Topological rigidity and actions on contractible manifolds with discrete singular set

By Frank Connolly, James F. Davis, and Qayum Khan

Abstract

The problem of equivariant rigidity is the -homeomorphism classification of -actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of . In other words, this is the classification of cocompact -manifolds.

We use surgery theory, algebraic -theory, and the Farrell–Jones Conjecture to give this classification for a family of groups which satisfy the property that the normalizers of nontrivial finite subgroups are themselves finite. More generally, we study cocompact proper actions of these groups on contractible manifolds and prove that the condition is always satisfied.

1. Introduction

Definition.

Let be a discrete group. Define to be the set of equivariant homeomorphism classes of contractible topological manifolds equipped with an effective cocompact proper -action. The singular set and free set are

When is a torsion-free group of isometries of a nonpositively curved Riemannian manifold , with , an inspirational result of Farrell and Jones Reference 13 says that consists of the single element . (See Reference 2 for the generalization to the case.) In these cases, is empty.

In this paper we compute when is a group of isometries of a nonpositively curved Riemannian manifold where is discrete (that is, the action is pseudo-free). One may speculate that consists of just one element. However, for many years experts have suspected that Cappell’s -groups (see Quinn Reference 28) obstruct this hope. We fully calculate .

Theorem 1.1 (Main result).

Let be a group satisfying Hypotheses (ABC) or more generally (ABC) below, with of virtual cohomological dimension . Write . There is a bijection

The set of conjugacy classes of maximal infinite dihedral subgroups of  is .

These unitary nilpotent groups have been computed by Banagl, Connolly, Davis, Koźniewski, and Ranicki Reference 1Reference 3Reference 5Reference 6. These abelian groups are

Thus is a singleton if or if has no elements of order 2.

Main Hypotheses.

Let be a discrete group.

(A)

There exists a torsion-free subgroup of with finite index.

(B)

The normalizer is finite for each nontrivial finite subgroup of .

(C)

There is a contractible Riemannian manifold of nonpositive sectional curvature with an effective cocompact proper action by isometries.

A more general but less geometric condition than (C) is (C), consisting of:

(Ci)

There exists an element for which the quotient space has the homotopy type of a finite CW complex.

(Cii)

Each infinite dihedral subgroup of lies in a unique maximal infinite dihedral subgroup.

(Ciii)

satisfies the Farrell–Jones Conjecture in -theory and -theory.

Remark 1.2.

For comparative purposes, we make some comments here.

•

Proposition 2.3 shows that if satisfies Hypothesis (A) and , then satisfies Hypothesis (B) if and only if is a discrete set.

•

Suppose satisfies Hypotheses (AB). Corollary 2.6 shows that any two -manifolds in model and hence are -homotopy equivalent.

•

(ABC) (ABC): Suppose satisfies Hypotheses (AB) and satisfies Hypothesis (C). Then clearly and has the homotopy type of the finite CW complex . Thus Hypothesis (Ci) holds. Let be an infinite dihedral subgroup of . Then stabilizes a unique geodesic , namely the geodesic which contains the fixed points of the involutions of , noting the fixed set of each involution in is a point. The stabilizer of is the unique maximal infinite dihedral subgroup of containing . Thus Hypothesis (Cii) holds. Finally Hypothesis (Ciii) holds by Reference 2 and Reference 35.

•

In the proof of Theorem 1.1 we will only refer to Hypotheses (ABC).

•

Theorem 1.1 was proven for the case in Reference 4.

We explain how is the isovariant structure set of any .

Definition.

Let be a cocompact -manifold. A structure on is a -equivariant homotopy equivalence of proper cocompact -manifolds. If is an isovariant homotopy equivalence, we call it an isovariant structure. Two such structures (resp. isovariant structures) and are equivalent if there is an equivariant homeomorphism and a -equivariant homotopy (resp. -isovariant homotopy) from to . Define the structure set (resp. isovariant structure set ) as the set of equivalence classes of structures (resp. isovariant structures) on .

Theorem 1.3.

Suppose satisfies Hypotheses (AB). Assume there is an element such that . Then the forgetful maps are bijections:

Corollary 1.4.

Suppose satisfies Hypotheses (ABC) with the dimension of at least five. Assume has no elements of order two. Then any contractible manifold equipped with an effective cocompact proper -action has the -homotopy type of , and any -homotopy equivalence is -homotopic to a -homeomorphism.

Proof.

This follows from Theorems 1.1 and 1.3, since is empty.

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When is a torsion-free group, Corollary 1.4 follows from Reference 2, Theorem B.

It is interesting to note that our theorems have no simple homotopy equivalence requirement; equivariant homotopy equivalence up to equivariant homeomorphism is enough. The reason is the isomorphism of the assembly map in the next result.

Proposition 1.5 (cf. Theorem 5.4).

Suppose satisfies Hypothesis (B). Then

This vanishing result can be interpreted geometrically, as follows. Recall from Reference 4, Section 1 that a cocompact manifold model for is a -space with the equivariant homotopy type of a -CW complex such that, for all subgroups of , the -fixed set is a contractible manifold if is finite and is empty if is infinite. A -space is locally flat if implies is a locally flat submanifold of . For our -cobordism rigidity, we do not assume that acts locally linearly.

Theorem 1.6.

Suppose satisfies Hypotheses (AB) and the Farrell–Jones Conjecture in -theory. Assume and are cocompact manifold models for of dimension at least six. If there exists a locally flat isovariant -cobordism, , between and , then is a product cobordism and and are equivariantly homeomorphic.

Here we outline the logical structure of arguments in the rest of the paper. In Sections 2 and 3 we establish the geometric topology needed to prove Theorem 1.3. In Section 4, we use the end theory of high-dimensional manifolds to reduce the isovariant structure set to a classical structure set of a certain compact pair, . Therein we use an algebraic result (Corollary 5.5) which depends on Proposition 1.5. Later, we further reduce to Ranicki’s algebraic structure groups. In Section 5, we systematically remove all of the -theory decorations and pass to . In Section 6, we calculate these structure groups using the Farrell–Jones Conjecture and the other axioms for , thus proving Theorem 1.1. In Section 7, we deduce Theorem 1.6 from Corollary 1.4 and general results of Quinn. In Section 8, we give a variety of examples of groups which satisfy the hypotheses of Theorem 1.1, and hence groups where the question of equivariant rigidity is completely answered.

2. -manifolds

Recall that for a discrete group , a model for is a -space which is -homotopy equivalent to a -CW complex and, for all subgroups of ,

Given any -CW complex with finite isotropy groups, there is an equivariant map , unique up to equivariant homotopy. It follows that any two models are -homotopy equivalent. Furthermore, a model exists for any group .

We show that if a group satisfies Hypotheses (AB) and acts effectively, cocompactly, and properly on a contractible manifold, then the manifold models .

The following lemma is not logically necessary, but nonetheless provides motivation.

Lemma 2.1.

Let be a virtually torsion-free group with an effective cocompact proper action by isometries on a contractible Riemannian manifold of nonpositive sectional curvature (that is, Hypotheses (AC) hold). Then models .

Proof.

Let be a finite index normal subgroup of with quotient group . The manifold is complete since is compact, hence complete. For an infinite subgroup of , the fixed set is empty by properness. For a finite subgroup of , the fixed set is nonempty by the Cartan Fixed Point Theorem Reference 19, Theorem 13.5, which states that a compact group of isometries of a complete nonpositively curved manifold has a fixed point. It is easy to show that the fixed set of group of isometries is geodesically convex, hence contractible if nonempty. Finally, since the finite group acts smoothly on the closed manifold , this is a -CW complex Reference 20, and so, by lifting cells, is a -CW complex.

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Now it remains to show that if satisfies Hypotheses (AB) and if is a contractible -manifold, then the fixed point set of each finite subgroup of is a point and has the -homotopy type of a -CW complex; hence will model . We start with a group-theoretic lemma.

Lemma 2.2.

Suppose a finite group acts on a set so that for all prime-order subgroups of , the fixed set is a singleton. Then is a singleton.

Proof.

Let be the collection of nontrivial subgroups of whose fixed set is a singleton. Note that satisfies the normal extension property: if with , then . Indeed since . On the other hand, since , we see the singleton is an -invariant by normality, so . So is a point and . In particular contains all cyclic subgroups of .

We next claim that, if and are elements of order 2 in , then , for the cyclic group is in , and . So . Therefore and are the same point . It follows that this point is the fixed set of the normal subgroup generated by all elements of order 2 in . This subgroup is nontrivial if has even order. So by the normal extension property, is a singleton if has even order.

Finally suppose has odd order. It that order is a prime, then is in . If not then by the Feit–Thompson Theorem Reference 14, there is a nontrivial proper normal subgroup and we proceed inductively to conclude that .

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Recall that a group action is pseudo-free if the singular set is discrete. Our next result shows that the finite normalizer condition is equivalent to pseudo-free.

Proposition 2.3.

Suppose satisfies Hypothesis (A). Assume is a contractible manifold equipped with an effective cocompact proper -action. The action is pseudo-free if and only if Hypothesis (B) holds. Furthermore, if either of these equivalent conditions holds, then for a subgroup of , the fixed point set is empty if is infinite and a point if is nontrivial and finite.

Proof.

First suppose is pseudo-free. Fix a nontrivial finite subgroup of . Let be a prime-order subgroup of . By Reference 4, Lemma 2.2(2), is nonempty and connected. Then, since is a subspace of the discrete space , must be a single point. Thus, by Lemma 2.2, is a single point. Therefore, since the restriction is proper, must be finite.

Now suppose, in addition to the existence of a torsion-free subgroup of finite index, that is a finite group for each nontrivial finite subgroup of . Let be a prime-order subgroup of . The covering map is -equivariant. By Reference 4, Lemma 2.3(3), is a regular -cover where . But , so is a homeomorphism. By Reference 4, Lemma 2.3(6), is compact, and hence so is . By Reference 4, Lemma 2.2(3), is a point since it is both compact and the fixed set of a -group action on a contractible manifold. Thus, by Lemma 2.2, is a point for each nontrivial finite subgroup of .

Note that since the action is proper, is empty for infinite subgroups and that we showed above that is a point when is nontrivial and finite.

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Remark 2.4.

The resort to the Feit–Thompson theorem in Lemma 2.2 is striking. But for our purposes (in the proof of Proposition 2.3) it is not actually necessary. For if is an odd order subgroup of , the argument in Lemma 2.2 shows that each nontrivial Sylow -subgroup of fixes exactly one point (use the normal extension property and the nilpotency of ). Since acts freely on the homology sphere , it follows that has periodic cohomology and is therefore cyclic, since is odd. As is well known, this implies that is metacyclic. So again by the normal extension property, since has a normal cyclic subgroup, we see and is a singleton.

Here is a general tameness result pertaining to all of the actions that we consider.

Proposition 2.5.

Let be a discrete group. Any topological manifold equipped with a cocompact pseudo-free -action has the -homotopy type of a -CW complex.

Proof.

Since is a discrete subset of the topological manifold , it is forward-tame (tame in the sense of Quinn Reference 29). Then, by Reference 29, Propositions 2.6 and 3.6, is forward-tame in . So, by Reference 29, Proposition 2.14, the complement is reverse-tame (tame in the sense of Siebenmann Reference 33).

Then by Siebenmann Reference 33, if , the ends of have collar neighborhoods. We will go ahead and cross with the 5-torus, and thereby guarantee both that the manifold is high-dimensional and the ends are collared. Hence there is a compact -manifold such that

Write and for the pullback -covers of and of the cover . Observe acts properly on the set of connected components of . Then, since is pseudo-free, we can define a -map

where is the connected component containing a point . Consider the equivalence relation on given by if . This is preserved by the -action. Write for the quotient. As -spaces,

The mapping cylinder has the quotient topology.

Since is a topological -manifold, by Kirby–Siebenmann Reference 22, p. 123, Theorem III:4.1.3, there is a proper homotopy equivalence for some CW pair . Also is a 0-dimensional -CW complex. Then has the -homotopy type of a -CW complex, via the -homotopy equivalence

Here denotes the -cover of a homotopy-inverse for .

Finally, choose a point and consider the -inclusion and corresponding -retraction

The mapping torus has the -homotopy type of a -CW complex. By cyclic permutation of composition factors, as -spaces, where means the infinite cyclic cover. This completes the proof.

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Corollary 2.6.

Suppose satisfies Hypotheses (AB). Any contractible manifold equipped with an effective cocompact proper -action is a model for . Hence the forgetful map of Theorem 1.3 is bijective.

Proof.

By Proposition 2.3, the -fixed sets of are single points if is a nontrivial finite subgroup of , otherwise the -fixed sets are empty if is infinite. Hence is a pseudo-free -manifold. Then, by Proposition 2.5, has the -homotopy type of a -CW complex. Therefore, since is contractible, it models . The remainder follows from universal properties of classifying spaces.

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3. From equivariance to isovariance

We recall our earlier result about improving equivariant maps:

Theorem 3.1 (Reference 4, Theorem A.3).

Let be a finite group. Let and be compact pseudo-free -manifolds. Assume . Let be a -map such that the restriction is bijective.

(1)

If is -connected, then is -homotopic to an isovariant map.

(2)

If and are -isovariant, -connected, and -homotopic, then is -isovariantly homotopic to .

Proposition 3.2.

The map of Theorem 1.3 is bijective.

Proof.

Let . Since is a -homotopy equivalence of -spaces, for every subgroup of , the restrictions are homotopy equivalences. Now, by Proposition 2.3, for each nontrivial finite subgroup of , and are single points. Thus is a bijection.

By hypothesis, there exists a normal torsion-free subgroup with finite. Since the restricted actions and are free, proper, and cocompact, the quotients and are compact -manifolds. By Theorem 3.1 applied to both the -homotopy equivalence and any choice of -homotopy-inverse, is -homotopic to a -isovariant homotopy equivalence , unique up to -isovariant homotopy.

By the Covering Homotopy Property applied to the cover , the -homotopy from to is covered by a unique homotopy from to some covering . By uniqueness of path lifting, it follows that is -equivariant. Then, since is -equivariant and is -isovariant, an elementary diagram chase shows that is a -isovariant map. Since is a -equivariant map covering a -isovariant homotopy equivalence , by similar reasoning it follows that is a -isovariant homotopy equivalence, unique up to -isovariant homotopy. Then and . Thus is surjective. Furthermore, since is unique up to -isovariant homotopy and is unique up to -isovariant homotopy, is injective.

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Proof of Theorem 1.3.

Immediate from Corollary 2.6 and Proposition 3.2.

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4. Reduction to classical surgery theory

In this section we show that each cocompact proper contractible -manifold in is well-behaved in a neighborhood of the discrete set . We use this to interpret as a structure set of a compact manifold-with-boundary, which we call a compact -manifold. We begin with a quick fact about uniqueness.

Lemma 4.1.

Let be a manifold of dimension greater than four, and let . Let be a compact neighborhood of in so that is homeomorphic to where is a manifold. Then is homeomorphic to a disc.

Denote the closed cone on a space by .

Proof.

There is a homeomorphism . By uniqueness of the one-point compactification of a space, there are basepoint-preserving homeomorphisms:

Observe that is a -manifold with , hence is compact. Note also that is simply-connected, since where the second isomorphism follows from the Seifert–van Kampen Theorem applied to the decomposition with a closed disc neighborhood of in the manifold . Excision shows that induces an isomorphism on homology. Thus is an -cobordism, and the -cobordism theorem Reference 15Reference 34 shows that is homeomorphic to the product . Hence , a sphere, and , a disc.

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An action of a group on a space is locally conelike if every orbit has a -neighborhood that is -homeomorphic to for some -space .

Lemma 4.2.

Suppose satisfies Hypotheses (ABC) with . Let . There is a compact -manifold with interior . Furthermore, the action is locally conelike.

Proof.

Since is compact and since is discrete by Proposition 2.3, is a finite set. Then the manifold has finitely many ends, one for each orbit of . Write for the isotropy groups.

Since the ends of are tame, and each end of is finitely covered by an end of , Propositions 2.6 and 3.6 of Reference 29 show the ends of are tame. If were smooth, then Siebenmann’s thesis Reference 33 gives a CW structure on which is a union of subcomplexes with a finite complex and each a connected, finitely dominated complex with . For a topological manifold, it follows from Reference 22, p. 123, Theorem III:4.1.3 that there is a proper homotopy equivalence with and as above. Let denote the reduced projective class group and the Wall finiteness obstruction.

Consider the inclusion-induced map of projective class groups:

Since is a finite complex, the sum theorem of Reference 33 shows

On the one hand, has the homotopy type of a finite CW complex by Hypothesis (Ci), and is a homotopy equivalence, so . On the other hand, is injective by Corollary 5.5 below. Hence for each . Therefore, by Siebenmann’s theorem (Reference 33, Reference 16, p. 214), there is a compact -manifold with interior .

Furthermore, is -homeomorphic to the universal cover . Consider the closure , which is a -cocompact neighborhood of in . By Lemma 4.1, each component of is a disc. Then

Therefore the action of on is locally conelike.

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By Lemma 4.2, we may choose a compact -manifold with interior . Note . Enumerate the connected components of the boundary:

Observe each has universal cover homeomorphic to and .

Definition.

A structure on is a pair where is a compact topological -manifold and is a homotopy equivalence of pairs. Two structures and are equivalent if there are an -cobordism of pairs from to and an extension of . The structure set is the set of equivalence classes of structures on .

Lemma 4.3.

Suppose satisfies Hypotheses (ABC) and . There is a bijection

Proof.

We define on representatives as follows. Let be a homotopy equivalence of pairs. Consider the -space defined by coning off:

Here denotes the universal cover. Since by Lemma 4.2 each has universal cover , we obtain that is a topological manifold. The map extends to a -isovariant map . Since and are homotopy equivalences and is a homeomorphism, it follows that is a -isovariant homotopy equivalence. We define

Next, we show that is constant on equivalence classes. Let in . By Corollary 5.5 with , and the fact that -cobordisms are determined up to homeomorphism by their torsion, observe for some -cobordism , and for some homotopy equivalence . By lifting to a -homotopy on universal covers, it follows that . By an Eilenberg swindle, which uses realization of -cobordisms and triviality of -cobordisms, there is a homeomorphism relative to . This extends to a homeomorphism relative to :

Here denotes one-point compactification. Then we obtain a homeomorphism relative to . So there is a -homeomorphism such that . Hence . Therefore is defined on .

We now show that has a two-sided inverse:

Let . Consider the proper homotopy equivalence

By Lemma 4.2, there is a compact -manifold with interior . Using a collar for in , after a small proper homotopy of , we may assume that extends to a homotopy equivalence of pairs. Define

Observe a different choice for the Siebenmann completion of the ends of would yield a pair such that , where is an -cobordism, and where the map extends and has image in . Therefore is well-defined. It is now straightforward to see that is a two-sided inverse of .

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The geometric structure set can be identified with a version of Ranicki’s algebraic structure group for the same pair. The connective algebraic structure groups are the homotopy groups of the homotopy cofiber of an assembly map , and so they fit into an exact sequence of abelian groups Reference 30:

where is the 1-connective algebraic -theory spectrum of the trivial group. Exactly as in Reference 4, for computations we shall use the nonconnective, periodic analogue:

where is the 4-periodic algebraic -theory spectrum of the trivial group.

Remark 4.4.

Following Ranicki Reference 32, Thm. 18.5, there is a pointed map

which is a bijection for . The map is called the total surgery obstruction for homotopy equivalences. In our case, it is also a bijection when , since each connected component of is a 4-dimensional spherical space form, with finite fundamental group, and so can be included with the high-dimensional () case by Freedman–Quinn topological surgery Reference 16. Since is -dimensional, by the Atiyah–Hirzebruch spectral sequence, we obtain:

Hence there is an exact sequence:

At the end of the proof of Theorem 1.1 in Section 6, we will show that the last map is zero. In the meantime, we will proceed to compute the nonconnective algebraic structure group .

5. Reduction from to structure groups

Our goal in this section is to replace the decoration by in the structure group by showing .

A group is virtually cyclic if there is a cyclic subgroup of finite index. There is a well-known trichotomy of types of virtually cyclic groups :

(I)

is finite.

(II)

There is an exact sequence with finite and the infinite cyclic group:

(III)

There is an exact sequence with finite and the infinite dihedral group:

For the rest of the paper, we consider the following increasing chain of classes :

•

denotes the class of all trivial groups.

•

denotes the class of all groups of type (I).

•

denotes the class of all groups of types (I, II) — the finite-by-cyclics.

•

denotes the class of all groups of types (I, II, III).

•

denotes the class of all groups.

Given a group and one of the above five classes , we shall consider the family of subgroups of . Each family is closed under conjugation and subgroups. One says that satisfies Property  if every element of is contained in a unique maximal element of   , and if, in addition, equals its normalizer in .

Lemma 5.1.

Suppose is a group satisfying Hypothesis (B). If , then is isomorphic to either or .

Proof.

By the trichotomy, the group contains a finite normal subgroup such that is isomorphic to either or . If , then is a subgroup of , contradicting (B). So . Therefore or .

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The maximal infinite dihedral subgroups are self-normalizing, as follows.

Lemma 5.2.

Let be a group satisfying Hypothesis (B).

(1)

If , then .

(2)

If also satisfies Hypothesis (Cii), then satisfies .

Proof.

(1) By Lemma 5.1, is infinite dihedral. That is, there exist subgroups such that and . There is an exact sequence

where is conjugation by .

First, note . By Hypothesis (B), both and are finite. Hence is finite. Next, note , where denotes the switch automorphism on . Thus, since contains an infinite cyclic subgroup of index , it is virtually cyclic. So the image is virtually cyclic. Therefore, since the kernel is finite, the normalizer is virtually cyclic.

(2) Any is infinite dihedral by part (1). By Hypothesis (Cii), is contained in a unique infinite dihedral group, which we will call . By Lemma 5.1, is also a maximal virtually cyclic subgroup. Thus by part (1), it is self-normalizing. Hence holds.

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Observe that Property  implies Hypothesis (B). A partial converse is:

Corollary 5.3.

Suppose satisfies Hypotheses (AB). If is nonempty, then satisfies Property .

Proof.

There exists a contractible manifold equipped with a cocompact proper -action. Let . By Proposition 2.3, the fixed set is a single point, say . Since the action is proper, the isotropy group is finite. Let contain . By Proposition 2.3 again, is nonempty. We must have . Hence . Thus satisfies .

By Hypothesis (B), . By Proposition 2.3, it fixes a single point, which is . Hence . Thus satisfies .

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For any group , ring , and integer , recall the generalized Whitehead groups

A direct sum decomposition of Whitehead groups is in Reference 9, Theorem 5.1(d).

Theorem 5.4.

Let be a group satisfying Property  and the Farrell–Jones Conjecture in algebraic -theory. Then, for each , the inclusion-induced map is an isomorphism:

Here is the set of conjugacy classes of maximal finite subgroups of .

For any connected space , we define .

Corollary 5.5.

Let be a group satisfying Hypotheses (AB) and the Farrell–Jones Conjecture in algebraic K-theory. Suppose is a contractible manifold of dimension equipped with an effective cocompact proper -action. Then, for each , the inclusion-induced map is an isomorphism:

Proof.

Since satisfies Hypotheses (AB), by Corollary 5.3, satisfies . By Theorem 5.4, the induced map is an isomorphism:

Since is a -space so that for , is empty for infinite and is a point for finite, the following function is a bijection:

Since , there are isomorphisms of fundamental groups

Finally, the map from Equation 1 induces a bijection

so that the bijection is given on representatives by group isomorphisms which are compatible with the inclusion maps to .

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Corollary 5.6.

Let be as above. The forgetful map is an isomorphism:

Proof.

For each , there is a commutative diagram with exact rows and :

By Reference 31, Proposition 2.5.1, there is a diagram with exact rows and columns:

Here is the isomorphism of Corollary 5.5 and is a certain group defined by Ranicki Reference 31, p. 166, which must vanish by exactness. Then, via the commutative diagram, is an isomorphism. So, by the Five Lemma, is an isomorphism. The result follows, since

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6. Calculation of the structure groups

For a group with orientation character , let be the Davis–Lück functor Reference 8. This defines a -homology theory which assigns an abelian group to a pair of -spaces and an integer. The “coefficients” are given by .

Our goal in this section is, for a group satisfying Hypotheses (ABC), to identify with and to compute this in terms of UNil-groups. As a byproduct of the computation we will see that the map is a bijection, as promised in Remark 4.4. This will complete the proof of Theorem 1.1.

Lemma 6.1.

Suppose satisfies Hypotheses (ABC) with . Let be a compact -manifold with interior . There is a commutative diagram with long exact rows and vertical isomorphisms:

Thus .

Proof.

The argument is closely analogous to that of Reference 4, Lemma 4.2.

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The next lemma allows us to simplify our families.

Lemma 6.2.

Let be a group.

(1)

.

(2)

if the Farrell–Jones Conjecture in -theory holds for .

Remark 6.3.

Part 2 is simply the modern statement of the Farrell–Jones Conjecture. Part 1 should be contrasted with the corresponding result in -theory: for any group and for any ring (see Reference 10, also Reference 7). Part 1 is given as Lemma 4.2 of Reference 23, however the proof lacks some details, so we give a complete proof in a special case.

Proof of Lemma 6.21 in the case where satisfies Hypothesis (B).

Suppose is a group satisfying Hypothesis (B). By the Farrell–Jones Transitivity Principle (see Reference 24, Theorem 65), and by Lemma 5.1, it suffices to show:

That is, we must show the following Davis–Lück assembly map is an isomorphism:

Since is a simply connected, free -CW complex, by Reference 4, Theorem B.1 (see also Reference 18), this is equivalent to the following Quinn–Ranicki assembly map being an isomorphism:

In other words, we must show the vanishing of Ranicki’s algebraic structure groups:

For all , by Reference 32, Theorem 18.5, there is a bijection:

For all , by the Farrell–Hsiang rigidity theorem Reference 12, Theorem 4.1, the structure set is a singleton. Thus for all . These structure groups are 4-periodic, so .

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Lastly, we recall the identification of Reference 4, Lemma 4.6, done for -theory.

Lemma 6.4 (Connolly–Davis–Khan).

Let and write . The following composite map, starting with Cappell’s inclusion, is an isomorphism:

The next lemma uses excision and is a more abstract version of Reference 4, Lemma 4.5.

Lemma 6.5.

Suppose is a group satisfying Hypotheses (BCii). Let . Assume the orientation character evaluates to on all elements of order two. Then there is an inclusion-induced isomorphism

Proof.

By Lemma 6.2, there are induced isomorphisms

Write for the set of conjugacy classes of maximal virtually cyclic subgroups of . By Lemma 5.2, satisfies . Then, by Reference 25, Corollary 2.8 and the Induction Axiom Reference 24, Prop. 157, Thm. 158(i), there is an isomorphism

By Lemma 5.1, we may replace the index set by . Let . By Lemma 6.2 again, there are induced isomorphisms

Thus there is an isomorphism

Finally, Lemma 6.4 gives the desired conclusion.

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Now we put the pieces together and prove our main theorem.

Proof of Theorem 1.1.

By Theorem 1.3 (whose proof was given in Section 3), the forgetful map is a bijection:

Since Hypothesis (Ciii) holds for -theory, by Lemma 4.3, there is a bijection:

By Remark 4.4 and Corollary 5.6, the following composition is injective:

Since Hypothesis (Ciii) holds for -theory, by Lemma 6.1, there is an isomorphism:

Consider the diagram:

The definitions of all the maps should be clear to the reader. Assuming momentarily that the diagram commutes, we complete the proof of Theorem 1.1. We have already shown that the composition from the upper left of the diagram to the lower left, to the lower middle, and finally to the lower right is a bijection. Furthermore, Remark 4.4 and Corollary 5.6 show that the right vertical map is an injection. It follows that the right vertical map is a bijection. This concludes the proof of Theorem 1.1. Note that, as a byproduct, we have also shown that

is an isomorphism, as promised in Remark 4.4.

Finally, we need to argue that the above diagram is commutative. The left hand square commutes by the definition of the maps. The commutativity of the right square follows from three commutative squares, the first of which is:

Here, the bottom map is the composite of and the identification of Reference 4, Appendix B. The commutativity of the square then follows from Reference 4, Appendix B and Ranicki’s identification of algebraic and geometric structure sets Reference 32, Theorem 18.5.

The second square is:

and the third square is:

These last two squares commute by naturality of all the maps involved.

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The map in Theorem 1.1 is given by applying the Wall realization theorem to the manifold . In particular, if one element of admits a smooth structure, then so do all elements of .

7. Equivariant -cobordisms

Proof of Theorem 1.6.

By Hypothesis (A), there exists a torsion-free subgroup of of finite index. By intersecting with its finitely many conjugates, we may assume is normal in . Write for the finite quotient group.

Let be a locally flat -isovariant -cobordism. By Reference 29, Corollary 1.6 applied to the locally flat -isovariant -cobordism , we obtain that is an -cobordism of manifold homotopically stratified spaces. Observe has only two strata:

Quinn’s stratified torsion for the quotient -cobordism is an element of his obstruction group . (This is a relative homology group with cosheaf coefficients in nonconnective pseudo-isotopy theory; see Reference 27, Definition 8.1.) This obstruction group fits into the exact sequence of the pair that simplifies to:

Here, for any point in the singular set , its orbit and isotropy group are:

By Corollary 5.5, is surjective and is injective. Then Quinn’s obstruction group vanishes: . So, by Reference 29, Theorem 1.8, is a stratified product cobordism.

Let be a homeomorphism of stratified spaces. Each stratum of is a covering space of the corresponding stratum in . So for each , the path,

lifts uniquely to a path in , with . This specifies a bijection,

Granting for a moment that is continuous, it follows that is a -homeomorphism and is a product cobordism. In particular, and are -homeomorphic.

We need only prove that is continuous at each point of . So let . Set .

A basis of neighborhoods of in is given by the collection of sets of the form

where:

(1)

is an open set of containing , and

(2)

is a single component of a -invariant open set of such that

Given such and , we seek a neighborhood of in such that

Let be a connected open neighborhood of in such that

Let be the component of containing . Then , and is connected (since and is connected). Therefore . Also So

as required. This proves is continuous.

■

8. Examples

In this section we give examples of groups satisfying the hypotheses of Theorem 1.1, and hence groups where the question of equivariant rigidity is completely answered.

We first give some examples of groups satisfying Hypotheses (ABC). First and foremost, we mention the group (with ) which was the subject of our previous paper Reference 4. Our second example is the generalization where and where the -action given by is free on . This generalization requires the -theory analysis of our current paper, and if is odd, give examples of groups satisfying the rigidity of Corollary 1.4.

Next suppose is a discrete cocompact subgroup of the isometries of the hyperbolic or Euclidean plane without reflections, thus the normalizers of nontrivial finite subgroups are finite. This is a subject of classical interest and there is a ready supply of examples. Note the product for satisfies Hypotheses (ABC).

The announcement Reference 21 gave examples satisfying Hypotheses (ABC) but not Hypotheses (ABC). The first of these constructions is due to Davis–Januskiewicz, and it is obtained from a triangulated homology sphere with nontrivial fundamental group. The dual cones of the triangulation decompose as a union of contractible closed subcomplexes (“mirrors”), any of whose intersections are either empty or contractible (“submirrors”). This structure gives a right-angled Coxeter group . The reflection trick of Mike Davis allows the construction of a contractible manifold obtained from on which a subgroup of acts pseudo-freely and has 2-torsion. This is not homeomorphic to Euclidean space; see Reference 21, Example 3.2. A second example is given there, derived from a Heisenberg-type group. Unlike the above examples, it is not , but does satisfy Hypotheses (ABC).

Finally, following a suggestion of the referee, we give examples based on Gromov’s technique of hyperbolization Reference 17, Section 3.4.

Lemma 8.1.

For any pseudo-free PL-action of a finite group on a closed PL-manifold , there is an effective, cocompact, proper, isometric PL-action of a discrete group on a complete CAT PL-manifold so that the following conditions are satisfied:

(1)

There is a short exact sequence

of groups with torsion-free.

(2)

The dimensions of and are equal and there is a -isovariant map .

(3)

The action of on satisfies Hypotheses (ABC) of the introduction and hence satisfies the conclusion of Theorem 1.1.

Proof.

We will use the hyperbolization of Davis–Januszkiewicz Reference 11. This hyperbolization is an assignment which, for every finite simplicial complex , assigns a compact polyhedron which is a locally CAT(0), piecewise Euclidean, geodesic metric space and a PL-map and for every simplicial embedding , assigns an isometric PL-embedding onto a totally geodesic subpolyhedron . It satisfies the following properties:

•

(Functoriality) The functorial identities and and the natural transformation identity .

•

(Local structure) If is a closed -simplex of , then is a compact -manifold with boundary and the link of in is PL-homeomorphic to the link of in .

•

.

Consider a pseudo-free PL-action of a finite group on a closed, PL-manifold . After subdivisions, assume that is a simplicial complex, that invariant simplicies are fixed and that the star of the singular set is a regular neighborhood of the singular set. Functoriality gives a -action by PL-isometries on the hyperbolization . Note that is an aspherical manifold: it is a manifold whose dimension equals that of by the local structure property and it is aspherical since its universal cover is a simply-connected complete CAT(0) space, hence contractible. Naturality shows that is -isovariant. Let be the group of homeomorphisms of the universal cover of which cover elements of and let be the fundamental group of . Then clearly the -action on is effective, cocompact, proper, isometric, and PL. Conditions (1) and (2) are satisfied.

Now we turn to Hypotheses (ABC). Part (1) shows Hypothesis (A). Since the hyperbolization of a -simplex is a point by the local structure property, the -action on is pseudo-free, hence so is the -action on . Proposition 2.3 then shows that Hypothesis (B) holds. To see that Hypothesis (Ci) holds, note that , which is a finite complex.

Hypothesis (Cii) states that every infinite dihedral subgroup of lies in a unique maximal infinite dihedral subgroup. Let be an infinite dihedral subgroup of and let where has infinite order and has order 2. Let be the fixed point of the involution . Let

where is the geodesic line segment joining the two points (there is a unique geodesic segment connecting any two points, since is CAT(0)). We now claim that for any and , one has . This holds since interchanges the endpoints of the geodesic segment and must leave the midpoint invariant. Since the fixed sets of all involutions are singletons, must be the midpoint of the geodesic segment. The claim implies that is a geodesic and is invariant under . Let . Then is the unique maximal infinite dihedral subgroup containing .

Hypothesis (Ciii) holds by Reference 2 and Reference 35.

■

Here is an example where the lemma applies. Let be a space form group, that is, a finite group so that for every prime , all subgroups of order and all subgroups of order are cyclic. Then according to Madsen–Thomas–Wall Reference 26, acts freely and smoothly on a sphere for some . By triangulating the quotient, we can assume this action is PL. Then acts semifreely on the suspension with a two-point fixed set.

This is of interest because the isotropy groups of any pseudo-free PL action on a manifold are space form groups, and the construction above shows that all such groups arise as isotropy groups.

Acknowledgments

The authors thank the referee for suggesting to use hyperbolization to find examples and also Mike Davis for discussions on this topic.

Mathematical Fragments

Theorem 1.1 (Main result).

Let be a group satisfying Hypotheses (ABC) or more generally (ABC) below, with of virtual cohomological dimension . Write . There is a bijection

The set of conjugacy classes of maximal infinite dihedral subgroups of  is .

Theorem 1.3.

Suppose satisfies Hypotheses (AB). Assume there is an element such that . Then the forgetful maps are bijections:

Corollary 1.4.

Suppose satisfies Hypotheses (ABC) with the dimension of at least five. Assume has no elements of order two. Then any contractible manifold equipped with an effective cocompact proper -action has the -homotopy type of , and any -homotopy equivalence is -homotopic to a -homeomorphism.

Proposition 1.5 (cf. Theorem 5.4).

Suppose satisfies Hypothesis (B). Then

Theorem 1.6.

Suppose satisfies Hypotheses (AB) and the Farrell–Jones Conjecture in -theory. Assume and are cocompact manifold models for of dimension at least six. If there exists a locally flat isovariant -cobordism, , between and , then is a product cobordism and and are equivariantly homeomorphic.

Lemma 2.2.

Suppose a finite group acts on a set so that for all prime-order subgroups of , the fixed set is a singleton. Then is a singleton.

Proposition 2.3.

Suppose satisfies Hypothesis (A). Assume is a contractible manifold equipped with an effective cocompact proper -action. The action is pseudo-free if and only if Hypothesis (B) holds. Furthermore, if either of these equivalent conditions holds, then for a subgroup of , the fixed point set is empty if is infinite and a point if is nontrivial and finite.

Proposition 2.5.

Let be a discrete group. Any topological manifold equipped with a cocompact pseudo-free -action has the -homotopy type of a -CW complex.

Equation (1)
Corollary 2.6.

Suppose satisfies Hypotheses (AB). Any contractible manifold equipped with an effective cocompact proper -action is a model for . Hence the forgetful map of Theorem 1.3 is bijective.

Theorem 3.1 (Reference 4, Theorem A.3).

Let be a finite group. Let and be compact pseudo-free -manifolds. Assume . Let be a -map such that the restriction is bijective.

(1)

If is -connected, then is -homotopic to an isovariant map.

(2)

If and are -isovariant, -connected, and -homotopic, then is -isovariantly homotopic to .

Proposition 3.2.

The map of Theorem 1.3 is bijective.

Lemma 4.1.

Let be a manifold of dimension greater than four, and let . Let be a compact neighborhood of in so that is homeomorphic to where is a manifold. Then is homeomorphic to a disc.

Lemma 4.2.

Suppose satisfies Hypotheses (ABC) with . Let . There is a compact -manifold with interior . Furthermore, the action is locally conelike.

Lemma 4.3.

Suppose satisfies Hypotheses (ABC) and . There is a bijection

Remark 4.4.

Following Ranicki Reference 32, Thm. 18.5, there is a pointed map

which is a bijection for . The map is called the total surgery obstruction for homotopy equivalences. In our case, it is also a bijection when , since each connected component of is a 4-dimensional spherical space form, with finite fundamental group, and so can be included with the high-dimensional () case by Freedman–Quinn topological surgery Reference 16. Since is -dimensional, by the Atiyah–Hirzebruch spectral sequence, we obtain:

Hence there is an exact sequence:

At the end of the proof of Theorem 1.1 in Section 6, we will show that the last map is zero. In the meantime, we will proceed to compute the nonconnective algebraic structure group .

Lemma 5.1.

Suppose is a group satisfying Hypothesis (B). If , then is isomorphic to either or .

Lemma 5.2.

Let be a group satisfying Hypothesis (B).

(1)

If , then .

(2)

If also satisfies Hypothesis (Cii), then satisfies .

Corollary 5.3.

Suppose satisfies Hypotheses (AB). If is nonempty, then satisfies Property .

Theorem 5.4.

Let be a group satisfying Property  and the Farrell–Jones Conjecture in algebraic -theory. Then, for each , the inclusion-induced map is an isomorphism:

Here is the set of conjugacy classes of maximal finite subgroups of .

Corollary 5.5.

Let be a group satisfying Hypotheses (AB) and the Farrell–Jones Conjecture in algebraic K-theory. Suppose is a contractible manifold of dimension equipped with an effective cocompact proper -action. Then, for each , the inclusion-induced map is an isomorphism:

Corollary 5.6.

Let be as above. The forgetful map is an isomorphism:

Lemma 6.1.

Suppose satisfies Hypotheses (ABC) with . Let be a compact -manifold with interior . There is a commutative diagram with long exact rows and vertical isomorphisms:

Thus .

Lemma 6.2.

Let be a group.

(1)

.

(2)

if the Farrell–Jones Conjecture in -theory holds for .

Lemma 6.4 (Connolly–Davis–Khan).

Let and write . The following composite map, starting with Cappell’s inclusion, is an isomorphism:

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Article Information

MSC 2010
Primary: 57S30 (Discontinuous groups of transformations), 57R91 (Equivariant algebraic topology of manifolds)
Secondary: 19J05 (Finiteness and other obstructions in ), 19J25 (Surgery obstructions)
Author Information
Frank Connolly
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
connolly.1@nd.edu
MathSciNet
James F. Davis
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
jfdavis@indiana.edu
MathSciNet
Qayum Khan
Department of Mathematics, Saint Louis University, St. Louis, Missouri 63103
khanq@slu.edu
ORCID
MathSciNet
Additional Notes

The authors were partly supported by the NSF (DMS-0601234, DMS-1210991, DMS-0904276).

Journal Information
Transactions of the American Mathematical Society, Series B, Volume 2, Issue 4, ISSN 2330-0000, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , , and published on .
Copyright Information
Copyright 2015 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
Article References
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  • DOI 10.1090/btran/9
  • MathSciNet Review: 3427570
  • Show rawAMSref \bib{3427570}{article}{ author={Connolly, Frank}, author={Davis, James}, author={Khan, Qayum}, title={Topological rigidity and actions on contractible manifolds with discrete singular set}, journal={Trans. Amer. Math. Soc. Ser. B}, volume={2}, number={4}, date={2015}, pages={113-133}, issn={2330-0000}, review={3427570}, doi={10.1090/btran/9}, }

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