aa r X i v : . [ m a t h . G T ] F e b TOPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP
KUN WANG
Abstract.
We call a group FJ if it satisfies the K - and L -theoretic Farrell-Jones conjecture withcoefficients in Z . We show that if G is FJ, then the simple Borel conjecture (in dimensions ≥ G ⋊ Z . If in addition W h ( G × Z ) = 0, which is true for all knowntorsion free FJ groups, then the bordism Borel conjecture (in dimensions n ≥
5) holds for G ⋊ Z .One of the key ingredients in proving these rigidity results is another main result, which says that ifa torsion free group G satisfies the L -theoretic Farrell-Jones conjecture with coefficients in Z , thenany semi-direct product G ⋊Z also satisfies the L -theoretic Farrell-Jones conjecture with coefficientsin Z . Our result is indeed more general and implies the L -theoretic Farrell-Jones conjecture withcoefficients in additive categories is closed under extensions of torsion free groups. This enables usto extend the class of groups which satisfy the Novikov conjecture. Introduction
In the area of manifold topology, one of the most intriguing unsolved problem is the Borelconjecture on topological rigidity of closed aspherical manifolds. Recall that a manifold is called aspherical if its universal cover is contractible.
The Borel Conjecture . Every closed aspherical manifold is topologically rigid. That is, every homo-topy equivalence between any two closed aspherical manifolds is homotopic to a homeomorphism.Note that every homeomorphism between two closed manifolds is simple, i.e. its Whiteheadtorsion vanishes. This follows from a theorem of Chapman [10] which says that every homeomor-phism between two finite CW -complexes is simple. Therefore, it is reasonable to study the simpleversion of the Borel conjecture, which is the converse (up to homotopy) to Chapman’s topologicalinvariance theorem of Whitehead torsion for closed aspherical manifolds: The Simple Borel Conjecture.
Every closed aspherical manifold is simply topologically rigid. Thatis, every simple homotopy equivalence between any two closed aspherical manifolds is homotopicto a homeomorphism.Clearly, the simple Borel conjecture is simpler than the Borel conjecture. The passage fromthe simple Borel conjecture to the Borel conjecture is the famous conjecture which states that theWhitehead group of any torsion free group vanishes (note that the fundamental group of a closedaspherical manifold is torsion free).
Another type of rigidity question one can ask for aspherical manifold is the following bordismtype rigidity:
The Bordism Borel Conjecture:
Every closed aspherical manifold is bordismly topologically rigid.That is, every homotopy equivalence f : N −→ M from another closed manifold N to M is h -cobordant to the identity 1 : M −→ M .Recall that a cobordism ( W ; M , M ) is called an h -corbordim if the inclusions M ֒ → W, M ֒ → W are homotopy equivalence. Two homotopy equivalences f i : M i −→ X, i = 1 , h-cobordant if there is an h -cobordism ( W ; M , M ) and a homotopy equivalence ( F ; f , f ) :( W ; M , M ) −→ ( X × [0 , X × , X × f to be simpleand h -cobordant to be s -cobordant in the bordism Borel conjecture, we then get the s -cobordismversion of the Borel conjecture, which by the s -cobordism theorem, is just the simple Borel conjec-ture in dimensions ≥ G if either G cannot be realized asthe fundamental group of a closed aspherical manifold, or the respective conjecture holds for everyclosed aspherical manifold with fundamental group isomorphic to G . It is an interesting problemto identify which groups can be realized as the fundamental group of a closed aspherical manifold.Conjecturally, a group has this property if and only if the group is a finitely presented Poincar´eduality group. This conjecture is usually referred to as Wall’s conjecture (though Wall [29] did notinclude the finitely presented condition, which would otherwise make the conjecture false). See [12]for a survey.In recent years, there has been significant progress on the proof of the Borel conjecture for a largeclass of groups, due to the solutions of the Farrell-Jones conjecture (FJC for short) for these groups.See the works by many authors [2],[5],[6],[7],[31],[30],[22],[18],[16],[15],[14],[28]. Roughly speaking,the conjecture says that the algebraic K - and L -groups K n ( Z [ G ]) , L n ( Z [ G ]) , n ∈ Z of the integralgroup ring Z [ G ] of a group G is determined by those of its virtually cyclic subgroups and the grouphomology of G . The conjecture was first formulated in [13] by Farrell and Jones with coefficientsin Z . Then in [11], Davis and L¨uck gave a general framework for the formulations of variousisomorphism conjectures in K -and L -theories. In these formulations, the coefficients are untwistedrings. Later on, the conjecture was extended by Bartels and Reich [8] to allow for coefficients inany additive category A with a right G -action. The precise formulation of the conjecture will begiven in Section 2.1. OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 3
The significance of the Farrell-Jones conjecture not only lies in the fact that it provides a tool forthe computations of algebraic K - and L -theories of groups rings, but also implies many importantconjectures in geometry, topology and algebra. In particular, if both the K - and L -theoretic FJCwith coefficients in Z hold for a group G , then the Borel conjecture holds for G in dimensionsgreater than or equal to 5. If the L -theoretic FJC with coefficients in Z holds for G , then theNovikov conjecture on homotopy invariance of higher signatures holds for G . See [25] for a surveyon the FJC and its applications.We call a group FJ if it satisfies the K - and L -theoretic FJC with coefficients in Z . We denotethe class of all FJ groups by F J . In general, in application of the FJC to the Borel conjecture,one needs to show G ∈ F J if one wants to prove the Borel conjecture for G in dimensions ≥ Theorem A. If G ∈ F J , then the simple Borel conjecture holds for every semi-direct product G ⋊ Z in dimensions ≥ . If in addition W h ( G × Z ) = 0 , then the bordism Borel conjecture holdsfor every semi-direct product G ⋊ Z in dimensions ≥ . An immediate corollary of Theorem A is the following:
Corollary 1.1.
Let M be a closed aspherical manifold which fibers over the unit circle with fiber N . If π ( N ) ∈ F J and dim ( M ) ≥ , then the simple Borel conjecture holds for M . If in addition W h ( π ( N ) × Z ) = 0 , then the bordism Borel conjecture holds for M . Following [5], we define a class of groups B in the following way, except we add more groups tothe list. Definition 1.2.
Let B be the smallest class of groups so that:(1) B contains all groups from the following classes of groups: CAT(0) groups, Gromov hyper-bolic groups, lattices in virtually connected Lie groups, virtually solvable groups, funda-mental groups of graphs of abelian groups, S -arithmetic groups.(2) B is closed under taking subgroups, finite direct products of groups, free products of groups,direct colimits (with not necessary injective structure maps) of groups.(3) For any group homomorphism φ : G → H . If H ∈ B and φ − ( V ) ∈ B for every virtuallycyclic subgroup V ⊆ H , then G ∈ B . Corollary 1.3. If G ∈ B , then the simple Borel conjecture and the bordism Borel conjecture holdsfor every semi-direct product G ⋊ Z in dimensions ≥ .Proof. Note that we can assume G is torsion free, otherwise G is not the fundamental group of anyclosed aspherical manifold. Now we have B ⊆ F J since the groups in part (1) of Definition 1.2and the groups obtained from these groups by the operations in parts (2) and (3) satisfy the K -and L -theoretic FJC with coefficients in Z (with coefficients in any additive category indeed), see KUN WANG [5],[6],[31],[30],[2],[22],[18],[16],[15],[28]. Therefore, Corollary 1.3 follows from Theorem A and thefact that
W h ( G × Z ) = 0 if G is torsion free and G ∈ B . This is because G × Z ∈ B ⊆ F J andWhitehead groups of torsion free FJ groups vanish. (cid:3) One of the main ingredients in proving Theorem A is the following result, which is of course ofindependent interest:
Theorem B.
Let G be a torsion free group and G ⋊ Z be any semi-direct product of G with Z . (1) Suppose the L -theoretic FJC with coefficients in Z holds for G , then it also holds for G ⋊ Z . (2) Suppose the K -theoretic FJC with coefficients in Z holds for G , then there are obstructiongroups, N il G ⋊Z n , n ∈ Z , associated to the given data, so that G ⋊ Z satisfies the K -theoreticFJC with coefficients in Z if and only if N il G ⋊Z n = 0 , ∀ n ∈ Z .Remark . As one sees, we are not able to prove the same result as in part (1) of Theorem Bfor the K -theoretic FJC with coefficients in Z . Therefore, some additional argument is needed inorder to deduce Theorem A from Theorem B. This is achieved by proving some general result on L -groups. See Lemma 4.1. Remark . Our result is indeed more general. It still holds if we replace the coefficient ring Z by any right G -additive category A with involution in part (1) and replace the coefficient ring Z by any associative ring with unit which is regular in part (2) (a ring is called regular if it is leftNoetherian and every finitely generated left R -module has a projective resolution of finite type).For example, let A be a right G ⋊ Z -additive category with involution, view A as a right G -additivecategory with involution in the natural way. Then if the L -theoretic FJC with coefficient in A holds for G , the L -theoretic FJC with coefficient in A also holds for G ⋊ Z . Remark . A key step in the proof of FJC for some classes of groups, including the class ofBaumslag-Solitar groups [16],[15] and the class of solvable groups [30], is to prove the conjecturefor a group of the form G ⋊ α Z , where G is a torsion free abelian group and α is some specialaction of Z on G . Complicated geometric arguments are used in these works in order to prove theconjecture for G ⋊ α Z . Therefore, a general result as in part (1) of Theorem B is very useful.In the literature, the term FJC may refer to different versions of the conjecture and they havedifferent inheritance properties. For convenience, we introduce the following. Let G be a groupand A be a right G -additive category. We say a group G satisfies FJC A if G satisfies FJC withcoefficient in A . We say G satisfies FJC Ad if it satisfies FJC A for every additive category A with aright G -action. When A is the category of finitely generated left free R -modules for some associativering R with trivial group action, we then denote FJC A by FJC R .The version FJC Ad has some nice inheritance properties. For example, FJC Ad is closed underthe operations in parts (2) and (3) of Definition 1.2, see [2, Section 2.3] for a summary. However,these inheritance properties are unknown for FJC A ; except it was proved in [3, Theorem 0.8] that OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 5
FJC A is closed under taking direct colimit of groups with injective structure maps. Theorem B isprobably the second inheritance property for the L -theoretic FJC A .Theorem B together with some nice inheritance properties of FJC Ad implies the following: Corollary 1.7.
Let −→ K −→ G −→ Q −→ be an extension of groups. Suppose both K and Q are torsion free and satisfy the L -theoretic FJC Ad , then G also satisfies the L -theoretic FJC Ad .Proof. It is well-known that an infinite virtually cyclic group either admits a surjection onto theinfinite cyclic group with finite kernel or admits a surjection onto the infinite Dihedral group withfinite kernel. Therefore a torsion free virtually cyclic group is either trivial or infinite cyclic. Hence,by the fact that FJC Ad is closed under operation (3) in Definition 1.2, see [2, Theorem 2.7][19,Theorem A], it suffices to show that for any infinite cyclic subgroup of Q , its inverse image in G satisfies the L -theoretic FJC Ad . Note that its inverse image in G is isomorphic to a group ofthe form K ⋊ α Z . Let A be any additive K ⋊ α Z category. By assumption, K is torsion freeand satisfies the L -theoretic FJC A , so by Theorem B and Remark 1.5, K ⋊ α Z also satisfies the L -theoretic FJC A . Since A can be arbitrary, we see that K ⋊ α Z satisfies FJC Ad . This completesthe proof. (cid:3) Let C be a class of groups. A group G is called poly- C if it has a filtration by subgroups1 = G ⊆ G ⊆ G ⊆ · · · ⊆ G n = G so that each G i − is normal in G i , i = 1 , , · · · , n and each G i /G i − is in C . Definition 1.8.
Let B be as in Definition 1.2. Define D ⊆ B to be the class of groups in B whichare torsion free. Corollary 1.9.
The class of poly- D -groups satisfies the L -theoretic FJC Ad . Therefore, they alsosatisfy the Novikov conjecture.Proof. By repeatedly applying Corollary 1.7, we see that the L -theoretic FJC Ad holds for poly- D groups. (cid:3) Remark . The best result so far for the Novikov conjecture is due to Guoliang Yu, who proved in[32] the coarse Baum-Connes conjecture for groups which admit a uniform embedding into Hilbertspace. This is a huge class of groups. However, there are groups, constructed as direct colimitsof Gromov hyperbolic groups, which do not admit a uniform embedding into Hilbert space, seefor example [20]. Note that these groups lie in the class B . Also it is still an open questionthat whether CAT(0)-groups and their extensions admit a uniform embedding into Hilbert space.Therefore, poly- D -groups contain new examples of groups that satisfy the Novikov conjecture.The paper is organized as follows. In Section 2, we briefly recall the formulation of the Farrell-Jones conjecture and the controlled algebra approach to the conjecture. We use this approach KUN WANG to prove Theorem B. We also prove some lemmas on the equivariantly continuously controlledcondition. These lemmas are important in proving the main theorem (Theorem 3.6) of Section 3,which is a key step in proving Theorem B. We prove our main theorems in the final section, inwhich we first prove an important lemma (Lemma 4.1) for L -groups. This lemma together withTheorem B enable us to prove Theorem A. Acknowledgement.
The author would like to thank Professor Bruce Hughes for his support andhelpful comments about the paper. The author would also like to thank his thesis advisor ProfessorJean-Fran¸cois Lafont for his enormous and continuous support throughout the years. Part of theproject was initiated when the author was a graduate student at the Ohio State University under theguidance of Jean. The author also thanks Professor Guoliang Yu for his support and encouragement.The author learned a lot through numerous discussions with Guoliang. The author also thanksShanghai Center for Mathematical Science for its hospitality during the author’s visit in Summer2015. Part of the project was done during this visit.2.
The Farrell-Jones conjecture and controlled algebra
In the first half of the section, we briefly recall the formulation of the FJC and the controlledalgebra approach to the conjecture. More details can be found in [13], [11], [4], [8]. This approachwill be important to our treatment later. In the second half of the section, we prove some lemmason the equivariantly continuously controlled condition (see Definition 2.2) that will be importantin later sections.2.1.
The Farrell-Jones conjecture.
Let G be a group and A be an additive category with aright G -action α (we always assume A comes with an involution, which is compatible with theright G -action, when we talk about L -theory). One can form the “twisted group additive category” A α [ G ] [8, Definition 2.1] (this category is denoted by A ∗ G pt in [8], but it is more enlighteningto denote this category by A α [ G ] for our purpose). Its definition is recalled in Definition 3.1and it will be important to our treatment later. When A is the additive category of finitelygenerated left free R -modules with the trivial G -action, A [ G ] is equivalent to the additive categoryof finitely generated left free R [ G ]-modules. Now let H G ∗ ( − ; K A ) and H G ∗ ( − ; L < −∞ > A ) be the two G -equivariant homology theories constructed by Bartels and Reich in [8], using the method ofDavis and L¨uck [11]. These two equivariant homology theories have the property that for anysubgroup H < G , H Gn ( G/H ; K A ) = K n ( A α [ H ]) , H Gn ( G/H ; L < −∞ > A ) = L < −∞ >n ( A α [ H ]) , ∀ n ∈ Z ,where the decoration −∞ on the L -groups means we are dealing with Ranicki’s ultimate L -groups.In particular, H Gn ( pt ; K A ) = K n ( A α [ G ]) and H Gn ( pt ; L < −∞ > A ) = L < −∞ >n ( A α [ G ]).We also need the notion of classifying space of a group relative to a family of its subgroups. For F , a family of subgroups of G which is closed under taking subgroups and conjugations, denote by E F G a model for the classifying space of G relative to the family F . It is a G -CW complex and OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 7 is characterized, up to G -equivariant homotopy equivalence, by the properties that every isotropygroup of the action lies in the family F , and the fixed point set of H is contractible if H ∈ F andempty if H / ∈ F . Examples of classifying spaces are
EG, E
FIN G , and E V C G , corresponding to thefamilies of trivial, finite, and virtually cyclic subgroups of G respectively. The Farrell-Jones Conjecture.
Let G be a group and A be an additive category with a right G -action. We say the group G satisfies the K -theoretic FJC with coefficient in A if the K -theoreticassembly map A A K : H Gn ( E VC G ; K A ) → H Gn ( pt ; K A ) = K n ( A α [ G ])(2.1)which is induced by the obvious map E VC G → pt , is an isomorphism for all n ∈ Z . We say the group G satisfies the L -theoretic FJC with coefficient in A if the corresponding L -theoretic assembly map A A L : H Gn ( E VC G ; L < −∞ > A ) → H Gn ( pt ; L < −∞ > A ) = L < −∞ >n ( A α [ G ])(2.2)is an isomorphism for all n ∈ Z .2.2. Controlled algebra approach to the FJC.
Let A be a (small) additive category with aright G -action and X be a left G -space. The additive category C ( X ; A ) of geometric modules over X with coefficient in A is defined as follows: objects are functions A : X → Ob ( A ) with locally finitesupport, i.e. supp A = { x ∈ X | A x = 0 } is a locally finite subset of X , meaning every point in X hasan open neighborhood whose intersection with supp A is finite. An object will usually be denotedby A = ( A x ) x ∈ X . A morphism φ : A → B is a matrix of morphisms ( φ y,x : A x → B y ) ( x,y ) ∈ X × X such that there are only finitely many nonzero entries in each row and each column. Compositionsof morphisms are given by matrix multiplications. More precisely( ψ z,y : B y → C z ) ◦ ( φ y,x : A x → B y ) = ( X y ∈ X ψ z,y ◦ φ y,x : A x → C z )Note that the sum on the right hand side is finite. There is a right G -action on C ( X ; A ), which isgiven by ( g ∗ A ) x = g ∗ ( A gx ) , ( g ∗ φ ) y,x = g ∗ ( φ gy,gx )and the fixed category is denoted by C G ( X ; A ).For any object A and morphism φ in C ( X ; A ), their supports are the following sets:supp A = { x ∈ X | A x = 0 } , supp φ = { ( x, y ) ∈ X × X | φ y,x = 0 } In order to get interesting subcategories of C ( X ; A ), one can prescribe certain support conditions onobjects and morphisms. The convenient language for this purpose is the notion of coarse structures on spaces introduced in [21]. We recall the definition here (with a slight modification for ourpurpose). KUN WANG
Definition 2.1. A coarse structure ( E , F ) on a set X is a collection E of subsets of X × X , and acollection F of subsets of X satisfying the following properties:(1) If E ′ , E ′′ ∈ E , then E ′ ∪ E ′′ ⊆ E for some E ∈ E ;(2) If E ′ , E ′′ ∈ E , then E ′ ◦ E ′′ = { ( x, y ) ∈ X × X | ∃ z ∈ X s.t.( x, z ) ∈ E ′ and ( z, y ) ∈ E ′′ } ⊆ E for some E ∈ E ;(3) The diagonal △ = { ( x, x ) | x ∈ X } is contained in some E ∈ E ;(4) If F ′ , F ′′ ∈ F , then F ′ ∪ F ′′ ⊆ F for some F ∈ F .If there is a G -action on the set X , we then require every member in E and F to be G -invariant,where G acts on X × X diagonally. If p : Y → X is a G -equivariant map, then the pullback(( p × p ) − E , p − F ) is a coarse structure on Y .Now if ( E , F ) is a coarse structure on a G -space X , one can define a subcategory C ( X, E , F ; A )of C ( X ; A ), with object and morphism supports contained in members of F and E respectively (inaddition to the general finiteness conditions on them). G acts on this additive subcategory and thefixed subcategory is denoted by C G ( X, E , F ; A ). The pair ( E , F ) are usually referred to as controlconditions on morphisms and objects.One of the control condition on morphisms, which is used to construct a model for the assemblymaps in the FJC, is the equivariantly continuously controlled condition introduced in [4]. It is ageneralization to the equivariant setting of the continuously controlled condition introduced in [1].We recall the definition here. Definition 2.2. ([4, Definition 2.7]) Let X be a topological space with a left G -action by home-omorphisms. Equip X × [1 , ∞ ] with the diagonal G -action, where G -acts trivially on [1 , ∞ ]. Asubset E ⊆ ( X × [1 , ∞ )) is called equivariantly continuously controlled if the following holds:(i) For every x ∈ X and G x -invariant open neighborhood U of ( x, ∞ ) in X × [1 , ∞ ], there existsa G x -invariant neighborhood V ⊆ U of ( x, ∞ ) in X × [1 , ∞ ] such that( U c × V ) ∩ E = ∅ where U c denotes the complement of U in X × [1 , ∞ ].(ii) There exists α >
0, depending on E , such that if ( x, s ) × ( x ′ , s ′ ) ∈ E , then | s − s ′ | < α ;(iii) E is symmetric, i.e. if ( p, q ) ∈ E , then ( q, p ) ∈ E ;(iv) E is invariant under the diagonal action of G .The collection of G -equivariantly continuously controlled subsets of ( X × [1 , ∞ )) will be denotedby E XGcc . It satisfies the conditions (1)-(3) in Definition 2.1.
Definition 2.3. ([4, Section 3.2][6, Section 3.3]) For any left G -space X and additive category A with a right G -action, one defines the following categories:(1) O G ( X ; A ) = C G ( G × X × [1 , ∞ ) , ( p × p ) − E XGcc ∩ ( r × r ) − E G , q − F Gc ; A ), where p : G × X × [1 , ∞ ) → X × [1 , ∞ ), q : G × X × [1 , ∞ ) → G × X and r : G × X × [1 , ∞ ) → G are projections, E G = OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 9 { E ⊆ G × G | gE = E for all g ∈ G and ∃ finite S ⊆ G s.t. for all ( g, g ′ ) ∈ E, we have g − g ′ ∈ S } and F Gc consists of G -cocompact subsets of G × X , i.e. subsets of the form G · K ⊆ G × X , where K ⊆ G × X is compact;(2) T G ( X ; A ) is the full subcategory of O G ( X ; A ) consisting of those objects A with the followingproperty: there exists C > A ( g,x,t ) = 0, then t < C ;(3) D G ( X ; A ) is the quotient category of O G ( X ; A ) by the full subcategory T G ( X ; A ): it hasthe same objects as O G ( X ; A ), and any morphism from A to B in D G ( X ; A ) is represented by amorphism φ : A → B in O G ( X ; A ), with two morphisms φ, ψ : A → B identified if their difference φ − ψ factors through an object in T G ( X ; A ). Remark . In [4], the morphism control condition E G was not required in the definitions. Itwas added into the definitions in [6][5] because it was important for proving the conjecture forhyperbolic and CAT(0)-groups. Although this addition changes the categories, it does not changethe theory. Later on, we will see that this is also important for our treatment.These constructions define functors from the category of G -CW complexes to the category ofadditive categories. The importance of these categories lies in the following facts: Theorem 2.5.
The following results hold:(i) The sequence T G ( X ; A ) → O G ( X ; A ) → D G ( X ; A ) is a Karoubi filtration, hence gives rise to a fibration sequence K −∞ ( T G ( X ; A )) → K −∞ ( O G ( X ; A )) → K −∞ ( D G ( X ; A )) of spectra after applying the non-connective K -theory, and therefore a long exact sequence on K -groups;(ii) The additive category O G ( pt ; A ) has trival K -groups;(iii) The functor π ∗ : T G ( X ; A ) → T G ( pt ; A ) induced by the obvious map π : X → { pt } is anequivalence of categories, hence induces isomorphisms on K -groups;(iv) There is a natural isomorphism between the two functors H G ∗ ( − ; K A ) and K ∗ +1 ( D G ( − ; A )) := π ∗ +1 ( K −∞ ( D G ( − ; A ))) from the category of G -CW complexes to the category of graded abeliangroups. In particular, the map K ∗ +1 ( D G ( E VC G ; A )) → K ∗ +1 ( D G ( pt ; A ))(2.3) is equivalent to the assembly map 2.1. For information about Karoubi filtrations, see [9]. Fact (ii) can be proved by an Eilenberg swindleargument. Fact (iii) can be checked directly. Fact (iv) is first proven in [4] for coefficients in rings,and proven for coefficients in additive categories in [8]. One has the same constructions and resultsfor L -theory and details can be found in [5]. The above facts implies the following [6][5]: Theorem 2.6.
The K -theoretic FJC A holds for G if and only if the K -theory of O G ( E VC G ; A ) istrivial, i.e. K n ( O G ( E VC G ; A )) = 0 , ∀ n ∈ Z . The corresponding statement is true for the L -theoreticFJC A . Because of this theorem, the category O G ( E VC G ; A ) is usually referred to as the obstructioncategory .2.3. Some lemmas on equivariant continuous control.
In this subsection, we prove somegeneral results about the equivariantly continuously controlled condition. These results will beimportant in proving Theorem 3.6, which is one of the key ingredients in proving Theorem B. Theproofs of some of these results and the proof of Theorem 3.6 in the next section are quite technicaland readers may want to jump to Section 4 to see the proof of the main theorems first (assumingTheorem 3.6.)Now let X be a G -CW complex and H < G be a subgroup. Then X is also an H -CW complexin a natural way. Thus we can consider E XGcc and E XHcc . In the following several lemmas we studythe relation between these two control conditions.
Lemma 2.7. E XGcc ⊆ E
XHcc if one of the following holds:(i) The action of G on X is free.(ii) H < G is of finite index.Proof. (i) This is obvious.(ii) The only nontrivial part is (i) of Definition 2.2. But this can be easily shown, by noting T g ∈ G x gU is a G x -invariant neighborhood of ( x, ∞ ) in X × [1 , ∞ ] for any H x -invariant neighborhood U of ( x, ∞ ) in X × [1 , ∞ ], since [ G x : H x ] ≤ [ G : H ] < ∞ . (cid:3) Lemma 2.8. If H is a normal subgroup of G , then for any E ∈ E XHcc and g ∈ G , we have gE ∈ E XHcc .Proof.
Since H is a normal subgroup of G , one sees easily that H · gE = gE , hence gE is H -invariant. One also easily sees gE is symmetric, bounded in the [1 , ∞ ) direction. Now for any x ∈ X , any H x -invariant neighborhood U of ( x, ∞ ) in X × [1 , ∞ ], we consider g − x . Since H < G is normal, we have H g − x = g − H x g . Hence g − U is an H g − x -invariant neighborhood of ( g − x, ∞ ).Therefore there exists an H g − x -invariant neighborhood V ′ ⊆ g − U of ( g − x, ∞ ) so that(( g − U ) c × V ′ ) ∩ E = ∅ Hence ( U c × gV ′ ) ∩ gE = ∅ This completes the proof by letting V = gV ′ and by noting V is H x -invariant. (cid:3) The proof of part (iv) in the proof of the following lemma is motivated from the proof of [4,Lemma 3.3].
OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 11
Lemma 2.9.
Let E ∈ E XHcc . For any compact subset K ⊆ X , define E ′ = { ( x, s ) × ( x ′ , s ′ ) | ∃ g, g ′ ∈ G s.t. g − g ′ ∈ H, g − x, g ′− x ′ ∈ K, ( g − x, s ) × ( g − x ′ , s ′ ) ∈ E } Then E ′ ∈ E XGcc .Proof. (i) E ′ is symmetric: suppose ( x, s ) × ( x ′ , s ′ ) ∈ E ′ , then we have g, g ′ ∈ G with the propertiesin the definition of E ′ . Now since E is symmetric, we have ( g − x ′ , s ′ ) × ( g − x, s ) ∈ E . Since E is H -invariant and g − g ′ ∈ H , we get ( g ′− x ′ , s ′ ) × ( g ′− x, s ) ∈ E . This shows g ′ , g fullfill theproperties for the pair ( x ′ , s ′ ) × ( x, s ) in the definition of E ′ . Hence E ′ is symmetric;(ii) E ′ is G -invariant: suppose ( x, s ) × ( x ′ , s ′ ) ∈ E ′ and l ∈ G . Let g, g ′ be as before. Oneeasily verifies that lg, lg ′ fulfill the properties in the definition of E ′ for ( lx, s ) × ( lx ′ , s ′ ). Hence( lx, s ) × ( lx ′ , s ′ ) ∈ E ′ , which shows E ′ is G -invariant;(iii) Bounded control in the s -direction: this is clear since E is controlled;(iv) E ′ is G -equivariantly continuously controlled: suppose not, then there exists x ∈ X , a G x -invariant open neighborhood U of x ∈ X and r >
0, such that for all G x -invariant openneighborhood V ⊆ U of x in X and all l > r , we have (cid:0) U × ( r, ∞ ] (cid:1) c × (cid:0) V × ( l, ∞ ] (cid:1) ∩ E ′ = ∅ Now since X is a G -CW complex, the Slice theorem (see [4, Proposition 3.4]) applies, so thatwe can find a descending sequence { V k } k ∈ N of open neighborhoods of x in X with the followingproperty:(a) Each V k is G x -invariant;(b) gV k ∩ V k = ∅ if g / ∈ G x ;(c) T k ≥ G · V k = G · x .We may assume V k ⊆ U, ∀ k ∈ N . Hence[( U × ( r, ∞ ]) c × ( V k × ( k + r, ∞ ])] ∩ E ′ = ∅ (2.4)thus we can find a sequence( x k , s k ) × ( x ′ k , s ′ k ) ∈ [( U × ( r, ∞ ]) c × ( V k × ( k + r, ∞ ])] ∩ E ′ (2.5)By the definition of E ′ , there exist g k , g ′ k such that g − k g ′ k ∈ H, g − k x k , g ′ k − x ′ k ∈ K, ( g − k x k , s k ) × ( g − k x ′ k , s ′ k ) ∈ E (2.6)Since E is H -invariant and g − k g ′ k ∈ H , the above also implies( g ′− k x k , s k ) × ( g ′− k x ′ k , s ′ k ) ∈ E (2.7)Now since K is compact, by passing to a subsequence, we may assume g ′ k − x ′ k → y . Now since x ′ k ∈ V k , we have for every n ∈ N and all k > n , g ′ k − x ′ k ∈ G · V k ⊆ G · V n , this implies y ∈ G · V n for all n , thus y ∈ T n ≥ G · V n = G · x . Hence there is g ∈ G so that y = gx . Therefore g ′ k − x ′ k → y = gx , so g − g ′ k − x ′ k → x ∈ V . Thus when k is large enough g − g ′ k − ∈ G x . Notethat s ′ k → ∞ , hence by 2.7 and the control condition in the s -direction, we have s k → ∞ . Nowby 2.5, when k is large enough, x k / ∈ U , hence g − g ′ k − x k / ∈ U since g − g ′ k − ∈ G x and U is G x -invariant. Thus when k is large enough, g ′ k − x k / ∈ gU .Now consider gU , it is G y -invariant since y = gx and U is G x -invariant. In particular, gU is H y -invariant. Now because E is H -equivariantly continuously controlled, we can find an H y -invariantopen neighborhood W ⊆ gU of y = gx in X and N > r such that[( gU × ( r, ∞ ]) c × ( W × ( N, ∞ ])] ∩ E = ∅ (2.8)However, on the one hand, ( g ′− k x k , s k ) × ( g ′− k x ′ k , s ′ k ) ∈ E for all k by 2.7, while on the other hand,when k is large enough, we showed g ′ k − x k / ∈ gU , g ′ k − x ′ k → y ∈ W and s k , s ′ k → ∞ , so we alsohave ( g ′− k x k , s k ) × ( g ′− k x ′ k , s ′ k ) ∈ ( gU × ( r, ∞ ]) c × ( W × ( N, ∞ ]). Thus when k is large enough, wehave ( g ′− k x k , s k ) × ( g ′− k x ′ k , s ′ k ) ∈ [( gU × ( r, ∞ ]) c × ( W × ( N, ∞ ])] ∩ E (2.9)This contradicts to 2.8 and we complete the proof. (cid:3) Equivalence of two categories
The major goal of this section is to prove Theorem 3.6. Let us firstly make some preparations.Let A be a right G -additive category. For any left G -set X , Bartels and Reich [8] defined a newadditive category A ∗ G X . The special case when X = pt will be important in our treatment andwe denote it by A α [ G ], where α denotes the right G -action on A . We now recall its definition here. Definition 3.1. ([8, Definition 2.1]) Objects of the category A α [ G ] are the same as the objectsof A . A morphism φ : A → B from A to B in A α [ G ] is a formal sum φ = P g ∈ G φ g · g , where φ g : A → g ∗ B is a morphism in A and there are only finitely many g ∈ G with φ g = 0. Additionof morphisms is defined in the obvious way. Composition of morphisms is defined as follows: let φ = P k ∈ G φ k · k : A → B be a morphism from A to B and ψ = P h ∈ G ψ h · h : B → C be a morphismfrom B to C , their composition is given by ψ ◦ φ := X g ∈ G (cid:0) X k,h ∈ G,g = hk k ∗ ( ψ h ) ◦ φ k (cid:1) · g Now let F and G be two groups. Suppose there is a group homomorphism α : F → Aut ( G ) , f α f . We then can form the associated semi-direct product Γ = G ⋊ α F . Every element γ ∈ Γ can beuniquely written as γ = gf for some g ∈ G and f ∈ F . We have gf g ′ f ′ = gα f ( g ′ ) f f ′ . In particular α f ( g ) = f gf − . Now let X be a left Γ-CW complex and A be a right Γ-additive category (with orwithout involution). Since F and G are subgroups of Γ, they naturally inherit actions on X and OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 13 A . So we can consider the obstruction category O G ( X ; A ). Recall from Definition 2.3 that O Γ ( X ; A ) = C Γ (Γ × X × [1 , ∞ ) , ( p × p ) − E X Γ cc ∩ ( r × r ) − E Γ , q − F Γ c ; A ) O G ( X ; A ) = C G ( G × X × [1 , ∞ ) , ( p × p ) − E XGcc ∩ ( r × r ) − E G , q − F Gc ; A )In what follows, we first show in Lemma 3.2 that the actions of F on G, X and A induce a right F -action on the additive category O G ( X ; A ) (which, by abuse of notation, will also be denoted by α ). We then can form the additive category O G ( X ; A ) α [ F ]. We prove in Theorem 3.6 that thetwo additive categories, O Γ ( X ; A ) and O G ( X ; A ) α [ F ], are equivalent under some assumptions. Inparticular, these assumptions are satisfied if F is finite or if the action of Γ on X is free.In the proofs of Lemma 3.2 and Theorem 3.6, we will check everything very carefully. One mightwant to jump to Remark 3.4 first for a simple interpretation of the somewhat complicated formulasin the lemma below. Lemma 3.2.
Let f ∈ F . For any objects A, B and morphism φ : A → B in O G ( X ; A ) , theformulas ( α f A ) ( g,x,s ) := f ∗ ( A ( α f ( g ) ,fx,s ) )( α f φ ) ( g ′ ,x ′ ,s ′ ) , ( g,x,s ) := f ∗ ( φ ( α f ( g ′ ) ,fx ′ ,s ′ ) , ( α f ( g ) ,fx,s ) ) where ( g, x, s ) , ( g ′ , x ′ , s ′ ) ∈ G × X × [1 , ∞ ) , define an additive functor α f : O G ( X ; A ) → O G ( X ; A ) .Moreover, the assignment f α f defines a right F -action on the obstruction category O G ( X ; A ) .Proof. The major part is to check for each f ∈ F , α f : O G ( X ; A ) → O G ( X ; A ) is a well-definedadditive functor. As soon as this is done, it is easy to see α defines a right F -action on O G ( X ; A ).We will omit the s -component in places where it is not important for the proof.(i) α f A is G -invariant: for any l ∈ G, ( g, x ) ∈ G × X , we have (cid:0) l ∗ ( α f A ) (cid:1) ( g,x ) = l ∗ (cid:0) ( α f A ) ( lg,lx ) (cid:1) = l ∗ (cid:0) f ∗ ( A ( α f ( lg ) ,flx ) ) (cid:1) = ( f l ) ∗ (cid:0) A ( α f ( lg ) ,flx ) (cid:1) = (cid:0) α f ( l ) f (cid:1) ∗ (cid:0) A ( α f ( lg ) ,flx ) (cid:1) = f ∗ (cid:0) α f ( l ) ∗ ( A ( α f ( lg ) ,flx ) ) (cid:1) = f ∗ (cid:0) ( α f ( l ) ∗ A ) ( α f ( g ) ,α f ( l − ) flx ) (cid:1) = f ∗ (cid:0) A ( α f ( g ) ,α f ( l − ) flx ) (cid:1) Since A is G -invariant= f ∗ (cid:0) A ( α f ( g ) ,fx ) (cid:1) = ( α f A ) ( g,x ) Thus l ∗ ( α f A ) = α f A for all l ∈ G . This proves α f A is G -invariant for all f ∈ F . (ii) α f φ is G -invariant: for any l ∈ G, ( g, x ) , ( g ′ , x ′ ) ∈ G × X , we have (cid:0) l ∗ ( α f φ ) (cid:1) ( g ′ ,x ′ ) , ( g,x ) = l ∗ [( α f φ ) ( lg ′ ,lx ′ ) , ( lg,lx ) ]= l ∗ [ f ∗ ( φ ( α f ( lg ′ ) ,flx ′ ) , ( α f ( lg ) ,flx ) )]= ( f l ) ∗ [ φ ( α f ( lg ′ ) ,flx ′ ) , ( α f ( lg ) ,flx ) ]= ( α f ( l ) f ) ∗ [ φ ( α f ( lg ′ ) ,flx ′ ) , ( α f ( lg ) ,flx ) ]= f ∗ [ α f ( l ) ∗ ( φ ( α f ( lg ′ ) ,flx ′ ) , ( α f ( lg ) ,flx ) )]= f ∗ [( α f ( l ) ∗ φ ) ( α f ( g ′ ) ,α f ( l − ) flx ′ ) , ( α f ( g ) ,α f ( l − ) flx ) ]= f ∗ [ φ ( α f ( g ′ ) ,fx ′ ) , ( α f ( g ) ,fx ) ]= ( α f φ ) ( g ′ ,x ′ ) , ( g,x ) Thus l ∗ ( α f φ ) = α f φ for all l ∈ G . This proves α f φ is G -invariant for all f ∈ F .(iii) α f ( id A ) = id α f A : this is easy.(iv) α f ( φ ◦ ψ ) = ( α f φ ) ◦ ( α f ψ ): in the following identities, we are taking sum over ( l, y ) ∈ G × X .We have [ α f ( φ ◦ ψ )] ( g ′ ,x ′ ) , ( g,x ) = f ∗ [( φ ◦ ψ ) ( α f ( g ′ ) ,fx ′ ) , ( α f ( g ) ,fx ) ]= f ∗ [ φ ( α f ( g ′ ) ,fx ′ ) , ( l,y ) ◦ ψ ( l,y ) , ( α f ( g ) ,fx ) ]= f ∗ [ φ ( α f ( g ′ ) ,fx ′ ) , ( l,y ) ] ◦ f ∗ [ ψ ( l,y ) , ( α f ( g ) ,fx ) ]= ( α f φ ) ( g ′ ,x ′ ) , ( α f − ( l ) ,f − y ) ◦ ( α f ψ ) ( α f − ( l ) ,f − y ) , ( g,x ) = [( α f φ ) ◦ ( α f ψ )] ( g ′ ,x ′ ) , ( g,x ) The last identity holds because ( α f − ( l ) , f − y ) runs over G × X as ( l, y ) runs over G × X . We thusget α f ( φ ◦ ψ ) = ( α f φ ) ◦ ( α f ψ )(v) Additivity: this is easy.(vi) Object support: we have to show { g − x | ( g, x, s ) ∈ supp( α f A ) } is contained in a compact subsetof X . Note that supp( α f A ) = { ( g, x, s ) | ( α f ( g ) , f x, s ) ∈ supp( A ) } . By definition, there exists acompact subset K ⊆ X , so that { α f ( g − ) f x = f g − x | ( α f ( g ) , f x, s ) ∈ supp( A ) } ⊆ K . This implies { g − x | ( g, x, s ) ∈ supp( α f A ) } = { g − x | ( α f ( g ) , f x, s ) ∈ supp( A ) } ⊆ f − · K , which completes theproof since f − · K ⊆ X is compact.(vii) Morphism support: by definition, there exists E ∈ E XGcc , so that the projection of supp( φ )in ( X × [1 , ∞ )) is contained in E . Now by definition of α f φ , the projection of supp( α f φ ) in OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 15 ( X × [1 , ∞ )) is contained in E ′ = { ( x, s ) × ( x ′ , s ′ ) | ( f x, s ) × ( f x ′ , s ′ ) ∈ E } = f − · E Now since
G <
Γ is a normal subgroup, by Lemma 2.8, we have E ′ = f − · E ∈ E XGcc . Hence α f φ respects the support condition in the X × [1 , ∞ )-direction. But clearly it respects the supportcondition in the G -direction since α f : G → G is an automorphism. Therefore α f φ respects themorphism support condition.Now if A has an involution, then α f commutes with the induced involution on O G ( X ; A ). Thistogether with the above verification work shows that for each f ∈ F , α f is a well-defined additivefunctor from O G ( X ; A ) to itself.Finally α e is the identity functor, where e ∈ F is the trivial element and for f, f ′ ∈ F , we have( α ff ′ A ) ( g,x ) = ( f f ′ ) ∗ [ A ( α ff ′ ( g ) ,ff ′ x ) ]= ( f ′ ) ∗ [ f ∗ ( A ( α ff ′ ( g ) ,ff ′ x ) )]= ( f ′ ) ∗ [( α f A ) ( α f ′ ( g )) ,f ′ x ) ]= (cid:0) α f ′ ( α f A ) (cid:1) ( g,x ) Therefore α ff ′ = α f ′ α f on objects. Similarly this identity holds on morphisms. Hence f α f defines a right F -action on O G ( X ; A ). We thus complete the proof of the lemma. (cid:3) Remark . We have used α to denote both the action of F on G and the action of F on O G ( X ; A ).Note however that the former action is from left while the later action is from right. It will be clearfrom the context which action α stands for. Remark . A convenient way to interpret the defining formulas in the above Lemma is as follows:for any object A in O G ( X ; A ), it is completely determined by its values at points of the form( e, x, s ) ∈ G × X × [1 , ∞ ), where e ∈ G is the identity element. But ( e, x, s ) is a point in Γ × X × [1 , ∞ ),using the Γ-action on Γ × X × [1 , ∞ ), A ( e,x,s ) can be used to define an object τ A living overΓ × X × [1 , ∞ ), i.e. by defining ( τ A ) ( γ,x,s ) := ( γ − ) ∗ ( A ( e,γ − x,s ) ) ( τ A is actually an object in O Γ ( X ; A ) that we will show shortly). Then A is just the restriction of τ A to G × X × [1 , ∞ ).Note for any f ∈ F , the map ( γ, x, s ) ( γf, x, s ) is a Γ-equivariant self-homeomorphism ofΓ × X × [1 , ∞ ), it thus induces an automorphism of O Γ ( X ; A ). Denote this automorphism by α f , then ( α f ( τ A )) ( γ,x,s ) = ( τ A ) ( γf − ,x,s ) = f ∗ (( τ A ) ( fγf − ,fx,s ) ), the second equality is due to theΓ-invariance of τ A . This formula restricts to G × X × [1 , ∞ ) is the defining formula on objects inthe above lemma. From this point of view, Lemma 3.2 is morally apparent. Remark . The naive definition ( α f A ) ( g,x,s ) := A ( α f ( g ) ,x,s ) doesn’t work since the map ( g, x, s ) ( α f ( g ) , x, s ) is not G -equivariant. We can now form the category O G ( X ; A ) α [ F ] and are going to show O Γ ( X ; A ) and O G ( X ; A ) α [ F ]are equivalent as additive categories (with involution) under some conditions. Forgetting aboutthe control conditions, this is then not very hard to see intuitively. The one-one correspondencebetween objects of these two categories has already been explained in Remark 3.4. Let us focus onmorphisms. By Γ-invariance, every morphism φ : A → B in O Γ ( X ; A ) is uniquely determined by itsvalues on ( e, x, s ) × ( γ ′ , x ′ , s ′ ), i.e. by φ ( γ ′ ,x ′ ,s ′ ) , ( e,x,s ) : A ( e,x,s ) → B ( γ ′ ,x ′ ,s ′ ) . They can be grouped intofamilies { φ ( gf,x ′ ,s ′ ) , ( e,x,s ) | g ∈ G } , f ∈ F . Due to the morphism control condition in the Γ-direction,see Definition 2.3, there are only finitely many f ∈ F whose corresponding family is non-trivial.For each of such f , the family indexed by f determines a morphism φ f : A → α f B in O G ( X ; A ),here we are viewing objects A and B as objects in O G ( X ; A ) by restriction to G × X × [1 , ∞ ).The map φ → P f ∈ F φ f · f then gives a one-one correspondence between morphisms of these twocategory. Theorem 3.6.
Let
Γ = G ⋊ α F be as before. Then for any Γ - CW complex X and any additivecategory (with involution) A with a right Γ -action, the two additive categories (with involutions) O Γ ( X ; A ) and O G ( X ; A ) α [ F ] are equivalent provided E X Γ cc ⊆ E XGcc . In particular, they are equivalentif F is finite or if the action of Γ on X is free.Proof. Define a functor τ : O G ( X ; A ) α [ F ] → O Γ ( X ; A ) as follows: on objects: ( τ A ) ( γ,x,s ) := f ∗ ( A ( α f ( g ) ,fx,s ) ) = ( α f A ) ( g,x,s ) where ( γ, x, s ) ∈ Γ × X × [1 , ∞ ) and g ∈ G, f ∈ F are the unique elements so that γ = gf − . Onecan easily check that ( τ A ) ( γ,x,s ) = ( γ − ) ∗ ( A ( e,γ − x,s ) ), from which we conclude τ A is Γ-invariant. on morphisms: Let φ = X f ∈ F φ f · f : A → B be a morphism in O G ( X ; A ) α [ F ], where φ f : A → α f B is a morphism in O G ( X ; A ). By definition, there are only finitely many f ∈ F with φ f = 0.Define τ φ : τ A → τ B to be the sum X f ∈ F τ f φ f , where τ f φ f : τ A → τ B is defined as follows:( τ f φ f ) ( γ ′ ,x ′ ,s ′ ) , ( γ,x,s ) := γ − γ ′ / ∈ Gf − ( γ − ) ∗ ( φ f ( γ − γ ′ f,γ − x ′ ,s ′ ) , ( e,γ − x,s ) ) if γ − γ ′ ∈ Gf − We have to check ( γ − ) ∗ (cid:0) φ f ( γ − γ ′ f,γ − x ′ ,s ′ ) , ( e,γ − x,s ) (cid:1) is indeed a map from ( τ A ) ( γ,x,s ) to ( τ B ) ( γ ′ ,x ′ ,s ′ ) .But this is because ( τ A ) ( γ,x,s ) = ( γ − ) ∗ ( A ( e,γ − x,s ) )( τ B ) ( γ ′ ,x ′ ,s ′ ) = ( γ − ) ∗ (cid:0) ( τ B ) ( γ − γ ′ ,γ − x ′ ,s ′ ) (cid:1) = ( γ − ) ∗ (cid:0) ( α f B ) ( γ − γ ′ f,γ − x ′ ,s ′ ) (cid:1) and φ f ( γ − γ ′ f,γ − x ′ ,s ′ ) , ( e,γ − x,s ) : A ( e,γ − x,s ) −→ ( α f B ) ( γ − γ ′ f,γ − x ′ ,s ′ ) The formulas above are motivated by the intuition explained in the paragraph preceding thistheorem. We firstly show τ is well-defined, i.e. it is a genuine additive functor and respects controlconditions on objects and morphisms. We then show τ actually gives an equivalence of two additivecategories. In what follows, we will again omit the s -component in appropriate places.(i) τ A is Γ-invariant: already checked.(ii) τ φ is Γ-invariant: it suffices to check for each f ∈ F , τ f φ f is Γ-invariant. For any l ∈ Γ, wehave [ l ∗ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) = l ∗ [( τ f φ f ) ( lγ ′ ,lx ′ ) , ( lγ,lx ) ]If γ − γ ′ / ∈ Gf − , then ( lγ ) − lγ ′ / ∈ Gf − , hence [ l ∗ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) = 0 = ( τ f φ f ) ( γ ′ ,x ′ ) , ( γ,x ) .If γ − γ ′ ∈ Gf − , then[ l ∗ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) = l ∗ [( τ f φ f ) ( lγ ′ ,lx ′ ) , ( lγ,lx ) ]= l ∗ [( γ − l − ) ∗ ( φ f ( γ − γ ′ f,γ − x ′ ) , ( e,γ − x ) )]= ( γ − ) ∗ ( φ f ( γ − γ ′ f,γ − x ′ ) , ( e,γ − x ) )= ( τ f φ f ) ( γ ′ ,x ′ ) , ( γ,x ) Thus l ∗ ( τ f φ f ) = τ f φ f , for all f ∈ F and l ∈ Γ. This shows τ φ is Γ-invariant.(iii) τ ( id A ) = id τA : note id A : A → A in O G ( X ; A ) α [ F ] is given by id A · e . Hence[ τ ( id A )] ( γ ′ ,x ′ ) , ( γ,x ) = γ − γ ′ / ∈ G ( γ − ) ∗ [( id A ) ( γ − γ ′ ,γ − x ′ ) , ( e,γ − x ) ] if γ − γ ′ ∈ G = γ, x ) = ( γ ′ , x ′ )( id τA )( γ ′ , x ′ ) , ( γ, x ) if ( γ, x ) = ( γ ′ , x ′ ) (iv) τ ( ψ ◦ φ ) = ( τ ψ ) ◦ ( τ φ ): Let φ = P f ∈ F φ f · f : A → B and ψ = P h ∈ F ψ h · h : B → C be twomorphisms in O G ( X ; A ) α [ F ]. We have τ ( ψ ◦ φ ) = τ (cid:16) X k ∈ F (cid:0) X hf = k ( α f ψ h ) ◦ φ f (cid:1) · k (cid:17) = X k ∈ F X hf = k τ k (cid:0) ( α f ψ h ) ◦ φ f (cid:1) and τ ( ψ ) ◦ τ ( φ ) = (cid:0) X h ∈ F τ h ψ h (cid:1) ◦ (cid:0) X f ∈ F τ f φ f (cid:1) = X k ∈ F X hf = k ( τ h ψ h ) ◦ ( τ f φ f )So it suffices to show τ k (cid:0) ( α f ψ h ) ◦ φ f (cid:1) = ( τ h ψ h ) ◦ ( τ f φ f ) when k = hf .Now by definition, when γ − γ ′ / ∈ Gk − , we have[ τ k (cid:0) ( α f ψ h ) ◦ φ f (cid:1) ] ( γ ′ ,x ′ ) , ( γ,x ) = 0and [( τ h ψ h ) ◦ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) = ( τ h ψ h ) ( γ ′ ,x ′ ) , ( γ ′′ ,x ′′ ) ◦ ( τ f φ f ) ( γ ′′ ,x ′′ ) , ( γ,x ) where on the right hand side of the above identity, we are taking sum over ( γ ′′ , x ′′ ) ∈ Γ × X .However a term in this sum can be non-zero only when γ ′′− γ ′ ∈ Gh − and γ − γ ′′ ∈ Gf − ,which implies when γ − γ ′ / ∈ Gk − , it must be zero. This verifies [ τ k (cid:0) ( α f ψ h ) ◦ φ f (cid:1) ] ( γ ′ ,x ′ ) , ( γ,x ) =[( τ h ψ h ) ◦ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) when γ − γ ′ / ∈ Gk − .When γ − γ ′ ∈ Gk − , on one hand, we have (in the following identities, we are taking sum over( g ′′ , x ′′ ) ∈ G × X .)[ τ k (cid:0) ( α f ψ h ) ◦ φ f (cid:1) ] ( γ ′ ,x ′ ) , ( γ,x ) = ( γ − ) ∗ (cid:2)(cid:0) α f ψ h ) ◦ φ f (cid:1) ( γ − γ ′ k,γ − x ′ ) , ( e,γ − x ) (cid:3) = ( γ − ) ∗ [( α f ψ h ) ( γ − γ ′ k,γ − x ′ ) , ( g ′′ ,x ′′ ) ◦ φ f ( g ′′ ,x ′′ ) , ( e,γ − x ) ]= ( γ − ) ∗ (cid:2) f ∗ ( ψ h ( α f ( γ − γ ′ k ) ,fγ − x ′ ) , ( α f ( g ′′ ) ,fx ′′ ) ) ◦ φ f ( g ′′ ,x ′′ ) , ( e,γ − x ) ]= ( γ − ) ∗ (cid:2)(cid:0) α f ( g ′′− ) f (cid:1) ∗ (cid:0) ψ h ( α f ( g ′′− γ − γ ′ k ) ,α f ( g ′′− ) fγ − x ′ ) , ( e,α f ( g ′′− ) fx ′′ ) ◦ φ f ( g ′′ ,x ′′ ) , ( e,γ − x ) ] (cid:1)(cid:3) = ( γ − ) ∗ (cid:2)(cid:0) f g ′′− (cid:1) ∗ (cid:0) ψ h ( α f ( g ′′− γ − γ ′ k ) ,fg ′′− γ − x ′ ) , ( e,fg ′′− x ′′ ) ◦ φ f ( g ′′ ,x ′′ ) , ( e,γ − x ) (cid:1)(cid:3) = (cid:0) f g ′′− γ − (cid:1) ∗ (cid:0) ψ h ( α f ( g ′′− γ − γ ′ k ) ,fg ′′− γ − x ′ ) , ( e,fg ′′− x ′′ ) (cid:1) ◦ (cid:0) γ − (cid:1) ∗ (cid:0) φ f ( g ′′ ,x ′′ ) , ( e,γ − x ) (cid:1) = ( τ h ψ h ) ( γg ′′ f − α f ( g ′′− γ − γ ′ k ) h − ,γg ′′ f − fg ′′− γ − x ′ ) , ( γg ′′ f − ,γg ′′ f − fg ′′− x ′′ ) ◦ ( τ f φ f ) ( γg ′′ f − ,γx ′′ ) , ( γ,x ) = ( τ h ψ h ) ( γ ′ kf − h − ,x ′ ) , ( γg ′′ f − ,γx ′′ ) ◦ ( τ f φ f ) ( γg ′′ f − ,γx ′′ ) , ( γ,x ) = ( τ h ψ h ) ( γ ′ ,x ′ ) , ( γg ′′ f − ,γx ′′ ) ◦ ( τ f φ f ) ( γg ′′ f − ,γx ′′ ) , ( γ,x ) = ( τ h ψ h ) ( γ ′ ,x ′ ) , ( γg ′′ f − ,x ′′ ) ◦ ( τ f φ f ) ( γg ′′ f − ,x ′′ ) , ( γ,x ) The last identity holds because when x ′′ runs over X , γx ′′ also runs over X . OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 19
Now on the other hand, we have[( τ h ψ h ) ◦ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) = ( τ h ψ h ) ( γ ′ ,x ′ ) , ( γ ′′ ,x ′′ ) ◦ ( τ f φ f ) ( γ ′′ ,x ′′ ) , ( γ,x ) Note we are taking sum over ( γ ′′ , x ′′ ) ∈ Γ × X on the right hand of the above identity. Howeveronly those terms with γ − γ ′′ ∈ Gf − can be non-zero (note γ − γ ′′ ∈ Gf − , γ − γ ′ ∈ Gk − togetherwith k = hf imply γ ′′− γ ′ ∈ Gh − ). So we only have to take sum over those terms with γ ′′ = γg ′′ f − , g ′′ ∈ G . Hence we have[( τ h ψ h ) ◦ ( τ f φ f )] ( γ ′ ,x ′ ) , ( γ,x ) = ( τ h ψ h ) ( γ ′ ,x ′ ) , ( γg ′′ f − ,x ′′ ) ◦ ( τ f φ f ) ( γg ′′ f − ,x ′′ ) , ( γ,x ) Therefore τ k (cid:0) ( α f ψ h ) ◦ φ f (cid:1) = ( τ h ψ h ) ◦ ( τ f φ f ) when k = hf . Thus τ ( ψ ◦ φ ) = ( τ ψ ) ◦ ( τ φ ).(v) Additivity: this is easy.(vi) Object support: ( γ, x, s ) ∈ supp( τ A ) if and only if ( e, γ − x, s ) ∈ supp( A ). By definition, thereis a compact set K ⊆ X , so that γ − x ∈ K for all ( e, γ − x, s ) ∈ supp( A ). Hence γ − x ∈ K for all( γ, x, s ) ∈ supp( τ A ). This shows τ A respects the object support condition.(vii) Morphism support: we firstly show each τ f φ f , f ∈ F respects the morphism support condition.By definition of τ f φ f , its morphism support condition in the Γ-direction is easily seen to be satisfied.Let us focus on its morphism support condition in the X × [1 , ∞ )-direction. By definition, for φ f : A → α f B , there exists E ∈ E XGcc , such that the projection of supp( φ f ) in ( X × [1 , ∞ )) iscontained in E . By definition of τ f φ f and the fact that supp( A ) and supp( α f B ) are G -cocompact inthe G × X -direction, there exists a compact subset K ⊆ X , such that the projection of supp( τ f φ f )in ( X × [1 , ∞ )) is contained in E ′ = { ( x, s ) × ( x ′ , s ′ ) | ∃ γ, γ ′ ∈ Γ s.t. γ − γ ′ ∈ Gf − , γ − x, f − γ ′− x ′ ∈ K, ( γ − x, s ) × ( γ − x ′ , s ′ ) ∈ E } Replacing γ ′ by γ ′ f − , one sees E ′ = { ( x, s ) × ( x ′ , s ′ ) | ∃ γ, γ ′ ∈ Γ s.t. γ − γ ′ ∈ G, γ − x, γ ′− x ′ ∈ K, ( γ − x, s ) × ( γ − x ′ , s ′ ) ∈ E } But by Lemma 2.9, E ′ ∈ E X Γ cc . This shows, for each f ∈ F , τ f φ f respects the morphism supportcondition for morphisms in O Γ ( X ; A ). Therefore τ φ = X f ∈ F τ f φ f respects the morphism supportcondition in O Γ ( X ; A ) since supp( τ φ ) is contained in the finite union S f ∈ F supp( τ f φ f ).We now complete the verification work that τ is a well-defined functor from O G ( X ; A ) α [ F ] to O Γ ( X ; A ). Next we show it is an equivalence of additive categories (with involution). (a) τ is full on objects: for any object A in O Γ ( X ; A ), let Res A denote its restriction to G × X × [1 , ∞ ), i.e. (Res A ) ( g,x,s ) := A ( g,x,s ) , ( g, x, s ) ∈ G × X × [1 , ∞ ). Clearly Res A is an object in O G ( X ; A ) α [ F ]. One also easily sees that τ (Res A ) = A .(b) τ is full on morphisms: for any objects A, B in O G ( X ; A ) α [ F ], morphism Φ : τ A → τ B in O Γ ( X ; A ), and f ∈ F , define φ f : A → α f B by defining φ f ( g ′ ,x ′ ,s ′ ) , ( g,x,s ) := Φ ( g ′ f − ,x ′ ,s ′ ) , ( g,x,s ) Note since Φ ( g ′ f − ,x ′ ,s ′ ) , ( g,x,s ) : ( τ A ) ( g,x,s ) = A ( g,x,s ) −→ ( τ B ) ( g ′ f − ,x ′ ,s ′ ) = ( α f B ) ( g ′ ,x ′ ,s ′ ) we see that φ f ( g ′ ,x ′ ,s ′ ) , ( g,x,s ) is indeed a map from A ( g,x,s ) to ( α f B ) ( g ′ ,x ′ ,s ′ ) .One easily checks φ f is G -invariant. We now check φ f respects the morphism support conditionfor morphisms in O G ( X ; A ). By definition, the projection of supp( φ f ) in the X × [1 , ∞ )-direction isthe same as the projection of supp(Φ) in the X × [1 , ∞ )-direction. Now by assumption, E X Γ cc ⊆ E XGcc .Therefore φ f respects the morphism support condition in the X × [1 , ∞ )-direction. By the morphismcontrol condition of Φ in the Γ-direction, there is a finite set S ⊆ Γ with the property thatΦ ( g ′ f − ,x ′ ,s ′ ) , ( g,x,s ) = 0 implies g − g ′ f − ∈ S . Therefore φ f ( g ′ ,x ′ ,s ′ ) , ( g,x,s ) = 0 implies g − g ′ f − ∈ S .This firstly shows φ f respects the morphism support condition in the G -direction. Therefore φ f isa morphism in O G ( X ; A ). Secondly, since S is finite, its image in F under the natural projectionΓ → F is finite. This implies there are only finitely many f ∈ F with the property that thereexist g, g ′ ∈ G so that g − g ′ f − ∈ S . Hence there are only finitely many f ∈ F so that φ f = 0.Therefore φ = X f ∈ F φ f · f : A → B defines a morphism in O G ( X ; A ) α [ F ].We now show τ φ = Φ. For any ( γ ′ , x ′ , s ′ ) , ( γ, x, s ) ∈ Γ × X × [1 , ∞ ), let γ − γ ′ = gf − , g ∈ G, f ∈ F . We then have ( τ φ ) ( γ ′ ,x ′ ,s ′ ) , ( γ,x,s ) = ( τ f φ f ) ( γ ′ ,x ′ ,s ′ ) , ( γ,x,s ) = ( γ − ) ∗ [ φ f ( g,γ − x ′ ,s ′ ) , ( e,γ − x,s ) ]= ( γ − ) ∗ [Φ ( gf − ,γ − x ′ ,s ′ ) , ( e,γ − x,s ) ]= Φ ( γ ′ ,x ′ ,s ′ ) , ( γ,x,s ) This shows τ φ = Φ and completes the proof that τ is full on morphisms.(c) τ is faithful on morphisms: this is easy using the identity φ f ( g ′ ,x ′ ,s ′ ) , ( g,x,s ) := ( τ φ ) ( g ′ f − ,x ′ ,s ′ ) , ( g,x,s ) . OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 21
Therefore τ is an equivalence between O G ( X ; A ) α [ F ] and O Γ ( X ; A ) provided E X Γ cc ⊆ E XGcc . Nowby Lemma 2.7, E X Γ cc ⊆ E XGcc if F is finite or the action of Γ on X is free. This completes the proofof the theorem. (cid:3) Remark . One sees from the proof that the assumption E X Γ cc ⊆ E XGcc is only used to show thefunctor τ is full. So τ is well-defined even without this assumption.4. Proof of the Main theorems
In this section, we prove Theorem A and Theorem B. We firstly prove a lemma on L -groupswhich is another ingredient in proving Theorem A.4.1. A Lemma on L -groups. In this subsection, we prove Lemma 4.1. We will need the followingwell-known fact: for every group G and every orientation map w : G −→ {± } , if W h ( G ) =0 , ˜ K ( Z [ G ]) = 0 and K i ( Z [ G ]) = 0 , i ≤ −
1, then all the L -groups of Z [ G ] associated to w withvarious decorations are naturally isomorphic (we will omit w from the notations): L sn ( Z [ G ]) ∼ = L hn ( Z [ G ]) = L n ( Z [ G ]) ∼ = L n ( Z [ G ]) ∼ = L < − >n ( Z [ G ]) ∼ = · · · · · · ∼ = L < −∞ >n ( Z [ G ])where W h ( G ) , ˜ K ( Z [ G ]) and K i ( Z [ G ] , i ≤ − K -group andthe negative K -groups of Z [ G ] respectively. For an explanation about the decorations of various L -groups and a proof of the above fact, see [25, Remmark 21 on page 720 and Proposition 23 onpage 721]. Lemma 4.1.
Let G be a group. Assume W h ( G ) = 0 , ˜ K ( Z [ G ]) = 0 and K i ( Z [ G ]) = 0 , i ≤ − .Then for every group of the form Γ = G ⋊ α Z and every orientation map w : Γ −→ {±} , its simple L -groups and ultimate L -groups are naturally isomorphic, i.e. L sn ( Z [Γ]) ∼ = L < −∞ >n ( Z [Γ]) , ∀ n ∈ Z ,naturally. If in addition W h ( G × Z ) = 0 , then L hn ( Z [Γ]) adds to the natural isomorphisms, i.e. L sn ( Z [Γ]) ∼ = L hn ( Z [Γ]) ∼ = L < −∞ >n ( Z [Γ]) , ∀ n ∈ Z , naturally.Proof. When α : G −→ G is trivial, then the result follows immediately from the above well-knownfact and the Shaneson splitting : L n ( Z [ G × Z ]) ∼ = L n ( Z [ G ]) ⊕ L hn − ( Z [ G ])which is natural with respect to the natural forgetful maps between them indexed by s → h → →· · · → −∞ .For the general case, we need to make use of the results of Ranicki [27]. Firstly, there is a longexact sequence [27, Page 413]: · · · −→ L sn ( Z [ G ]) −→ L sn ( Z [Γ]) −→ L ′ n − ( Z [ G ]) −→ L sn − ( Z [ G ]) −→ · · · (4.1) where L ′ n − ( Z [ G ]) is a certain intermediate L -group with torsion lies in (1 − α ∗ ) − ( G ), where1 − α ∗ : ˜ K ( Z [ G ]) −→ ˜ K ( Z [ G ])is the map induced by α and G is the image of G in ˜ K ( Z [ G ]). Now by assumption, W h ( G ) = 0, thisimplies G = ˜ K ( Z [ G ]). Hence (1 − α ∗ ) − ( G ) = ˜ K ( Z [ G ]). Therefore L ′ n ( Z [ G ]) = L hn ( Z [ G ]) , ∀ n ∈ Z and 4.1 becomes: · · · −→ L sn ( Z [ G ]) −→ L sn ( Z [Γ]) −→ L hn − ( Z [ G ]) −→ L sn − ( Z [ G ]) −→ · · · (4.2)Now there is also a well-known Wang type long exact sequence for the ultimate L -groups ([23, theproof of Lemma 4.2]): · · · −→ L < −∞ >n ( Z [ G ]) −→ L < −∞ >n ( Z [Γ]) −→ L −∞ n − ( Z [ G ]) −→ L < −∞ >n − ( Z [ G ]) −→ · · · (4.3)The natural forgetful maps between these groups give rise to a commutative diagram of the abovelong exact sequences. By assumption, all these forgetful maps are isomorphisms for the L -groupsof Z [ G ]. Therefore, by the five lemma, L sn ( Z [Γ]) ∼ = L < −∞ >n ( Z [Γ]).If in addition W h ( G × Z ) = 0, then by replacing G by G × Z in 4.1, where we use the trivialaction of Z on Z (so ( G × Z ) ⋊ Z ∼ = Γ × Z ), and use the same argument as we did for G , we get · · · −→ L sn ( Z [ G × Z ]) −→ L sn ( Z [Γ × Z ]) −→ L hn − ( Z [ G × Z ]) −→ L sn − ( Z [ G × Z ]) −→ · · · (4.4)By Shaneson splitting, the above sequence naturally splits into two long exact sequences, one ofwhich is 4.2 and another one is: · · · −→ L hn − ( Z [ G ]) −→ L hn − ( Z [Γ]) −→ L n − ( Z [ G ]) −→ L n − ( Z [ G ]) −→ · · · (4.5)Now as before, 4.2, 4.3, and 4.5 together imply the natural isomorphisms L sn ( Z [Γ]) ∼ = L hn ( Z [Γ]) ∼ = L < −∞ >n ( Z [Γ]). (cid:3) Remark . We cannot conclude isomorphisms for the L -groups of Z [Γ] with other decorations.There is the issue of intermediate L -groups. But if we assume W h ( G × Z n ) = 0 , ∀ n ∈ N , then allthe L -groups of Z [Γ] are naturally isomorphic.4.2. Proof of Theorem B.
In this subsection, we prove Theorem B first. Theorem A will be aconsequence of Theorem B and Lemma 4.1. We will prove the more general version of Theorem Bas explained in Remark 1.5. We firstly treat the L -theory case. Let A be a right G ⋊ β Z additivecategory with involution. View it as a G -additive category in the natural way. Since G is torsionfree, G ⋊ β Z is also torsion free. This implies the L -theoretic assembly map A A L for G ⋊ β Z in 2.2is equivalent to the following assembly map: H G ⋊ β Z n ( E ( G ⋊ β Z ); L < −∞ > A ) → L < −∞ >n ( A α [ G ⋊ β Z ]) OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 23 where E ( G ⋊ β Z ) is the classifying space for free G × β Z -actions. See [25, Proposition 66 on page743]. Compare [2, Theorem 8.14]. Therefore, by Theorems 2.5, 2.6, A A L for G × β Z is an isomorphismif and only if L < −∞ >n ( O G ⋊ β Z ( E ( G ⋊ β Z ) , A )) = 0 , ∀ n ∈ Z Now by Theorem 3.6, we have an equivalence of additive categories with involutions O G ⋊ β Z ( E ( G ⋊ β Z ) , A ) ∼ = O G ( E ( G ⋊ β Z ) , A ) β [ Z ]since the action of G ⋊ β Z on E ( G ⋊ β Z ) is free. Denote O G ( E ( G ⋊ β Z ) , A ) by ˜ A . Note that bythe characterization properties of classifying spaces, E ( G ⋊ β Z ) is also a model for the classifyingspace for free G -actions. Now by assumption, the L -theoretic FJC holds for G with coefficient in A , therefore, by Theorems 2.5, 2.6, we have L < −∞ >n ( ˜ A ) = 0 , ∀ n ∈ Z Now by applying the Wang type long exact sequence 4.3 to L < −∞ > ∗ ( ˜ A β [ Z ]), which is well-knownwhen the coefficient is a ring and can be extended to with twisted coefficient in any additive category(see [2, proof of Theorem 8.14]), we immediately get that L < −∞ >n ( ˜ A β [ Z ]) = 0 , ∀ n ∈ Z This implies L < −∞ >n ( O G ⋊ β Z ( E ( G ⋊ β Z ) , A )) = 0 , ∀ n ∈ Z Therefore the L -theoretic FJC with coefficient in A holds for G ⋊ β Z . This proves part (1) ofTheorem B.We now turn into part (2) of Theorem B. When R is regular and G ⋊ β Z is torsion free, the K -theoretic assembly map A RK for G ⋊ β Z in 2.1 is equivalent to the following assembly map: H G ⋊ β Z n ( E ( G ⋊ β Z ); K A ) → K n ( A α [ G ⋊ β Z ])Similar to the argument as in the L -theoretic case, we see that the above assembly map is anisomorphism if and only if K n ( O G ( E ( G ⋊ β Z ) , R ) β [ Z ]) = 0 , ∀ n ∈ Z Now, there is a Wang type long exact sequence for the non-connective K -theory of O G ( E ( G ⋊ β Z ) , R ) β [ Z ]. This is well-known when the coefficient is a ring and has been recently generalized byL¨uck-Steimle [26, Theorem 0.1, Remark 0.2] to with coefficient in any additive category. Using thissequence and the assumption that the K -theoretic FJC holds for G with coefficient in R , we seethat K n ( O G ( E ( G ⋊ β Z ) , R ) β [ Z ]) ∼ = N K n ( O G ( E ( G ⋊ β Z ) , R ); β ) ⊕ N K n ( O G ( E ( G ⋊ β Z ) , R ); β ) Denote the Nil-groups
N K n ( O G ( E ( G ⋊ β Z ) , R ); β ) as defined in [26] by N il G ⋊ β Z n,R , then K n ( O G ( E ( G ⋊ β Z ) , R ) β [ Z ]) = 0if and only if N il G ⋊ β Z n,R = 0. This proves part (2) of Theorem B.4.3. Proof of Theorem A.
We now prove Theorem A. Let us assume G ∈ F J . If G is nottorsion free, then there is nothing to prove, since it cannot be realized as the fundamental group ofa closed aspherical manifold. So let us assume G is torsion free. Since G satisfies the K -theoreticFJC with coefficient in Z , it follows that W h ( G ) = 0 , ˜ K ( Z [ G ]) = 0 , K i ( Z [ G ]) = 0 , i ≤ − w : G ⋊ Z −→ {±} , natural isomorphisms L sn ( Z [ G ⋊ Z ]) ∼ = L < −∞ >n ( Z [ G ⋊ Z ]) , ∀ n ∈ Z (4.6)Because G is torsion free and satisfies the L -theoretic FJC with coefficient in Z , Theorem B appliesand the L -theoretic FJC with coefficient in Z holds for G ⋊ Z . Therefore, by a standard surgerylong exact sequence argument, we see that the simple Borel conjecture holds for G ⋊ Z . For theconvenience of the reader, let us sketch the main idea, more details can be found in [25, Theorem28 on page 723], [17, pages 26-28] . Let M be a closed aspherical manifold of dimension n ≥ π ( M ) ∼ = G ⋊ Z . Let S s ( M ) be its simple topological structure set (an abelian group indeed). Itconsists of the equivalence classes of all simple homotopy equivalence f : N −→ M , from anotherclosed manifold N to M . Two such maps f : N −→ M, f ′ : N ′ −→ M are equivalent if thereis a homeomorphism g : N −→ N ′ so that f ′ ◦ g is homotopic to f . Therefore the simple Borelconjecture holds for G if and only if S s ( M ) consists of one point. Now there is a surgery long exactsequence for S s ( M ): · · · / / N n +1 ( M ) σ n +1 / / L sn +1 ( Z [ G ⋊ Z ]) ∂ n +1 / / S s ( M ) η n / / N n ( M ) σ n / / L sn ( Z [ G ⋊ Z ])For each i ≥ n , the assembly map 2.2 and the above sequence fit into the following commutativediagram N i ( M ) σ i / / p i (cid:15) (cid:15) L si ( Z [ G ⋊ Z ]) id (cid:15) (cid:15) H i ( M ; L s Z ) q i (cid:15) (cid:15) / / L si ( Z [ G ⋊ Z ]) f i (cid:15) (cid:15) H G ⋊Z i ( E ( G ⋊ Z ); L < −∞ > Z ) A Z L / / L < −∞ >i ( Z [ G ⋊ Z ]) OPOLOGICAL RIGIDITY FOR FJ BY THE INFINITE CYCLIC GROUP 25
We have showed that f i and A Z L are isomorphisms for all i ≥ n . q i is also an isomorphism since H G ⋊Z i ( E ( G ⋊ Z ); L < −∞ > Z ) ∼ = H i ( M ; L < −∞ > Z ) ∼ = H i ( M ; L s Z )where the first isomorphism comes from, by viewing M = B ( G ⋊Z ) = E ( G ⋊Z ) /G ⋊Z , the inductionstructure of an equivariant homology, see [25, Remark 61 on page 736], and the second isomorphismis because the two homology theories H ∗ ( − ; L s Z ) , H ∗ ( − ; L < −∞ > Z ) are naturally isomorphic (sincethey are naturally isomorphic on a point). Now it is a fact from surgery theory that p i is injectivewhen i = n , and bijective when i > n . This implies that σ n is injective and σ n +1 is bijective.Therefore S s ( M ) is trivial and the simple Borel conjecture holds for G ⋊ Z .If in addition W h ( G × Z ) = 0, then by Lemma 4.1, we have L hn ( Z [ G ⋊ Z ]) ∼ = L < −∞ >n ( Z [ G ⋊ Z ]) , ∀ n ∈ Z (4.7)This implies, by using the surgery long exact sequence for the structure set S h ( M ), which consistsof h -cobordant equivalence classes of all homotopy equivalent maps to M , and a similar argumentas above, that S h ( M ) consists of a single element. This implies the bordism Borel conjecture holdsfor G ⋊ Z . This completes the proof of Theorem A. References [1] Douglas R. Anderson, Francis X. Connolly, Steven C. Ferry, and Erik K. Pedersen. Algebraic K -theory withcontinuous control at infinity. J. Pure Appl. Algebra , 94(1):25–47, 1994.[2] A. Bartels, F. T. Farrell, and W. L¨uck. The Farrell-Jones conjecture for cocompact lattices in virtually connectedLie groups.
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