⊂ Int C i +1 , and(iii) if W i denotes C i +1 \ C i , then ( W i , Fr C i ) ≈ (Fr C i × [0 , , Fr C i × { } ) .Proof. For the forward implication, suppose c M m is a compact manifold, A is closedsubset set of ∂ c M m , and M m = c M m \ A . Write A as ∩ i F i , where { F i } ∞ i =1 is a sequenceof compact clean codimension 0 submanifolds of ∂ c M m with F i +1 ⊆ Int F i . Let c : ∂ c M m × [0 , → c M m be a collar on ∂ c M m with c (cid:16) ∂ c M m × { } (cid:17) = ∂ c M m and, for each i , let C i = c M m \ c (Int( F i ) × [0 , /i )). Assertions (i) and (ii) are clear. Moreover, W i ≈ F i × [0 , /i ] \ (Int F i +1 × [0 , / ( i + 1))) ≈ F i × [0 , /i ]via a homeomorphism taking c ( F i × { /i } ) onto F i × { /i } . Then, since Fr C i = c ( F i × { /i } ∪ ∂F i × [0 , /i ]) ≈ F i , an application of relative regular neighborhoodtheory allows an adjustment of that homeomorphism so that Fr C i is taken onto F i × { /i } . A reparametrization of the closed interval completes the proof of assertion(iii). (Note that this works even when the F i have multiple and varying numbers ofcomponents. See Figure 2.) C W A Figure 2.
Decomposing completed M m into product cobordisms.For the converse, we reverse the above procedure to embed M m in a copy of C .Details can be found in [Tuc74, Lemma 1]. (cid:3) The above lemma shows that a strategy for completing a manifold is to fill up aneighborhood of infinity in M m with a sequence of cobordisms, then modify thosecobordisms (when possible) so they become products.Recall that an (absolute) cobordism is a triple ( W, A, B ), where W is a mani-fold with boundary and A and B are disjoint manifolds without boundary for which A ∪ B = ∂W . The triple ( W, A, B ) is a relative cobordism if A and B are dis-joint codimension 0 clean submanifolds of ∂W . In that case, there is an associatedabsolute cobordism ( V, ∂A, ∂B ) where V = ∂W \ (int A ∪ int B ). We view absolutecobordisms as special cases of relative cobordisms where V = ∅ . A relative cobor-dism is an h-cobordism if each of the inclusions A ֒ → W , B ֒ → W , ∂A ֒ → V , and ∂B ֒ → V is a homotopy equivalence; it is an s-cobordism if each of these inclu-sions is a simple homotopy equivalence. (For convenience, ∅ ֒ → ∅ is considereda simple homotopy equivalence.) A relative cobordism is nice if it is absolute orif ( V, ∂A, ∂B ) ≈ ( ∂A × [0 , , ∂A × { } , ∂A × { } ). The crucial result, proof (andadditional discussion) of which may be found in [RS82] , is the following. Theorem 8.2 (Relative s-cobordism Theorem) . A compact nice relative cobordism ( W, A, B ) with dim W ≥ is a product, i.e., ( W, A, B ) ≈ ( A × [0 , , A × { } , A × { } ) ,if and only if it is an s-cobordism. Remark 7.
A situation similar to a nice relative cobordism occurs when ∂W = A ∪ B ′ , where A and B ′ are codimension 0 clean submanifolds of ∂W with a commonnonempty boundary ∂A = ∂B ′ . By choosing a clean codimension 0 submanifold B ⊆ B ′ with the property that B ′ \ Int B ≈ ∂B × [0 ,
1] we arrive at a nice relativecobordism (
W, A, B ). When this procedure is applied, we will refer to (
W, A, B ) as
OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 17 a corresponding nice relative cobordism. For notational consistency, we will alwaysadjust the term B ′ on the far right of the triple ( W, A, B ′ ), leaving A alone.For our purposes, the following lemma will be crucial. Lemma 8.3.
Let W be a compact manifold with ∂W = A ∪ B ′ , where A and B ′ arecodimension clean submanifolds of ∂W with a common boundary. Suppose A ֒ → W is a homotopy equivalence and that there is a homotopy J : W × [0 , → W suchthat J = id W , J is fixed on ∂B ′ , and J ( W ) ⊆ B ′ . Then B ′ ֒ → W is a homotopyequivalence, so the corresponding nice relative cobordism ( W, A, B ) is an h-cobordism.Proof. Choose p ∈ ∂A = ∂B ′ , to be used as the basepoint for A , B ′ and W . Let i : A ֒ → W and ι : B ′ ֒ → W denote inclusions and define f : A → B ′ by f ( x ) = J ( x ).Then(8.1) ι ◦ f = J ◦ i Clearly J : W → W induces the identity isomorphism on π ( W, p ), and since i is a homotopy equivalence, it induces a π -isomorphism. So, from (8.1), we maydeduce that f ∗ : π ( A, p ) → π ( B ′ , p ) is injective. Moreover, since f restricts tothe identity function mapping ∂A onto ∂B ′ , [Eps66] allows us to conclude that f ∗ is an isomorphism. From there it follows that ι ∗ : π ( B ′ , p ) → π ( W, p ) is also anisomorphism.Let p : f W → W be the universal covering projection, e A = p − ( A ), and e B ′ = p − ( B ′ ). Since i ∗ and ι ∗ are both π -isomorphisms these are the universal covers of A and B ′ , respectively. By generalized Poincar´e duality for non-compact manifolds, H k ( f W , e B ′ ; Z ) ∼ = H n − kc ( f W , e A ; Z ) , where cohomology is with compact supports. Since e A ֒ → f W is a proper homotopyequivalence, all of these relative cohomology groups vanish, so H k ( f W , e B ′ ; Z ) = 0 forall k . By the relative Hurewicz theorem, π k ( f W , e B ′ ) = 0 for all k , so the same is truefor π k ( W, B ′ ). An application of Whitehead’s theorem allows us to conclude that B ′ ֒ → W is a homotopy equivalence. (cid:3) Proof of the Manifold Completion Theorem: necessity
We will prove necessity of the conditions in Theorem 1.1 by a straightforwardapplication of Lemma 8.1.
Proof of Theorem 1.1 (necessity).
Suppose c M m is a compact manifold and A is closedsubset set of ∂ c M m such that M m = c M m \ A . As in the proof of Lemma 8.1 write A = ∩ i F i , where { F i } is a sequence of compact clean codimension 0 submanifoldsof ∂ c M m with F i +1 ⊆ Int F i , and let c : ∂ c M m × [0 , → c M m be a collar on ∂ c M m with c (cid:16) ∂ c M m × { } (cid:17) = ∂ c M m . For each i , let b N i = c ( F i × [0 , /i ]) and N i = b N i \ A .Then { N i } is cofinal sequence of clean neighborhoods of infinity in M m with Fr N i = c ( F i × { /i } ∪ ∂F i × [0 , /i ]). Since F i × { /i } ∪ ∂F i × [0 , /i ] ֒ → F i × [0 , /i ] and N i ֒ → b N i are both homotopy equivalences, then so is Fr N i ֒ → N i ; and since each N i has finite homotopy type, conditions (a) and (c) of Theorem 1.1 both hold (by thediscussion in § W i = N i \ N i +1 , then τ ∞ ( M m ) is determined by the Whitehead torsionsof inclusions Fr N i ֒ → W i (see § W i with F i × [0 , /i ] and Fr N i with F i × { /i } ∪ ∂F i × [0 , /i ], as in the proof of Lemma 8.1. Then, the fact that both F i × { /i } ֒ → F i × [0 , /i ] and F i × { /i } ֒ → F i × { /i } ∪ ∂F i × [0 , /i ] are simplehomotopy equivalences ensures that τ ( W i , Fr N i ) = 0. So condition (d) is satisfied.It remains to verify the peripheral π -stability condition. Fix i ≥ F ji be one component of F i , b N ji = c (cid:0) F ji × [0 , /i ] (cid:1) and N ji = b N ji \ A . Then ∂ M N ji = c ( F i × { } ) \ A and N ji is clearly ∂ M N ji -connected at infinity. For each k > i , let F ′ k be the union of all components of F k contained in F ji , c N ′ k = c ( F ′ k × [0 , /k ]) and N ′ k = c N ′ k \ A . By definition, we may consider the sequence(9.1) π (cid:0) ∂ M N ji ∪ N ′ i +1 (cid:1) µ ←− π (cid:0) ∂ M N ji ∪ N ′ i +2 (cid:1) µ ←− π (cid:0) ∂ M N ji ∪ N ′ i +3 (cid:1) µ ←− · · · where basepoints are suppressed and bonding homomorphisms are compositions ofmaps induced by inclusions and change-of-basepoint isomorphisms. Each of thoseinclusions is the top row of a commutative diagram ∂ M N ji ∪ N ′ k ← ֓ ∂ M N ji ∪ N ′ k +1 ↓ incl ↓ incl ∂ M N ji ∪ c N ′ k ∂ M N ji ∪ [ N ′ k +1 ↓ ≈ ↓ ≈ ( F ji × { } ) ∪ ( F ′ k × [0 , /k ]) ← ֓ ( F ji × { } ) ∪ (cid:0) F ′ k +1 × [0 , /k + 1] (cid:1) where the bottom row is an obvious homotopy equivalence, as are all vertical maps.It follows that the initial inclusion is a homotopy equivalence as well. As a result, allbonding homomorphisms in (9.1) are isomorphisms, so the sequence is stable. (cid:3) Proof of the Manifold Completion Theorem: sufficiency
Throughout this section { C i } ∞ i =1 will denote a clean compact exhaustion of M m with a corresponding cofinal sequence of clean 0-neighborhoods of infinity { N i } ∞ i =1 ,each of which has a finite set of connected components (cid:8) N ji (cid:9) k i j =1 . For each i we willlet W i = N i \ N i +1 , a compact clean codimension 0 submanifold of M m . Note that ∂W i may be expressed as Fr N i ∪ ( ∂ M W i ∪ Fr N i +1 ), a union of two clean codimension0 submanifolds of ∂W i intersecting in a common boundary ∂ (Fr N i ). (Figures 2 and1 contain useful schematics.) The proof of Theorem 1.1 will be accomplished bygradually improving the exhaustion of M m so that ultimately, conditions (i)-(iii) ofLemma 8.1 are all satisfied. Lemma 10.1. If M m is inward tame and σ ∞ ( M m ) vanishes, then for each i , σ ( N i ) and σ ( N i \ ∂M m ) are both zero. OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 19
Proof.
By our discussion in §
6, if M m is inward tame and σ ∞ ( M m ) = 0, then each N i has finite homotopy type. Since N i ֒ → N i \ ∂M m is a homotopy equivalence, so does N i \ ∂M m . (cid:3) Proposition 10.2. If M m satisfies Conditions (a)-(c) of Theorem 1.1 then the { C i } and the corresponding { N i } can be chosen so that, for each i ,(1) Fr N i ֒ → N i is a homotopy equivalence, and(2) ∂ M W i ∪ Fr N i +1 ֒ → N i is a homotopy equivalence; therefore,(3) the nice relative cobordisms corresponding to ( W i , Fr N i , ∂ M W i ∪ Fr N i +1 ) areh-cobordisms.Proof. By Lemma 10.1 and the definition of peripheral π -stability at infinity, we canbegin with a clean compact exhaustion { C i } ∞ i =1 of M m and a corresponding sequenceof neighborhoods of infinity { N i } ∞ i =1 , each with a finite set of connected components (cid:8) N ji (cid:9) k i j =1 , so that for all i ≥ ≤ j ≤ k i ,i) N ji is inward tame,ii) N ji is (cid:0) ∂ M N ji (cid:1) -connected and ( ∂ M N ji )- π -stable at infinity, andiii) σ ∞ (cid:0) N ji (cid:1) = 0.By Lemmas 5.3, 4.2, and 6.1, this implies thati ′ ) N ji \ ∂ M N ji is inward tame,ii ′ ) N ji \ ∂M m is 1-ended and has stable fundamental group at infinity, andiii ′ ) σ ∞ (cid:0) N ji \ ∂M m (cid:1) = 0.These are precisely the hypotheses of Siebenmann’s Relativized Main Theorem([Sie65, Th.10.1]), so N ji \ ∂M m contains an open collar neighborhood of infinity V ji ≈ ∂V ji × [0 , ∞ ). Following the proof in [Sie65] (similar to what is done in[O’B83, Th.3.2]), this can be done so that ∂N ji \ ∂M m (= int(Fr N ji )) and ∂V ji con-tain clean compact codimension 0 submanifolds A ji and B ji , respectively, so that( ∂N ji \ ∂M m ) \ int A ji = ∂V ji \ int B ji ≈ ∂A ji × [0 , K ji = N ji \ V ji is a clean codimension 0 submanifold of M m which intersects C i in A ji . To save on notation, replace C i with C i ∪ (cid:0) ∪ K ji (cid:1) , which is still a cleancompact codimension 0 submanifold of M m , but with the added property that(10.1) N i \ ∂M m ≈ int(Fr N i ) × [0 , ∞ ) . Since adding ∂ M N i back in does not affect homotopy types, we also have that(10.2) Fr N i ֒ → N i is a homotopy equivalence.Having enlarged the C i , pass to a subsequence if necessary to regain the propertythat C i ⊆ Int C i +1 for all i .Letting N i = M m \ C i gives a nested cofinal sequence of clean neighborhoods ofinfinity { N i } with the property that each inclusion Fr N i ֒ → N i is a homotopy equiv-alence; in other words, we have obtained a pseudo-collar structure on M m . For each i ≥
1, let W i = N i \ N i +1 , a clean compact codimension 0 submanifold of M m with ∂W i = Fr N i ∪ ( ∂ M W i ∪ Fr N i +1 ). K B VA
Figure 3. V ji ≈ ∂V ji × [0 ,
1) contained in N ji \ ∂M m . Claim 1. Fr N i ֒ → W i is a homotopy equivalence. Condition (10.2) applied to N i ensures the existence a strong deformation retraction H t of N i onto Fr N i . That same condition applied to N i +1 ensures the existence ofa retraction r : N i +1 → Fr N i +1 , which extends to a retraction b r : N i → W i . Thecomposition b rH t , restricted to W i , gives a deformation retraction of W i onto Fr N i . Claim 2. ∂ M W i ∪ Fr N i +1 ֒ → W i is a homotopy equivalence. By applying Lemma 8.3, it is enough to show that there exists a homotopy H : W i × [0 , → W i , fixed on ∂ (Fr N i ), with the property that H ( W i ) ⊆ ∂ M W i ∪ Fr N i +1 .Toward that end, let B be a collar neighborhood of ∂ M W i in W i and let D = W i \ B .Use the collar structure on N i \ ∂M m to obtain a homotopy K : N i × [0 , → N i , fixedon ∂ (Fr N i ), which pushes N i into the complement of D ; in other words K ( N i ) ⊆ B ∪ N i +1 . Compose this homotopy with the retraction b r : N i → W i used in theprevious claim to get a homotopy b rK t of W i (still fixed on ∂ (Fr N i )) with b rK ( W i ) ⊆ B ∪ Fr N i +1 . Follow this with a homotopy that deformation retracts B onto ∂ M W i while sending Fr N i +1 into itself to complete the desired homotopy and prove Claim2. We can now write M m = C ∪ W ∪ W ∪ W ∪ · · · where, for each i , • W i is a compact clean codimension 0 submanifold of M m , • ∂W i = Fr N i ∪ ( ∂ M W i ∪ Fr N i +1 ), and • both Fr N i ֒ → W i and ∂ M W i ∪ Fr N i +1 ֒ → W i are homotopy equivalences.As such, the corresponding nice relative cobordisms (as described in Remark 7) areh-cobordisms. (cid:3) OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 21
Proposition 10.3. If M m satisfies Conditions (b)-(d) of Theorem 1.1 the conclu-sion of Proposition 10.2 can be improved so that, for each i , the nice relative cobor-disms corresponding to ( W i , Fr N i , ∂ M W i ∪ Fr N i +1 ) are s-cobordisms. In that case, ( W i , Fr N i ) ≈ (Fr N i × [0 , , Fr N i × { } ) for all i , and M m is completable.Proof. By the triviality of τ ∞ ( M m ), it is possible to adjust the choices of the N i sothat each inclusion Fr N i ֒ → W i has trivial Whitehead torsion, i.e., τ ( W i , Fr N i ) = 0,and hence is a simple homotopy equivalence. As was discussed in §
7, the adjustmentinvolves “lending and borrowing torsion to and from immediate neighbors of the W i ”as described in [CS76, § W i thenapply Lemma 8.1. (cid:3) Z -compactifications and the proof of Theorem 1.2 In this section we prove Theorem 1.2. Since M m × [0 ,
1] satisfies Conditions (a), (c)and (d) of Theorem 1.1 if and only if M m satisfies those same conditions (see [CS76]),it suffices to prove the following proposition which is based on work found in [Gui07]. Proposition 11.1.
If a manifold M m is inward tame at infinity, then M m × [0 , isperipherally π -stable at infinity.Proof. Apply Corollary 5.5 to obtain a cofinal sequence { N i } of clean neighborhoodsof infinity for M m with the property that, for all i , each component N ji of N i is ∂ M N ji -connected at infinity. Since { N i × [0 , } is a cofinal sequence of clean neighborhoodsof infinity for M m × [0 ,
1] it suffices to show that the corresponding connected com-ponents, N ji × [0 , ∂ M × [0 , ( N ji × [0 , ∂ M × [0 , ( N ji × [0 , π -stable at infinity. By Lemmas 4.1 and 4.2, that is equivalent to showing that, foreach N ji , int M ( N ji ) × (0 ,
1) is 1-ended and has stable pro- π at that end. Every con-nected topological space becomes 1-ended upon crossing with (0 , π -stability property is proved with a small variation on the maintechnical argument from [Gui07]; in particular, Corollary 3.6 from that paper. The“small variation” is necessary because the earlier argument assumed the product of an open manifold with (0 , K t used in [Gui07, Prop.3.3] sends the manifold interior of Int M ( N ji )into itself and sends Fr N ji into itself for all t . That is easily accomplished since Fr N ji has an open collar neighborhood at infinity. (cid:3) A counterexample to a question of O’Brien
We now give a negative answer to a question posed by O’Brien [O’B83, p.308].
Question. (For a -ended manifold M m with -ended boundary), let { V i } be a cofinalsequence of clean 0-neighborhoods of infinity. If { π ( ∂M m ∪ V i ) } i ≥ is stable, does itfollow that M m is peripherally π -stable at infinity? The key ingredient in our counterexamples is a collection of contractible open n -manifolds W n (one for each n ≥ W n has the property that it cannot be embedded inany compact n -manifold. Although these W n have finite homotopy type, they arenot inward tame, since they contain arbitrarily small clean connected neighborhoodsof infinity with non-finitely generated fundamental groups. Our counterexamples willbe the ( n + 1)-manifolds W n × [0 , Proposition 12.1.
Let W n be a connected open n -manifold. If W n has finite homo-topy type, then W n × [0 , is -ended and inward tame, with σ ∞ ( W n × [0 , .Proof. It suffices to exhibit arbitrarily small connected clean neighborhood of infinityin W n with finite homotopy type. Let N ⊆ W n be a clean neighborhood of infinityand a ∈ (0 , N small and a close to 1, we can obtain arbitrarily smallneighborhoods of infinity in W n × [0 ,
1) of the form V ( N, a ) = ( N × [0 , ∪ ( W n × [ a, . Since V ( N, a ) deformation retracts onto W n × { a } , it is connected and has finitehomotopy type. (cid:3) Example 2.
Consider the ( n + 1)-manifold M n +1 = W n × [0 , W n is theSternfeld n -manifold ( n ≥
3) described above. Then ∂M n +1 = W n × { } . A standardduality argument shows that every contractible open manifold of dimension ≥ { N i } be a cofinal sequence of clean connected neighborhoods of infinityin W n , and for each i ≥
1, let V i = V (cid:0) N i , ii +1 (cid:1) , as defined in the previous proof. BySeifert-van Kampen, each V i ∪ ∂M n +1 is simply connected, so the inverse sequence { π ( ∂M n +1 ∪ V i ) } i ≥ is pro-trivial, hence, stable.To see that M n +1 is not peripherally π -stable at infinity, first assume that n ≥
5. Then, if M n +1 were peripherally π -stable at infinity, it would be completableby Theorem 1.1. (The triviality of τ ∞ ( M n +1 ) is immediate since M n +1 is simplyconnected at infinity, which follows from the simple connectivity of the V i .) But, if c M n +1 were a completion, then W n × { } ֒ → ∂ c M n +1 would be an embedding into aclosed n -manifold, contradicting Sternfeld’s theorem.To obtain analogous examples when n = 3 or n = 4, we cannot rely on the ManifoldCompletion Theorem. But a direct analysis of the fundamental group calculations inSternfeld’s proof reveals that the peripheral pro- π -systems arising in W n × [0 ,
1) arenonstable in those dimensions as well.13.
Proof of Lemma 4.6
We now return to Lemma 4.6, which asserts that the two natural candidates forthe definition of “peripherally π -stable at infinity” (the global versus the local ap-proach) are equivalent for inward tame manifolds. The intuition behind the lemmais fairly simple. If M m contains arbitrarily small 0-neighborhoods of infinity N withthe property that each component N j is ∂ M N j - π -stable at infinity, then those com-ponents provide arbitrarily small neighborhoods of the ends satisfying the necessary OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 23 π -stability condition. Conversely, if each end ε has arbitrarily small strong 0-neigh-borhoods P that are ∂ M P - π -stable at infinity, we can use the compactness of theset of ends (in the Freudenthal compactification) to find, within any neighborhood ofinfinity, a finite collection { P , · · · , P k } of such neighborhoods which cover the endof M m . If we can do this so the P i are pairwise disjoint, we are finished—just let N = ∪ P i . That is not as easy as one might hope, but we are able to attain the desiredconclusion by proving the following proposition. Proposition 13.1.
Suppose M m is inward tame and each end ε has arbitrarily smallstrong -neighborhoods P ε that are ∂ M P ε - π -stable at infinity. Then every strongpartial -neighborhood of infinity Q ⊆ M m is ∂ M Q - π -stable at infinity. Our proof requires that we break the stability condition into a pair of weakerconditions. An inverse sequence of groups is: • semistable (sometimes called pro-epimorphic ) if it is pro-isomorphic to aninverse sequence of surjective homomorphisms; • pro-monomorphic of it is pro-isomorphic to an inverse sequence of injectivehomomorphisms.It is an elementary fact that an inverse sequence is stable if and only if it is bothsemistable and pro-monomorphic.We will make use of the following topological characterizations of the above proper-ties, when applied to pro- π . In these theorems, a “space” should be locally compact,locally connected, and metrizable. Proposition 13.2.
Let X be a -ended space and r : [0 , ∞ ) → X a proper ray. Thenpro- π ( X, r ) is(1) semistable if and only if, for every compact set C ⊆ X , there exists a largercompact set D ⊆ X such that for any compact set E with D ⊆ E ⊆ X , everyloop in X \ D with base point on r can be pushed into X \ E by a homotopy withimage in X \ C keeping the base point on r , and(2) pro-monomorphic if and only if X contains a compact set C with the propertythat, for every compact set D with C ⊆ D ⊆ X , there exists a compact set E ⊇ D with the property that every loop in X \ E that contracts in X \ C alsocontracts in X \ D . These are standard. See, for example [Geo08] or [Gui16]. In the case that pro- π ( X, r ) is pro-monomorphic, the compact set C in the above proposition is called a π -core for X . Notice that, by Proposition 13.2, the property of (1-ended) X havingpro-monomorphic pro- π ( X, r ) is independent of the choice of r .It is a non-obvious (but standard) fact that having semistable pro- π ( X, r ) is alsoindependent of the choice of r . As for the characterization of semistable pro- π ( X, r ),we are mostly interested in the following easy corollary.
Corollary 13.3. If X is a -ended space and pro- π ( X, r ) is semistable for some(hence every) proper ray r , then for each compact set C ⊆ X , there is a larger compact set D ⊆ X such that, for every compact set E ⊆ X and every path λ : [0 , → X \ D with λ ( { , } ) ⊆ E , there is a path homotopy in X \ C taking λ to a path λ ′ in X \ E. We are now ready for our primary task.
Proof of Proposition 13.1.
Let Q be a strong partial 0-neighborhood of infinity in M m . By Lemma 4.2, proving that Q is ∂ M Q - π -stable at infinity is equivalent toproving that the 1-ended space Q \ ∂M m has stable pro- π . We will take the latterapproach.By Lemma 5.3 Q \ ∂M m is inward tame, so a modification of the argument in [GT03,Prop. 3.2] ensures that pro- π ( Q \ ∂M m , r ) is semistable. It is therefore enough toshow that pro- π ( Q \ ∂M m , r ) is pro-monomorphic. We will do that by verifying thecondition described in Proposition 13.2, i.e., we will show that Q \ ∂M m contains a π -core.By hypothesis, each end ε of Q has a strong 0-neighborhood P ε which is ∂ M P ε - π -stable at infinity and lies in Int M Q . Since the set of ends of Q is compact in theFreudenthal compactification, there is a finite subcollection { P ε i } ki =1 whose union is aneighborhood of infinity in Q . Place the collection of submanifolds { P ε i } ki =1 in generalposition. Claim 1.
For each Ω ⊆ { , · · · , k } the set ∩ j ∈ Ω P ε j has finitely many components,each of which is a clean codimension submanifold of M m . General position ensures that each component is a clean codimension 0 submanifoldof M m . Since each P ε j is a closed subset of M m each component T of ∩ j ∈ Ω P ε j is closedin M m , and since T cannot also be open in M m it must have nonempty frontier.Since (cid:8) P ε j (cid:9) j ∈ Ω is in general position, so also is the collection of (compact) frontiers, (cid:8) Fr P ε j (cid:9) j ∈ Ω . So, for each i = j in Ω, ∆ i,j = Fr P ε i ∩ Fr P ε j is a clean codimension1 submanifold of Fr P ε i and Fr P ε j . The union of these ∆ i,j separate ∪ kj =1 Fr P ε j intofinitely many pieces, and since the frontier of each T is a union of these pieces, therecan only be finitely many such T .Choose an embedding b : ∂M m × [0 , → M m with b ( x,
0) = x for all x ∈ ∂M m and whose image B is a regular neighborhood of ∂M m in M m . With some additionalcare, arrange that B intersects: Q in b ( ∂ M Q × [0 , P ε i in b ( ∂ M P ε i × [0 , T of each finite intersection ∩ j ∈ Ω P ε j in b ( ∂ M T × [0 , ≤ s < t ≤
1, let B [ s,t ] = b ( ∂M m × [ s, t ]), B ( s,t ) = b ( ∂M m × ( s, t )), etc. For A ⊆ ∂M m , let B A = b ( A × [0 , B [ s,t ] A , B ( s,t ) A ,etc. analogously.By hypothesis and Proposition 13.2 we can choose a clean codimension 0 compact π -core C i for each P ε i \ ∂M m . Then choose t so small that B [0 ,t ] ∩ ( ∪ ki =1 C i ) = ∅ . Let C ′ ≡ Q \ ∪ ki =1 P ε i , then let C = C ′ \ B [0 ,t ) so that C is a compact clean codimension0 submanifold of Q \ ∂M m . Let C = ∪ ki =0 C i ⊆ Q \ ∂M m . Notice that the collection n B [0 ,t ] ∂ M Q , P ε , · · · , P ε k o covers Q \ Int Q C .Choose a clean codimension 0 compact submanifold of D ′ ⊆ Q \ ∂M m so large that OMPACTIFICATIONS OF MANIFOLDS WITH BOUNDARY 25 i) Int Q D ′ ⊇ C ,ii) D ′ contains every compact component of ∩ j ∈ Ω P ε j for all Ω ⊆ { , · · · , k } , andiii) for any compact set E ⊆ Q \ ∂M m such that D ′ ⊆ E , if λ is a path in T \ ∂M m ,where T is an unbounded component of P ε i ∩ P ε j for some i, j ∈ { , · · · , k } ,and λ lies outside D ′ with endpoints outside E , then there is a path homotopyof λ in ( T \ ∂M m ) \ C pushing λ outside E . (This uses Corollary 13.3 and thefact that each T , being a clean partial neighborhood of infinity in M m , has theproperty that T \ ∂M m has finitely many ends, each with semistable pro- π .)Now choose a compact set D ⊆ Q \ ∂M m such thati ′ ) D ⊇ D ′ ,ii ′ ) for every Ω ⊆ { , · · · , k } and every unbounded component T of ∩ j ∈ Ω P ε j , each x ∈ ( T \ ∂M m ) \ D can be pushed to infinity in ( T \ ∂M m ) \ D ′ . (This is possiblesince there are only finitely such T .)iii ′ ) if x = b ( y, t ) ∈ B \ D , then b ( y × [0 , t ]) ∩ D ′ = ∅ . Claim 2. D is a π -core for Q \ ∂M m . Toward that end, let F be a compact subset of Q \ ∂M m containing D , then choose G ⊆ Q \ ∂M m to be an even larger compact set with the following property:( † ) for each i ∈ { , · · · , k } , loops in P ε i \ ∂M m lying outside G which contract in( P ε i \ ∂M m ) \ C , also contract in ( P ε i \ ∂M m ) \ F .Let α : [0 , × [0 , → ( Q \ ∂M m ) \ D . The interiors of sets n B [0 ,t ] ∂ M Q , P ε , · · · , P ε k o cover ( Q \ ∂M m ) \ D , so we can subdivide [0 , into subsquares { R t } so small thatthe image of each R t lies in B (0 ,t ) or one of the P ε i \ ∂M m and hence, in B (0 ,t ) \ D orone of the ( P ε i \ ∂M m ) \ D . Since each vertex of this subdivision is sent to a point x in B (0 ,t ) \ D and/or T \ D , where T is an unbounded component of the intersection ofthe P ε i which contain the images of the subsquares containing that vertex, then bythe choice of D we can push x into ( Q \ ∂M m ) \ G along a path that does not leave T and does not intersect D ′ . In those cases where x = b ( y, t ) ∈ B (0 ,t ) \ D , push x outof G along b ( y × (0 , B (0 ,t ) \ D ′ , by property (iii ′ ).Doing the above for each vertex adjusts α up to homotopy in ( Q \ ∂M m ) \ D ′ so thateach vertex of the subdivision is taken into ( Q \ ∂M m ) \ G and each R t is still takeninto the same P ε i (or B (0 ,t ) ) as before.Next we move to the 1-skeleton of our subdivision of [0 , . If an edge e is theintersection R t ∩ R t ′ of two squares, i.e., e is not in ∂ ([0 , ), we use property (iii) toadjust α up to homotopy so e is mapped into ( Q \ ∂M m ) \ G , noting that this homotopymay causes the “new” α to drift into ( Q \ ∂M m ) \ C . (If e is sent into B (0 ,t ) , we canuse (iii ′ ) to ensure that the push stays in B (0 ,t ) \ D ′ as well.)Do the above for each edge until the entire 1-skeleton of the subdivision of [0 , is mapped into ( Q \ ∂M m ) \ G . The image of α now lies in ( Q \ ∂M m ) \ C . Notice thatthe restriction of α to each R t is a map of a disk into a single P ε i (or B (0 ,t ) ) missing C i with boundary being mapped into P ε i \ G . So by the choice of G , we may redefine α on R t to be the same on its boundary, but to take R t into P ε i \ F or B (0 ,t ) \ F . Assembling the α | R t we get a map α ′ : [0 , × [0 , → ( Q \ ∂M m ) \ F that agrees with α on ∂ ([0 , ). (cid:3) References [ADG97] F. D. Ancel, M. W. Davis, and C. R. Guilbault, CAT(0) reflection manifolds , Geometrictopology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc.,Providence, RI, 1997, pp. 441–445. MR 1470741[AG99] Fredric D. Ancel and Craig R. Guilbault, Z -compactifications of open manifolds , Topology (1999), no. 6, 1265–1280. MR 1690157[BM91] Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups , J. Amer.Math. Soc. (1991), no. 3, 469–481. MR 1096169[CKS12] Jack S. Calcut, Henry C. King, and Laurent C. Siebenmann, Connected sum at infinityand Cantrell-Stallings hyperplane unknotting , Rocky Mountain J. Math. (2012), no. 6,1803–1862. MR 3028764[Coh73] Marshall M. Cohen, A course in simple-homotopy theory , Springer-Verlag, New York-Berlin, 1973, Graduate Texts in Mathematics, Vol. 10. MR 0362320[CP95] Gunnar Carlsson and Erik Kjær Pedersen,
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