Topological rigidity and actions on contractible manifolds with discrete singular set

The problem of equivariant rigidity is the $\Gamma$-homeomorphism classification of $\Gamma$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $\Gamma$. In other words, this is the classification of cocompact $E_{fin}\Gamma$-manifolds. We use surgery theory, algebraic $K$-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 $E_{fin}$-condition is always satisfied.

When Γ is a torsion-free group of isometries of a nonpositively curved Riemannian manifold M , with [M, Γ] ∈ S(Γ), an inspirational result of Farrell and Jones [13] says that S(Γ) consists of the single element [M, Γ]. (See [2] for the generalization to the CAT(0) case.) In these cases, M sing is empty.
In this paper we compute S(Γ) when Γ is a group of isometries of a nonpositively curved Riemannian manifold M where M sing is discrete (that is, the action Γ M is pseudo-free). One may speculate that S(Γ) consists of just one element. However, for many years experts have suspected that Cappell's UNil-groups (see Quinn [28]) obstruct this hope. We fully calculate S(Γ). (C) There is a contractible Riemannian manifold X of nonpositive sectional curvature with an effective cocompact proper action Γ X by isometries. A more general but less geometric condition than (C) is (C ′ ), consisting of: (C ′ i) There exists an element [X, Γ] ∈ S(Γ) for which the quotient space X f ree /Γ has the homotopy type of a finite CW complex. (C ′ ii) Each infinite dihedral subgroup of Γ lies in a unique maximal infinite dihedral subgroup. (C ′ iii) Γ satisfies the Farrell-Jones Conjecture in K-theory and L-theory. Remark 1.2. For comparative purposes, we make some comments here.
• Proposition 2.3 shows that if Γ satisfies Hypothesis (A) and [M, Γ] ∈ S(Γ), then Γ satisfies Hypothesis (B) if and only if M sing is a discrete set. • Suppose Γ satisfies Hypotheses (AB). Corollary 2.6 shows that any two Γ-manifolds in S(Γ) model E fin Γ and hence are Γ-homotopy equivalent. • (ABC) =⇒ (ABC ′ ): Suppose Γ satisfies Hypotheses (AB) and X satisfies Hypothesis (C). Then clearly [X, Γ] ∈ S(Γ) and X f ree /Γ has the homotopy type of the finite CW complex (X − N (X sing ))/Γ. Thus Hypothesis (C ′ i) holds. Let ∆ be a infinite dihedral subgroup of Γ. Then ∆ stabilizes a unique geodesic γ, 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 γ is the unique maximal infinite dihedral subgroup of Γ containing ∆. Thus Hypothesis (C ′ ii) holds. Finally Hypothesis (C ′ iii) holds by [2] and [35]. • In the proof of Theorem 1.1 we will only refer to Hypotheses (ABC ′ ).
We explain how S(Γ) is the isovariant structure set S iso (X, Γ) of any [X, Γ].
Definition. Let (X, Γ) be a cocompact Γ-manifold. A structure on (X, Γ) is a Γ-equivariant homotopy equivalence f : M → X of proper cocompact Γ-manifolds. If f is an isovariant homotopy equivalence, we call it an isovariant structure. Two such structures (resp. isovariant structures), (M, f, Γ) and (M ′ , f ′ , Γ) are equivalent if there is an equivariant homeomorphism h : M → M ′ and a Γ-equivariant homotopy (resp. Γ-isovariant homotopy) from f ′ • h to f . Define the structure set S(X, Γ) (resp. isovariant structure set S iso (X, Γ)) as the set of equivalence classes of structures (resp. isovariant structures) on (X, Γ). Theorem 1.3. Suppose Γ satisfies Hypotheses (AB). Assume there is an element [X n , Γ] ∈ S(Γ) such that n 4. Then the forgetful maps are bijections: Corollary 1.4. Suppose Γ satisfy Hypotheses (ABC ′ ) with the dimension of X at least five. Assume Γ has no elements of order two. Then any contractible manifold M equipped with an effective cocompact proper Γ-action has the Γ-homotopy type of X, and any Γ-homotopy equivalence f : M → X is Γ-homotopic to a Γhomeomorphism.
Proof. This follows from Theorems 1.
This vanishing result can be interpreted geometrically, as follows. Recall from [4, Section 1] that a cocompact manifold model for E fin Γ is a Γ-space with the equivariant homotopy type of a Γ-CW complex such that, for all subgroups H of Γ, the H-fixed set is a contractible manifold if H is finite and is empty if H is infinite. A Γ-space W is locally flat if W H ⊂ W K implies W H is a locally flat submanifold of W K . For our h-cobordism rigidity, we do not assume that Γ acts locally linearly. Theorem 1.6. Suppose Γ satisfies Hypotheses (AB) and the Farrell-Jones Conjecture in K-theory. Assume M and M ′ are cocompact manifold models for E fin Γ of dimension at least six. If there exists a locally flat isovariant h-cobordism, W , between M and M ′ , then W is a product cobordism and M and M ′ 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, (X, ∂X). 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 K-theory decorations and pass to S −∞ . In Section 6, we calculate these structure groups using the Farrell-Jones Conjecture and the other axioms for Γ, 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.
Given any Γ-CW complex X with finite isotropy groups, there is an equivariant map X → E fin Γ, unique up to equivariant homotopy. It follows that any two models are Γ-homotopy equivalent. Furthermore, a model E fin Γ exists for any group Γ. We show that if a group Γ satisfies Hypotheses (AB) and acts effectively, cocompactly, and properly on a contractible manifold, then the manifold models E fin Γ.
The following lemma is not logically necessary, but nonetheless provides motivation.
Lemma 2.1. Let Γ be a virtually torsion-free group with an effective cocompact proper action by isometries on a contractible Riemannian manifold X of nonpositive sectional curvature (that is, Hypotheses (AC) hold). Then X models E fin Γ.
Proof. Let Γ 0 be a finite index normal subgroup of Γ with quotient group G. The manifold X is complete since X/Γ 0 is compact, hence complete. For an infinite subgroup H of Γ, the fixed set X H is empty by properness. For a finite subgroup H of Γ, the fixed set X H is nonempty by the Cartan Fixed Point Theorem [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 G acts smoothly on the closed manifold X/Γ 0 , this is a G-CW complex [20], and so, by lifting cells, X is a Γ-CW complex.
Now it remains to show that if Γ satisfies Hypotheses (AB) and if M is a contractible Γ-manifold, then the fixed point set of each finite subgroup of Γ is a point and M has the Γ-homotopy type of a Γ-CW complex; hence M will model E fin Γ. We start with a group-theoretic lemma.
Lemma 2.2. Suppose a finite group G acts on a set Y so that for all prime-order subgroups P of G, the fixed set Y P is a singleton. Then Y G is a singleton.
Proof. Let C be the collection of nontrivial subgroups of G whose fixed set is a singleton. Note that C satisfies the normal extension property: On the other hand, since N ∈ C, we see the singleton Y N is an H-invariant by normality, so Y N ⊂ Y H . So Y H is a point and H ∈ C. In particular C contains all cyclic subgroups of G.
We next claim that, if a and b are elements of order 2 in G, then X a = X b = X a,b . For the cyclic group ab is in C, and ab ⊳ a, b . So a, b ∈ C. Therefore X a and X b are the same point X a,b . It follows that this point is the fixed set of the normal subgroup N generated by all elements of order 2 in G. This subgroup is nontrivial if G has even order. So by the normal extension property, Y G is a singleton if G has even order.
Finally suppose G has odd order. It that order is a prime, then G is in C. If not then by the Feit-Thompson Theorem [14], there is a nontrivial proper normal subgroup and we proceed inductively to conclude that G ∈ C.
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.   For if G is an odd order subgroup of Γ, the argument in Lemma 2.2 shows that each nontrivial Sylow p-subgroup P of G fixes exactly one point (use the normal extension property and the nilpotency of P ). Since P acts freely on the homology sphere X − X P , it follows that P has periodic cohomology and is therefore cyclic, since p is odd. As is well known, this implies that G is metacyclic. So again by the normal extension property, since G has a normal cyclic subgroup, we see G ∈ C and X G is a singleton.
Here is a general tameness result pertaining to all of the actions that we consider. Proposition 2.5. Let Γ be a discrete group. Any topological manifold M equipped with a cocompact pseudo-free Γ-action has the Γ-homotopy type of a Γ-CW complex.
Then by Siebenmann [33], if dim M 5, the ends of M f ree /Γ × S 1 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 W such that Write W and ∂W for the pullback Γ × Z 5 -covers of W and ∂W of the cover M f ree ×R 5 → M f ree /Γ×T 5 . Observe Γ acts properly on the set Π 0 ∂W of connected components of ∂W . Then, since Γ M is pseudo-free, we can define a Γ-map where C(s) ∈ Π 0 ∂W is the connected component containing a point s. Consider the equivalence relation on M × T 5 given by (s, z) ∼ (s, z ′ ) if s ∈ M sing . This is preserved by the Γ-action. Write M 1 = (M × T 5 )/ ∼ for the quotient. As Γ-spaces, Here (∂f ) −1 denotes the Γ-cover of a homotopy-inverse for ∂f .
Finally, choose a point * ∈ T 5 and consider the Γ-inclusion and corresponding Γ-retraction  Proof. By Proposition 2.3, the H-fixed sets of M are single points if H is a nontrivial finite subgroup of Γ, otherwise the H-fixed sets are empty if H is infinite. Hence M is a pseudo-free Γ-manifold. Then, by Proposition 2.5, M has the Γ-homotopy type of a Γ-CW complex. Therefore, since M is contractible, it models E fin Γ. The remainder follows from universal properties of classifying spaces.

From Equivariance to Isovariance
We recall our earlier result about improving equivariant maps: . Let G be a finite group. Let A n and B n be compact pseudo-free G-manifolds. Assume n 4. Let f : A → B be a G-map such that the restriction f | Asing : A sing → B sing is bijective.
(1) If f is 1-connected, then f is G-homotopic to an isovariant map.
(2) If f = f 0 and f 0 , f 1 : A → B are G-isovariant, and 2-connected, and G-homotopic, then f 0 is G-isovariantly homotopic to f 1 . M and Γ 0 X are free, proper, and cocompact, the quotients M/Γ 0 and X/Γ 0 are compact G-manifolds. By Theorem 3.1 applied to both the G-homotopy equivalence f /Γ 0 : M/Γ 0 → X/Γ 0 and any choice of By the Covering Homotopy Property applied to the cover M → M/Γ 0 , the G-homotopy from f /Γ 0 to g is covered by a unique homotopy F : M × I → X from f : M → X, to someĝ : M → X covering g. By uniqueness of path lifting, it follows that F is Γ-equivariant. Then, sinceĝ is Γ-equivariant and g is G-isovariant, an elementary diagram chase shows thatĝ is a Γ-isovariant map. Sinceĝ is a Γ-equivariant map covering a G-isovariant homotopy equivalence g, by similar reasoning it follows thatĝ is a Γ-isovariant homotopy equivalence, unique up to Γ-isovariant homotopy.
Thus ψ is surjective. Furthermore, since g is unique up to G-isovariant homotopy andĝ is unique up to Γ-isovariant homotopy, ψ is injective.
Proof of Theorem 1.3. Immediate from Corollary 2.6 and Proposition 3.2.

Reduction to classical surgery theory
In this section we show that each cocompact proper contractible Γ-manifold [M, Γ] in S(Γ) is well-behaved in a neighborhood of the discrete set M sing . We use this to interpret S iso (X, Γ) 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 A be a manifold of dimension greater than four, and let a ∈ A.
Denote the closed cone on a space S by cS : Proof. There is a homeomorphism ϕ : B × [0, ∞) → C − {a}. By uniqueness of the one-point compactification of a space, there are basepoint-preserving homeomorphisms: where the second isomorphism follows from the Seifert-van Kampen Theorem applied to the decomposition C = Int D ∪ (C − {a}) with D a closed disc neighborhood of a in the manifold Int C. Excision shows that ∂D ֒→ (C − Int D) induces an isomorphism on homology. Thus (C − Int D; ∂D, B) is an h-cobordism, and the h-cobordism theorem [34,16] shows that is C − Int D is homeomorphic to the product ∂D × I. Hence B ∼ = ∂D, a sphere and C ∼ = D, a disc.
An action of a group Γ on a space X is locally conelike if every orbit Γx has a Γ-neighborhood that is Γ-homeomorphic to Γ × Γx cS(x) for some Γ x -space S(x).
Proof. Since M sing /Γ is compact and since M sing is discrete by Proposition 2.3, M sing /Γ is a finite set. Then the manifold M 0 := M f ree /Γ has finitely many ends, one for each orbit Γp 1 , . . . , Γp m of M sing . Write Γ j := Γ pj for the isotropy groups.
Since the ends of M f ree are tame, and each end of M 0 is finitely covered by an end of M f ree , Propositions 2.6 and 3.6 of [29] shows the ends of M 0 are tame. If M 0 were smooth, then Siebenmann's thesis [33] gives a CW structure on M 0 which is a union of subcomplexes M 0 = K ∪ E 1 ∪ · · · ∪ E m with K a finite complex and each E i a connected, finitely dominated complex with π 1 E j = Γ j . For M 0 a topological manifold, it follows from [22, p123, Theorem III:4.1.3] that there is a proper homotopy equivalence M 0 ≃ K ∪ E 1 ∪ · · · ∪ E m with K and E j as above. Let Wh 0 (Γ j ) denote the reduced projective class group K 0 (ZΓ j ) and σ(E j ) ∈ Wh 0 (Γ j ) the Wall finiteness obstruction.
Consider inclusion-induced map of projective class groups: Since K is a finite complex, the sum theorem of [33] shows On the one hand, X 0 := X f ree /Γ has the homotopy type of a finite CW complex by Therefore the action of Γ on M is locally conelike.
By Lemma 4.2, we may choose a compact ∂-manifold X with interior X f ree /Γ. Note π 1 (X) ∼ = Γ. Enumerate the connected components of the boundary: Observe each ∂ j X has universal cover homeomorphic to S n−1 and π 1 (∂ j X) ∼ = Γ j .  Here + denotes one-point compactification. Then we obtain a homeomorphism . We now show that Φ has a two-sided inverse: where the map f ′ extends f and has image in ∂X. Therefore Ψ is well-defined. It is now straightforward to see that Ψ is a two-sided inverse of Φ.
The geometric structure set S h TOP (X, ∂X) can be identified with a version of Ranicki's algebraic structure group for the same pair. The connective algebraic structure groups S h * are the homotopy groups of the homotopy cofiber of a assembly map α 1 , and so they fit into an exact sequence of abelian groups [31]: where L 1 is the 1-connective algebraic L-theory spectrum of the trivial group. Exactly as in [4], for computations we shall use the non-connective, periodic analogue: where L is the 4-periodic algebraic L-theory spectrum of the trivial group.  which is a bijection for n > 5. The map s is called the total surgery obstruction for homotopy equivalences. In our case, it is also a bijection when n = 5, since each connected component of ∂X is a 4-dimensional spherical space form, with finite fundamental group, and so can be included with the high-dimensional (n > 5) case by Freedman-Quinn topological surgery [15]. Since X is n-dimensional, by the Atiyah-Hirzebruch spectral sequence, we obtain: 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 non-connective algebraic structure group S per,h n+1 (X, ∂X).

Reduction from h to −∞ structure groups
Our goal in this section is to replace the h decoration by −∞ in the structure group by showing S per,h n+1 (X, ∂X) ∼ = S per,−∞ n+1 (X, ∂X). 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 V : (I) V is finite (II) there is an exact sequence with F finite and C ∞ the infinite cyclic group: (III) there is an exact sequence with F finite and D ∞ the infinite dihedral group: For the rest of the paper, we consider the following increasing chain of classes F : Proof. By the trichotomy, the group V contains a finite normal subgroup F such that V /F is isomorphic to either C ∞ or D ∞ . If F = 1, then V is a subgroup of The maximal infinite dihedral subgroups are self-normalizing, as follows.
(1) By Lemma 5.1, V is infinite dihedral. That is, there exist subgroups There is an exact sequence . By Hypothesis (B), both C Γ (V 1 ) and C Γ (V 2 ) are finite. Hence C Γ (V ) is finite. Next, note Aut(V ) = Inn(V ) ⋊ sw ∼ = (C 2 * C 2 )⋊ sw C 2 , where sw denotes the switch automorphism on V ∼ = C 2 * C 2 . Thus, since Aut(V ) contains an infinite cyclic subgroup of index 4, it is virtually cyclic. So the image Im(p) is virtually cyclic. Therefore, since the kernel Ker(p) = C Γ (V ) is finite, the normalizer N Γ (V ) is virtually cyclic.
(2) Any V ∈ vc(Γ) − fbc(Γ) is infinite dihedral by part (1). By Hypothesis (C ′ ii), V is contained in a unique infinite dihedral group, which we will call V max . By Lemma 5.1, V max is also a maximal virtually cyclic subgroup Thus by part (1), it self-normalizing. Hence N M fbc⊂vc holds.
Observe that Property N M 1⊂fin implies Hypothesis (B). A partial converse is: Theorem 5.4. Let Γ be a group satisfying Property N M 1⊂fin and the Farrell-Jones Conjecture in algebraic K-theory. Then, for each q ∈ Z, the inclusion-induced map is an isomorphism: Here (mfin)(Γ) is the set of conjugacy classes of maximal finite subgroups of Γ.
Corollary 5.5. Let Γ be a group satisfying Hypotheses (AB) and the Farrell-Jones Conjecture in algebraic K-theory. Suppose X is a contractible manifold of dimension 3 equipped with an effective cocompact proper Γ-action. Then, for each q ∈ Z, the inclusion-induced map is an isomorphism: Wh q (X).
Proof. Since Γ satisfies Hypotheses (AB), by Corollary 5.3, Γ satisfies N M 1⊂fin . By Theorem 5.4, the induced map is an isomorphism: Since X is a Γ-space so that for 1 = H < Γ, X H is empty for H infinite and X H is a point for H finite, the following function is a bijection: Since dim X 3, there are isomorphisms of fundamental groups Finally, the map e from (1) induces a bijection so that the bijection α • β is given on representatives by group isomorphisms which are compatible with the inclusion maps to Γ.  Lemma 6.1. Suppose Γ satisfy Hypothesis (ABC ′ ) with dim X 5. Let X be a compact ∂-manifold with interior X f ree /Γ. There is a commutative diagram with long exact rows and vertical isomorphisms: Thus S per,−∞ * (X, ∂X) ∼ = H Γ * (E all Γ, E fin Γ; L). Proof. The argument is closely analogous to that of [4,Lemma 4.2].
The next lemma allows us to simplify our families.   (2) is simply the modern statement of the Farrell-Jones Conjecture. Part (1) should be contrasted with the corresponding result in K-theory: H Γ * (E vc Γ, E fbc Γ; K R ) = 0 for any group Γ and for any ring R (see [10], also [7]). Part (1) is given as Lemma 4.2 of [23], however the proof lacks some details, so we give a complete proof in a special case.
Proof of Lemma 6.2 (1) in the case where Γ satisfies Hypothesis (B). Suppose Γ is a group satisfying Hypothesis (B). By the Farrell-Jones Transitivity Principle (see [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 R is a simply connected, free C ∞ -CW complex, by [4, Theorem B.1] (see also [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: S per * (S 1 ) = 0. For all n 4, by [30,Theorem 18.5], there is a bijection: For all n 4, by the Farrell-Hsiang rigidity theorem [12,Theorem 4.1], the structure set S TOP (S 1 × D n rel ∂) is a singleton. Thus S per k (S 1 ) = S k (S 1 ) = 0 for all k 6. These structure groups are 4-periodic, so S per * (S 1 ) = 0.
Lastly, we recall the identification of [4,Lemma 4.6], done for L-theory.
Lemma 6.4 (Connolly-Davis-Khan). Let n ∈ Z and write ε := (−1) n . The following composite map, starting with Cappell's inclusion, is an isomorphism: The next lemma uses excision and is a more abstract version of [4, Lemma 4.5].
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 (C ′ iii) holds for K-theory, by Lemma 4.3, there is a bijection: S h TOP (X, ∂X) −→ S iso (X, Γ). By Remark 4.4 and Corollary 5.6, the following composition is injective: (X, ∂X).
Since Hypothesis (C ′ iii) holds for L-theory, by Lemma 6.1, there is an isomorphism: Consider the diagram: is an isomorphism, as promised in Remark 4.4.

F CONNOLLY, J F DAVIS, AND Q KHAN
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 right square follows from three commutative squares, the first of which is: Here, the bottom map is the composite of The commutativity of the square then follows from [4, Appendix B] and Ranicki's identification of algebraic and geometric structure sets [30,Theorem 18.5].
The second square is and the third square is These last two squares commute by naturality of all the maps involved.
The map ∂ in Theorem 1.1 is given by applying the Wall realization theorem to the manifold X. In particular, if one element of S(Γ) admits a smooth structure then so do all elements of S(Γ).

Equivariant h-cobordisms
Proof of Theorem 1.6. By Hypothesis (A), there exists a torsion-free subgroup Γ 0 of Γ of finite index. By intersecting Γ 0 with its finitely many conjugates, we may assume Γ 0 is normal in Γ. Write G := Γ/Γ 0 for the finite quotient group.
Let Quinn's stratified torsion for the quotient h-cobordism is an element of his obstruction group H 1 (cδM n , δM n ; P(q n )). (This is a relative homology group with cosheaf coefficients in non-connective pseudo-isotopy theory; see [27,Definition 8.1].) This obstruction group fits into the exact sequence of the pair that simplifies to: Here, for any point x in the singular set M sing , its orbit and isotropy group are: By Corollary 5.5, A 1 is surjective and A 0 is injective. Then Quinn's obstruction group vanishes: H 1 (cδM n , δM n ; P(q n )) = 0. So, by [29,Theorem 1.8], W/Γ is a stratified product cobordism.
Let f : M/Γ × (I; 0, 1) → (W/Γ; M/Γ, M ′ /Γ) be a homeomorphism of stratified spaces. Each stratum of W is a covering space of the corresponding stratum in W/Γ. So for each x ∈ M , the path, lifts uniquely to a path in F x : I → W , with F x (0) = x. This specifies a bijection, Granting for a moment that F is continuous, it follows that F is a Γ-homeomorphism and W is a product cobordism. In particular, M and M ′ are Γ-homeomorphic.
We need only prove that F is continuous at each point of M sing × I. So let (x 0 , t) ∈ M sing × I. Set y 0 = F (x 0 , t).
A basis of neighborhoods of y 0 in W is given by the collection of sets of form where: (1) O is an open set of I containing t, and Given such U 0 and O then, we seek a neighborhood N 0 of x 0 in M such that Let N 0 be the component of (π M ) −1 (N ) containing x 0 . Then F (N 0 × I) ⊂ U , and F (N 0 × I) is connected (since F (N 0 × 0) = N 0 and N 0 is connected). Therefore as required. This proves F is continuous.

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 Γ satisfying Hypotheses (ABC). First and foremost, we mention the group Γ = Z n ⋊ −1 C 2 (with n 5) which was the subject of our previous paper [4]. Our second example is the generalization Γ = Z n ⋊ ρ C m where n 5 and where the C m -action given by ρ is free on Z n − {0}. This generalization requires the K-theory analysis of our current paper, and if m is odd, give examples of groups satisfying the rigidity of Corollary 1.4.
Next suppose G 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 Γ = G r for r 3 satisfies Hypotheses (ABC).
The announcement [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 Σ with nontrivial fundamental group. The dual cones of the triangulation decompose Σ 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 W . The reflection trick of Mike Davis allows the construction of a contractible manifold X obtained from Σ on which a subgroup Γ of W acts pseudo-freely and has 2-torsion. This X is not homeomorphic to Euclidean space; see [21, Lemma 8.1. For any pseudo-free PL-action of a finite group G on a closed PLmanifold K, there is a effective, cocompact, proper, isometric PL-action of a discrete group Γ on a complete CAT(0) PL-manifold X so that the following conditions are satisfied.
(1) There is a short exact sequence of groups with Γ 0 torsion-free. Proof. We will use the hyperbolization of Davis-Januszkiewicz [11]. This hyperbolization is an assignment which, for every finite simplicial complex K, assigns a compact polyhedron h(K) which is a locally CAT(0), piecewise Euclidean, geodesic metric space and a PL-map p(K) : h(K) → K and for every simplicial embedding f : K → L, assigns an isometric PL-embedding onto a totally geodesic subpolyhedron h(f ) : h(K) → h(L). It satisfies the following properties: • (Functoriality) The functorial identities h(id K ) = id h(K) and h(f • g) = h(f ) • h(g) and the natural transformation identity p(L) • h(f ) = f • p(K).
• (Local structure) If σ n is a closed n-simplex of K then h(σ n ) is a compact nmanifold with boundary and the link of h(σ n ) in h(K) is PL-homeomorphic to the link of σ n in K. • h(point) = point.
Consider a pseudo-free PL-action of a finite group G on a closed, PL-manifold K. After subdivisions, assume that K 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 G-action by PL-isometries on the hyperbolization h(K). Note that h(K) is an aspherical manifold: it is a manifold whose dimension equals that of K 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 p(K) : h(K) → K is G-isovariant. Let Γ be the group of homeomorphisms of the universal cover X of h(K) which cover elements of G and let Γ 0 be the fundamental group of X. Then clearly the Γ-action on X 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 0-simplex is a point by the local structure property, the Gaction on h(K) is pseudo-free, hence so is the Γ-action on X. Proposition 2.3 then shows that hypothesis (B) holds. To see that hypothesis (C ′ i) holds, note that X f ree /Γ = h(K) f ree /G = (h(K) − h(K) sing )/G ≃ (h(K) − int star h(K) sing )/G, which is a finite complex.
Hypothesis (C ′ ii) states that every infinite dihedral subgroup of Γ lies in a unique maximal infinite dihedral subgroup. Let ∆ be an infinite dihedral subgroup of Γ and let ∆ = t ⋊ a where t has infinite order and a has order 2. Let x n ∈ X be the fixed point of the involution t n a ∈ ∆ ⊂ Γ. Let γ = n∈Z x n x n+1 , where x n x n+1 is the geodesic line segment joining the two points (there is a unique geodesic segment connecting any two points, since X is CAT(0)). We now claim that for any n and k, one has x n+k ∈ x n x n+2k . This holds since t n+k a interchanges the end points of the geodesic segment x n x n+2k and must leave the midpoint invariant. Since the fixed sets of all involutions are singletons, x n+k must be the midpoint of the geodesic segment. The claim implies that γ is a geodesic and is invariant under ∆. Let ∆ = {g ∈ Γ | g(γ) = γ}. Then ∆ is the unique maximal infinite dihedral subgroup containing ∆.
Here is an example where the lemma applies. Let G be a space form group, that is, a finite group so that for every prime p, all subgroups of order p 2 and all subgroups of order 2p are cyclic. Then according to Madsen-Thomas-Wall [26], G acts freely and smoothly on a sphere S n for some n. By triangulating the quotient, we can assume this action is PL. Then G acts semifreely on the suspension K = ΣS n 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.