The isomorphism conjecture in L-theory: graphs of groups

THE ISOMORPHISM CONJECTUR E IN L -THEOR Y: GRAPHS OF GROUPS S. K. ROUSHON Abstract. W e study the Fiber ed Is o morphism Co njecture of F ar - rell and Jones in L -theory for g roups a cting on trees. In se veral cases w e pr ov e the conjecture. This includes wreath pr o ducts of ab elian gro ups and free metab elian gro ups. W e als o deduce the conjecture in pseudoisotopy theory for these gr o ups. Finally in B of Theo rem 1.1 we prove the L -theory version of [[7], Theor em 1.2]. 1. Intro duction and st a teme nts of res ul ts The classification problem for manifolds needs the study of tw o classes of obstruction groups. One is the lo w e r K -groups (that is K - theory in dimension ≤ 1 ) (pseudoisotop y-theory) and t he o t her is the surgery L -groups (surgery theory) of the gro up ring o f the fundamen- tal group. The F arrell-Jo nes Isomorphism Conjecture giv es an unified approac h for computations and understanding of b oth these classes of groups. If this conjecture is true for the pseudoisotop y theory as well as f or the surgery theory then a mong other results, for example, the Borel Conjecture, the No vik o v Conjecture and the Hsiang Conjecture will be immediate conseque nces (see [9]). The F arrell-Jones Conjecture predicts that one needs to consider only virtually cyclic subgroups of a group for computations o f the ab o v e obstruction gro ups of the group. In this second article we a re concerned ab out the Fib ered Isomor- phism Conjecture in surgery theory for gro ups acting (without in v er- sion) on trees or equiv alen tly for the fundamen ta l groups of graphs of groups. Also w e prov e the conjecture f o r a certain class of virtually solv able gro ups in b oth the pseudoisotop y and surgery theory . The Fib ered Conjecture in surgery theory for v a rious classes of groups were pro v ed in [17]. Also some mac hinery w as set up in [17] whic h are crucial in this pap er. 2000 Mathematics Subje ct Classific ation. P rimary: 19G24, 19J25 . Secondary: 55N91. Key wor ds and phr ases. gr oup a ction on trees, gr a ph of gr o ups, fibere d is omor- phism conjecture, L -theory , surgery groups. April 06, 2010 . 1 2 S.K. ROUSHON The Fib ered Isomorphism Conjecture is strong er and has hereditary prop ert y . Also it allows one to consider groups with torsion in induc- tion steps, although the final aim is to pro v e r esults f or torsion free groups. This tec hnique w as first used in [8] to prov e the conjecture in the pseudoisotop y case for Artin full braid groups. The general metho ds in [16] and [17] extend this feature further by considering the conjecture alwa ys for groups wreath pro duct with finite groups. This simplifies pro ofs a nd pro v e stronger results. In most of our results o f the Fib ered Isomorphism Conjecture in the equiv arian t homo lo gy theory ([2]) w e need the assumption that w t T V C , P V C and L V C (see Definition 2 .2) are satisfied. W e c hec k ed b efore that these conditions are satisfied for the L h−∞i and for the pseudoisotop y v ersion of the conjecture. (See [16] and [1 7]). In [[4], Theorem 0.1] it is included that L V C and P V C are satisfied f or the K - theory case o f the conjecture. F ormally , the conjecture in surgery theory say s t ha t certain assem- bly map in L h−∞i -theory is an isomorphism. A w eak er v ersion of the conjecture is that the assem bly map is a n isomorphism af ter tensoring with Z [ 1 2 ]. This eliminates the UNiL groups o f Capp ell and the ten- sored assem bly map can b e prov en to b e an isomorphism for a larger class of groups. In additio n to some general results we also prov e the isomorphism o f t his tensored assem bly map for a larg e class of groups acting on trees. F or tw o groups G a nd H , G ≀ H denotes the (restricted) wreath pro duct with respect to the regular action of H on G H . By definition G H = L h ∈ H G h where G h ’s ar e copies of G indexed by H . And the action of H on G H is suc h that h ′ ∈ H sends an elemen t of G h to the corresp onding elem en t o f G h ( h ′ ) − 1 . If the Fib ered Isomorphism Conjecture is true for G ≀ F f o r all finite groups F for the L h−∞i , L h−∞i = L h−∞i ⊗ Z Z [ 1 2 ] or for the pseudoiso- top y theory then w e sa y resp ectiv ely that t he F I C w F L , F I C w F L or F I C w F P is true for G . Throughout the article a ‘graph’ is assumed to b e connected and lo cally finite. And groups are assumed to b e discrete and coun table. Definition 1.1. A finitely generated group G is called closely crystal- lo gr aphic if it is of the fo rm A ⋊ C where A is torsion free ab elian, C is infinite cyclic and A is irreducible as a Q [ C ]- mo dule. When C is virtually cyclic then G w as defined as n e arly crystal lo- gr aphic in [[7], Definition]. Our fir st theorem is the following. THE ISOMORPHISM CONJECTURE IN L -THEOR Y 3 Theorem 1.1. A. L et G b e a gr oup whic h c ontains a sub gr oup H of finite index so that H b elongs to one of the fol lowing classes. a. A ≀ B wher e A an d B ar e b oth ab elian. b. F r e e metab elian gr oups. That is , it is a quotient of a fr e e gr oup by the se c ond de rive d sub gr oup. c. A ⋊ Z wher e A is torsion ab elian . Then the F I C w F L and the F I C w F P ar e sa tisfie d for G . B. I f the F I C w F L ( F I C w F P ) is true fo r al l clo sely crystal lo gr aphic gr oups then the F I C w F L ( F I C w F P ) is true for al l virtual ly solvable gr oups. Remark 1.1. It is not yet know n if the Fib ered Isomorphism Conjec- ture is tr ue fo r all metab elian groups. The simplest case for whic h it is unkno wn is Z [ 1 2 ] ⋊ Z where the action of Z on Z [ 1 2 ] is m ultiplication b y 2. One can sho w that Z [ 1 2 ] ⋊ Z can not b e em b edded in A ≀ B where A and B are b oth ab elian. I thank Ch uc k Miller for explaining this fact to me. On the other hand b y a result of Magnus free metab elian groups can be em bedded in suc h a wreath pro duct. Although our metho d do es not w ork to deduce the Fib ered Isomor- phism Conjecture in the closely crystallographic case, the Isomorphism Conjecture can b e pro v en for these gro ups in surgery theory for all the decorations. Theorem 1.2. The Isomorphi s m Conje c tur e in L i -the ory is true for closely crystal lo gr aphic gr oups wher e i = h−∞i , h or s . Remark 1.2. Here w e recall tha t in [[7], Theorem 1.2] it w as pro v ed that the Fib ered Isomorphism Conjecture in the pseudoisotop y theory is true for a ny virtually solv able groups if the same is true for an y nearly crystallographic groups. Th us B of Theorem 1 .1 is t he L -theory v ersion of [[7], Theorem 1 .2]. The follow ing is an Imp ortant Assertion in the Fib ered Isomorphism Conjecture. In general it is not y et kno wn. IA(K) . K is a normal subgroup of a gr o up G with infinite cyclic quotien t. If the F I C w F V C ( K ) is satisfied then the F I C w F V C ( G ) is also satisfied. Theorem 1.3. L e t G b e a gr aph of g r oups with finite e dge gr oups. A. I f the vertex gr oups ar e r e sidual ly finite and the F I C w F L is true for the vertex gr oups of G then the F I C w F L is true for π 1 ( G ) . B. Assume that ther e is a homom o rphism f : π 1 ( G ) → Q . The n the fol lowing statements hold. 4 S.K. ROUSHON i. I f the k e rn els of the r estriction of f to the ve rtex gr oups of G ar e finitely gener ate d, r esidual ly finite and satisfies the F I C w F L then the F I C w F L is true f o r π 1 ( G ) pr ovide d the same is true fo r Q and the IA(V) is satisfie d for a l l vertex gr oups V of G in the L -the ory c ase. ii. If the kernels of the r estriction of f to the vertex gr oups of G ar e virtual ly p olycyclic and the F I C w F L is true for Q then the F I C w F L is true for π 1 ( G ) . Let us now recall from [1 6] the fo llo wing definitions. V G and E G denotes resp ectiv ely the set of all v ertices and edges of a graph of groups G . G x denotes a vertex or an edge group fo r x a ve rtex o r an edge resp ectiv ely . An edge e of G is called a finite e dg e if the edge group G e is finite. G is called almost a tr e e of gr oups if there are finite edges e 1 , e 2 , . . . so that the components of G − { e 1 , e 2 , . . . } are tr ee. If w e remov e all the finite edges from a graph of groups then we call the comp onen ts of the resulting graph as c o mp one nt sub g r aphs . A graph of groups G is said to satisfy the interse ction pr op erty if each connected subgraph of groups G ′ of G , ∩ e ∈ E G ′ G ′ e con tains a subgro up whic h is normal in π 1 ( G ′ ) and is of finite index in some edge group. A gr o up G is called sub gr oup sep ar able if for an y finitely generated subgroup H of G and for an y g ∈ G − H there is a finite index normal subgroup N of G so that H ⊂ N and g ∈ G − N . Theorem 1.4. The F I C w F L is true for π 1 ( G ) wher e G s a tisfies one of the fol lowing. A. G is a gr a p h of p oly-cyclic gr oups with interse ction pr op erty. B. G is a gr aph of finitely gener ate d nilp otent gr oups with π 1 ( G ) sub- gr oup sep ar a ble. C. The vertex gr oups ar e v i rtual l y cyclic and any c omp onent sub gr aph is either a single vertex or a tr e e of ab elian gr oups. D. The vertex and e dge gr oups of any c omp onent sub gr aph ar e finitely gener ate d ab elian and of the same r ank and any c omp onent sub gr ap h i s a tr e e. Finally we state our results in the L h−∞i -theory case . Let D b e a class of groups whic h is closed under isomorphism. F or a graph G w e denote b y D G the class of graphs of g roups whose v ertex and edge groups b elong to D and the underlying graph is G . Theorem 1.5. A. If the F I C w F L ( π 1 ( T )) is satisfie d for al l tr e e of gr oups T then the F I C w F L ( π 1 ( G )) is satisfie d for al l gr ap h of g r oups G . THE ISOMORPHISM CONJECTURE IN L -THEOR Y 5 B. I f the F I C w F L ( π 1 ( H )) is satisfie d for al l H ∈ D T and fo r al l tr e e T then the F I C w F L ( π 1 ( H )) i s satisfie d for al l H ∈ D G and for al l gr aph G . Theorem 1.6. L et G b e a gr aph o f finitely gene r ate d ab elian g r oups. Then the F I C w F L is true for π 1 ( G ) . Theorem 1.7. L e t G b e a gr aph of g r oups with finite e dge gr oups. A. I f the vertex gr oups ar e r e sidual ly finite and the F I C w F L is true for the vertex gr oups of G then the F I C w F L is true for π 1 ( G ) . B. Assume that ther e is a homom orphism f : π 1 ( G ) → Q and the F I C w F L is true for Q . If the kernels o f the r estriction of f to the vertex gr o ups of G ar e r esidual l y finite and satisfies the F I C w F L then the F I C w F L is true for π 1 ( G ) . 2. S t a tement of the Isomorphism Conjecture and some basic resul ts No w w e pro ceed to describ e the formal statemen t of the conjecture (see [2]) and in tro duce some notations. Let H ? ∗ b e an equiv ariant homology theory with v alues in R -mo dules for R a comm utativ e asso ciativ e ring with unit. In this article w e are considering the sp ecial case R = Z . In this section we alwa ys assume that a class of groups C is closed under isomorphisms, ta king subgroups and taking quotien ts. W e de- note b y C ( G ) the set of all subgroups of a group G whic h b elong to C . In this case C ( G ) is said t o b e a family of sub gr oups of G . It follo ws that C ( G ) is closed under taking subgroup and conjug a tion. Giv en a group homomorphism φ : G → H and C a family of sub- groups of H define φ ∗ C b y the family of subgroups { K < G | φ ( K ) ∈ C } of G . F or a family C of subgroups of a group G there is a G -CW complex E C ( G ) whic h is unique up to G -equiv a lence satisfying the prop ert y that for eac h H ∈ C the fixp oint set E C ( G ) H is con tractible and E C ( G ) H = ∅ for H not in C . The Isomorphism Conje ctur e for the pair ( G, C ) states that the pro- jection p : E C ( G ) → pt to the p oint pt induces an isomorphism H G n ( p ) : H G n ( E C ( G )) ≃ H G n ( pt ) for n ∈ Z . And the Fib er e d Isomo rphism C onje ctur e f or the pair ( G, C ) states that for an y group homomorphism φ : K → G the Isomorphism Con- jecture is true for the pair ( K, φ ∗ C ). 6 S.K. ROUSHON Definition 2.1. ([ [16] , Defi n ition 2.1]) Let C b e a class o f groups. If the (Fib ered) Isomorphism Conjecture is true f or the pair ( G, C ( G )) w e sa y that the (F)IC C is tru e for G or simply sa y (F)IC C ( G ) is sa tisfi e d . Also w e sa y that the (F)I CwF C ( G ) is sa tisfi e d if the (F)IC C is true for G ≀ H f or any finite group H . Clearly , if H ∈ C t hen the (F)IC C ( H ) is satisfied. Let us denote b y P , L and L , t he equiv ar ia n t homology theories arise for the pseudoisotopy theory , L h−∞i -theory a nd for the L h−∞i -theory resp ectiv ely . W e also denote the corr esp o nding conjectures with resp ect to the class of groups V C by (F)IC X ((F)ICwF X ) where X = P , L or L . Definition 2.2. ([ [16] , Definition 2.2 ]) W e sa y that w t T C is satisfied if for a g r a ph o f groups G with trivial edge gro ups and the vertex gr oups b elonging to the class C , the FICwF C for π 1 ( G ) is true . And w e sa y that P C is satisfied if fo r G 1 , G 2 ∈ C the pro duct G 1 × G 2 satisfies the FIC C . W e further say that L C is satisfied if for an y directed sequence of groups { G } i ∈ I for whic h the FIC C ( G i ) is satisfied for i ∈ I then the FIC C (lim i ∈ I G i ) is satisfied. W e denote the a b o v e prop erties for the equiv aria nt homolog y theo- ries P , L and L with a sup er-script b y the corr esp o nding theory . F or example P C for L is denoted b y P L C . W e now recall some results w e need to pro v e the The orems. Lemma 2.1. Assume that L C is satisfie d. If f o r a dir e cte d se quenc e of gr oups { G } i ∈ I the FI C wF C ( G i ) is satisfie d for i ∈ I then the FICwF C is true for lim i ∈ I G i . Pr o of. Giv en a finite gro up F note t he follo wing equality . (lim i ∈ I ( G i )) ≀ F = lim i ∈ I ( G i ≀ F ) . The pro o f now follo ws.  The f o llo wing is easy to pro v e and is kno wn as the hereditary prop- ert y of t he Fib ered Isomorphism Conjecture. Lemma 2.2. If the FI C C (FICwF C ) is true for a gr oup G then the FIC C (FICwF C ) is true for any sub g r oup H o f G . Lemma 2.3. ([ [17] , L emma 2.2]) Assume that P C is satisfie d. (1). If G 1 and G 2 satisfy the FI C C (FICwF C ) then G 1 × G 2 satisfies the FIC C ( FICwF C ). THE ISOMORPHISM CONJECTURE IN L -THEOR Y 7 (2). L et G b e a finite index sub gr oup of a gr oup K . If the gr oup G satisfies the FICwF C then K also satisfies the FIC wF C . (3). L et p : G → Q b e a gr o up homom orphism. If the FI CwF C is true for Q and for p − 1 ( H ) for al l H ∈ C ( Q ) then the FIC wF C is true for G . If C = V C then using (2) it is enough to c onsider H ∈ C ( Q ) to b e infinite cyclic. Lemma 2.4. ([ [16] , Cor ol lary 5. 3 ] and [ [17] , L emma 2 . 11]) The pr op- erties P P V C , P L V C , P L V C and P L F I N ar e satisfie d. Lemma 2.5. ([ [16] , Cor ol lary 2. 1 ] and [ [17] , L emma 2 . 14]) The pr op- erties w t T P V C , w T L F I N , w t T L V C and w t T L V C ar e sa tisfie d. The pro ofs of the prop erties P and T in the L -theory case w ere giv en in [17] as referred using [[6 ], Theorem 2.1 and Remark 2.1.3]. See Remark 5.1 regarding the presen t status of the pro o f of [[6], Theorem 2.1 and Remark 2.1.3]. Here w e sk etch alternate pro ofs of the ab o v e prop erties using some recen t result of Bartels and L¨ uc k in [3]. In fa ct w e can even prov e w f T L V C . A lternate pr o o fs of P L V C , w T L F I N and w t T L V C . The pro of of these fa cts in the pseudoisotop y case of the Fib ered Isomorphism Conjecture w ere giv en in [16]. The same pro ofs also apply in the L -theory case if we use [3]. W e describe b elo w the c hanges required. F or P L V C replace P b y L a nd use [[3], Theorem B] in the pro of of [[1 6], Corollary 5.3]. Also see [[1 8], Section 3] for some more on this matter. F or a pro of of w t T L V C use the last paragraph of the pro of of [[16], Prop osition 2.4] after replacing P by L and use [[3], Theorem B]. W e also need to use some basic deductions from [[18], Section 3]. The pro o f of the second prop ert y is immediate.  F or the pro of of w f T L V C w e again use the pro of of [[16], Prop osition 2.4]. The follo wing Lemma is another ingredien t for the pro ofs of the Theorems. Lemma 2.6. ([ [7] , T he or em 7.1]) The pr op erties L P V C , L L V C , L L V C and L L F I N ar e sa tisfie d. Finally we recall the followin g t w o lemmas. Lemma 2.7. ([ [16] , L emm a 6.3]) Assume P C and w t T C ar e satisfie d. If the F ICwF C is true for G 1 and G 2 then the FIC wF C is true for G 1 ∗ G 2 . Lemma 2.8. L et F I N ⊂ C . Assume P C , L C and w t T C ar e satisfie d. Then the FICw F C is true for G wher e G is either a v i rtual l y ab elian gr oup or a virtual l y f r e e gr oup. 8 S.K. ROUSHON Pr o of. By Lemma 2.1 w e can assume that the group G is finitely gener- ated. A t first assume tha t G is virtually ab elian. Using (2) of Lemma 2.3 w e r educe to the case of finitely generated a b elian groups. Since F I N ⊂ C and since the conjecture is true for mem bers of C it is enough to prov e the Lemma for finitely g enerated f ree ab elian g roups. No w (1 ) of Lemma 2.3 implies that we need to consider o nly the infinite cyclic group. Since the fundamen tal group of a graph of g roups with t rivial stabilizers is a free group, w e are done using w t T C and Lemma 2.2. Next assume that G is finitely generated and virtually f ree. Aga in using (2) of Lemma 2.3 it is enough to assume that G is a finitely generated free g r o ups. No w note tha t a fr ee group is isomorphic to the fundamen tal group of a graph o f groups whose v ertex gro ups are trivial. Hence using w t T C w e complete the pro of.  3. P ro ofs of the Theorems F or the pro of of Theorem 1.1 w e prov e the following g eneral Theorem for the conjecture in equiv a r ian t homolo gy t heory . The adv antage of this general statemen t is that it works for the conjecture in an y equi- v ariant homology theory and to prov e Theorem 1.1 w e j ust hav e to sho w that the h ypotheses are satisfied b oth for the pseudoisotopy and for the L h−∞i -theory case. Theorem 3.1. Assume that w t T V C , P V C and L V C ar e satisfie d. Then the fol lowing hold. 1. The F I C w F V C is true fo r G if G c ontains A ≀ B as a sub gr oup of finite index, wher e A and B ar e ab elian gr oups. 2. The F I C w F V C is true for any virtual ly fr e e metab elian gr oup. 3. The F I C w F V C is true for A ⋊ Z w her e A is torsion ab elian. 4. Th e F I C w F V C is true for a ny virtual ly solvable gr oups pr ovide d it is true for any closely crystal lo gr aphic gr o ups. Remark 3.1. In the algebraic K -theory v ersion of the conjecture P V C and L V C are known. See [[4], Theorem 0.1]. But it is not y et kno wn if w t T V C is also satisfied. If this is the case then t o gether with the results in this pap er most of the results from [16] and [17] will be true for the Fib ered Isomorphis m Conjecture in algebraic K -theory . Here we should recall that it w a s pro v ed in [[7], Lemma 4.3 ] that the pseudoisotop y ve rsion of the Fib ered Isomorphism Conjecture is true for ( Z n ≀ Z ) ≀ F , where F is a finite group. That is, the F I C w F P is true for Z n ≀ Z . Pr o of of The o r em 3.1. Using (2) of Lemma 2.3 it is enough to prov e the F I C w F V C for A ≀ B where B is infinite, for free metabelian groups and THE ISOMORPHISM CONJECTURE IN L -THEOR Y 9 for solv able groups under the resp ectiv e hypotheses as in (1), (2) and (3). Also we will use the fa ct that the F I C w F V C is tr ue fo r virtually ab elian gro ups during the pro of. See Lemma 2.8. Pro of of 1. At first w e reduce the situatio n to the case A ≀ Z . Let B = lim j ∈ J B i where B j are increasing sequence of finitely generated subgroups of B . Then w e get the follow ing equality . A ≀ B = lim j ∈ J ( A ≀ B j ) . Therefore from no w on we can assume that B is finitely generated. If B ha s rank equal to k then B = Z k × F 1 where F 1 is finite. He nce A ≀ B con tains A Z k × F 1 ⋊ Z k = A F 1 ≀ Z k as a subgroup of finite index. Therefore w e can use (2) of Lemma 2.3 to reduce the situation to the case A ≀ Z k . If k ≥ 2 then note the following equalit y . Let Z k = B 1 × B 2 where B 1 and B 2 are b oth nontrivial. Then A ≀ ( B 1 × B 2 ) = A B 1 × B 2 ⋊ ( B 1 × B 2 ) < ( A B 1 × B 2 ⋊ B 1 ) × ( A B 1 × B 2 ⋊ B 2 ) ≃ ( A B 2 ≀ B 1 ) × ( A B 1 ≀ B 2 ) . In the ab ov e displa y , for i = 1 , 2, the action of B i on A B 1 × B 2 is the restriction of the regular action of B 1 × B 2 on A B 1 × B 2 . Note that the restricted action o f B 1 ( B 2 ) is aga in regular on ( A B 2 ) B 1 (( A B 1 ) B 2 ). And the second inequ alit y is easily c hec k ed b y show ing that the map A B 1 × B 2 ⋊ ( B 1 × B 2 ) → ( A B 1 × B 2 ⋊ B 1 ) × ( A B 1 × B 2 ⋊ B 2 ) defined by ( x, ( b 1 , b 2 )) 7→ (( x, b 1 ) , ( x, b 2 )) for x ∈ A B 1 × B 2 and ( b 1 , b 2 ) ∈ B 1 × B 2 is an injectiv e homomorphism. Therefore using (1) of Lemma 2.3 and b y the hereditary prop ert y it is enough to prov e the F I C w F V C for groups of t he form A ≀ Z where A is ab elian. Since A is coun table ab elian w e can write it as a limit of finitely generated ab elian subgroups A i . No w note that A ≀ Z = (lim i ∈ I A i ) ≀ Z = lim i ∈ I ( A i ≀ Z ). Hence b y Lemma 2 .1 it is enough to prov e the F I C w F V C for A i ≀ Z . Therefore f rom now on we can assume that A is finitely generated. Next note the equalit y in the followin g Lemma. This w as obtained in the pro o f of [[7], Lemma 4.3]. Lemma 3.1. L et A b e an ab elian gr oup. Then the fol lowing e quality holds. A ≀ Z = lim n →∞ ( A n +1 ∗ A n ) . 10 S.K. ROUSHON Wher e the HNN extensio n A n +1 ∗ A n = H n (say) is obtaine d using the fol lowing two inclusions. i j : A n → A n +1 . i 1 ( a 1 , . . . , a n ) 7→ ( a 1 , . . . , a n , 0) . i 2 ( a 1 , . . . , a n ) 7→ (0 , a 1 , . . . , a n ) . Again b y Lemma 2.1 w e need to pro v e the F I C w F V C for H n . W e hav e a surjectiv e homomorphism p : H n → A n +1 ⋊ α Z = H n (sa y) where α ( a 1 , . . . , a n +1 ) = ( a n +1 , a 1 , . . . , a n ). Recall that A is a finitely generated ab elian group. Let B b e a finitely generated free ab elian subgroup of A of finite index. Clearly α lea v es B n +1 in v arian t. Therefore B n +1 ⋊ α Z = G n (sa y) is a finite index subgroup o f H n . Hence p − 1 ( G n ) is a finite index subgroup of H n . Ob viously p − 1 ( G n ) = B n +1 ∗ B n = G n (sa y). Where the HNN extension G n is obtained b y the same maps i j as w e defined ab ov e. W e now use (2) of Lemma 2.3 t o reduce the situation to G n . That is w e need to pro v e the F I C w F V C for G n . W e w o uld lik e to apply (3) of Lemma 2.3 to p : G n → G n . F rom no w on w e follow the pro of of Lemma 4.3 in page 314 in [7 ]. Let C b e a virtually cyclic subgroup of G n . Since G n is torsion free C is either trivial or infinite cyclic. Since G n is an H N N -extension it acts on a tree with v ertex stabilizer conjugates of B n +1 and edge stabilizers conjugates of B n . Therefore k er( p ) also acts on this tree and it follows that the stabilizers of this restricted action are trivial. Hence k er( p ) is a f ree group b y [[16], Lemma 3.2 ]. When C is infinite cyclic then in the pro of o f [[7], Lemma 4.3] (see paragraphs 2 and 3 in pag e 31 6 in [7]) it w as deduced that p − 1 ( C ) is a direct limit of finitely generated subgroups C i (sa y) so that eac h C i is a subgroup of a finite free pro duct K ∗ · · · ∗ K where K is isomorphic to a dir ect pro duct of a finitely generated free group and an infinite cyclic gro up. No w since w t T V C is satisfied the F I C w F V C is true for fr ee groups. See Lemma 2.8 . Therefore the F I C w F V C is true for ker( p ). Also by Lemma 2.7 the F I C w F V C is true for the f ree pro duct of t w o gr o ups if the F I C w F V C is true for eac h f ree summand and w t T V C and P V C are satisfied. Therefore, in addition, using (1) of L emma 2 .3 w e deduce that the F I C w F V C is true for K ∗ · · · ∗ K and hence for C i also by Lemma 2.2. Finally b y Lemma 2.1 w e conclude that the F I C w F V C is true fo r p − 1 ( C ). Therefore G n satisfies the F I C w F V C for each n . THE ISOMORPHISM CONJECTURE IN L -THEOR Y 11 This completes the pro of of (1). Pro of of 2. Let G b e a free metab elian group. Then the Magnus Em bedding Theorem ([12]) sa ys that G can b e embedded as a subgroup of a gro up of the form A ≀ B where A and B are abelian. The pro of of (2) now follo ws from (1) using Lemma 2.2. Pro of of 3. The pro of follo ws the steps of the pro of of [[7], Corollary 4.2]. Using Lemma 2.1 w e assume that G = A ≀ Z is finitely generated. This ma kes A a finitely generated Z [ Z ]-mo dule via the conjugation action of G on A . Hence A has finite exp onen t. Let us first assume that we ha v e pro v ed the r esult when this exp o- nen t is a prime. T o complete the pro of w e no w use induction on the exp o nen t, sa y τ . If τ = 1 then there is nothing to pro v e. So assume τ = pq ≥ 2 and p is a prime. Note that pA is a normal subgroup of G and hence w e hav e the following t w o exact seq uences. 1 → pA → G → G 1 → 1 1 → A/pA → G 1 → Z → 1 . Note that the exp onen t of A/pA is p and hence the F I C w F V C is true for G 1 b y assumption. Next, the exp onen t of pA is q < τ a nd hence b y the induction hypothesis and applying (3) of Lemma 2.3 to the homomorphism G → G 1 w e are done. Let us no w assume that the exponent of A is a prime p and complete the pro of. This mak es A a finitely generated Z p [ Z ]-mo dule. Since Z p [ Z ] is a PID A has a decompo sition in free part and torsion part as Z p [ Z ]- mo dule. Let A 0 b e the free part. T hen A 0 is a normal subgroup o f G . Let C b e an infinite cyclic subgroup o f G whic h go es on to Z under the map G → Z . Then A 0 C is a finite index subgroup o f G and also A 0 C ≃ Z n p ≀ Z where n is the rank of A 0 as a free Z p [ Z ]-mo dule. Hence using (2) of Lemma 2.3 w e are done once w e sho w that the F I C w F V C is true for Z n p ≀ Z . Let B = Z n p then by Lemma 3.1 w e ha v e the follow ing equalit y . B ≀ Z ≃ lim k →∞ B k +1 ∗ B k . Next note that B k +1 ∗ B k is finitely generated and isomorphic to the fundamen tal group of a graph of finite g roups and hence contains a free subgroup of finite index (see [[16], Lemma 3.2]. F ina lly using Lemma 1.6 w e complete the pro of. Pro of of 4. The pro of uses the metho d of the pro of of [[7], Corolla ry 4.4]. F or a solv able gro up G we say that it is n - step solv able if G ( n ) = (1) and G ( n − 1) 6 = ( 1 ), where G ( i ) denotes the i -th deriv ed subgroup of G . 12 S.K. ROUSHON Let G b e an n -step solv a ble group. The pro of of (3) is by induction on n . So assume that if the F I C w F V C is true f o r all closely crys- tallographic g roups then it is true for all k -step solv able g r oups for k ≤ n − 1. W e hav e an exact sequence 1 → G (2) → G → G / G (2) → 1. By (3) of Lemma 2.3 and b y the induction hy p othesis it is enough to pro v e the F I C w F V C for 2- step solv able groups, since for an y infinite cyclic subgroup of G/G (2) the in v ers e image under the quotien t map G → G/G (2) is an ( n − 1)-step solv able group. Therefore we hav e reduced the pro of to the follow ing situation. 1 → G (1) → G → G/ G (1) → 1 . Here G (1) and G/ G (1) are b oth ab elian. By Lemma 2.8 w e can assume t hat G/G (1) is infinite. Again applying (3) of Lemma 2.3 to the map G → G/G (1) w e see that it is enough to pro v e the F I C w F V C for the group G = A ⋊ Z where A is an ab elian group. Let A T b e the subgroup of A consisting of all elemen ts of finite o rder. Then A T is a c haracteristic subgroup o f A and hence we ha v e an exact sequence . 1 → A T → G → G/ A T → 1 . Note that G/ A T ≃ ( A/ A T ) ⋊ Z . Therefore b y (2) and (3) of Lemma 2.3 it is enough to prov e the F I C w F V C for G for the follo wing tw o individual cases. C ase ( a ) . A is tor sion ab elian. C ase ( b ) . A is torsion free ab elian. Pro of of C ase ( a ) : This case is same as (2 ). Pro of of C ase ( b ) : Note that by Lemma 2.1 w e may a ssume that A is finitely generated as a Q [ Z ]-mo dule (see the pro of of [[7], Corollary 4.4]). As Q [ Z ] is a principal ideal domain, A ≃ X ⊕ Y where X is the sum of free a nd Y is sum of finite Q -dimensional Q [ Z ]-submo dule of A . Let m = dim Y and n is the n um b er of free parts in X . Not e that Y is a normal subgroup of G . The pro of is no w by induction first on n a nd t hen on m . If m = n = 0 then G is infinite cyclic so there is nothing to pro v e. So assume n = 0 and m > 0. Let Y 0 b e an irreducible Q [ Z ]-submo dule of Y . Then w e ha v e an exact sequence. 1 → Y 0 → G → G/ Y 0 → 1 . By induction and by (2) and (3) of Lemma 2.3 it is enough to prov e the F I C w F V C for Y 0 ⋊ Z , whic h is tr ue by hypothesis since Y 0 ⋊ Z is a closely crystallographic group. THE ISOMORPHISM CONJECTURE IN L -THEOR Y 13 Next assume that n > 0. Then it follows that G/ Y is isomorphic to Q n ≀ Z for whic h (1) sho ws that the F I C w F V C is true. No w again we apply (2) a nd (3) of Lemma 2.3 to the homomorphism G → G/ Y and hence w e need only to show the F I C w F V C when G/ Y is infinite cyclic . But this is again the case n = 0 treated ab o v e.  Pr o of of The o r em 1.1. The pro of is immediate from Theorem 3.1, Lem- mas 2.4, 2.5 and 2.6.  Pr o of of The o r em 1.2. Let G b e a closely crystallographic group. Re- call that then G is nearly crystallographic. Since nearly crystallo- graphic groups are linear (see the paragraph after [[7], D efinition]) the follo wing hold b y [[7], Theorem 1.1] and the discussion follo wing it. W h ( G ) = ˜ K 0 ( Z [ G ]) = K i ( Z [ G ]) = 0 for all negativ e integers i . Let G = A ⋊ Z where A is torsion free a b elian. Since A is a direct limit of its finitely generated subgroups and since the functors in the ab ov e display commu te with direct limit, the displa y also holds if w e replace G b y A . As the Whitehead groups of the g roups ( G a nd A ) we are considering v anish the surgery L -groups of these groups with differen t decorations coincide. Therefore w e denote the surgery groups b y the simple nota- tion L n ( − ). Let us first sho w that the non-connectiv e assem bly map in L -theory is an isomorphism for G . That is H n ( K ( G, 1) , L 0 ) → L n ( Z [ G ]) is an isomorphism. Since the isomorphism of the ab ov e assem bly map is inv ar ia n t un- der ta king direct limit of groups and since the map is an isomor - phism for finitely generated free ab elian groups ([5]), it follows that H n ( K ( A, 1) , L 0 ) → L n ( Z [ A ]) is also an isomorphism. Let us no w recall the follow ing Ranic ki’s Ma y er-Vietoris t ype exact sequence of surgery groups ([14]) for G . · · · → L n +1 ( G ) → L n ( A ) → L n ( A ) → L n ( G ) → · · · . There is a similar exact sequenc e for the homolog y theory H n ( − , L 0 ). No w, since the assem bly map is natural, a fiv e lemma argumen t imply that H n ( K ( G, 1) , L 0 ) → L n ( Z [ G ]) is an isomorphism. Next, the ab o v e K -t heoretic v anishing result and an application of the Rothenberg’s exact sequence (see the pro of o f Corollary 5 .3) implies that the IC L i ( G ) is satisfied f o r i = h−∞i , h or s .  14 S.K. ROUSHON Pr o of of The o r em 1.3. Pro of of A. ( A ) is an immediate conseque nce of [[16 ], (1) o f Prop osition 2.2] and Lemmas 2.4, 2.5 and 2 .6. Recall that in [[16 ], (1) of Prop osition 2.2 ] w e a ssumed that the equiv arian t homology theory should b e contin uo us when the graph of g r o ups is infinite. But there this con tin uity a ssumption was used to get Lemma 2.6 for the corresp onding homolog y theory . Since we ha v e noted that for L -theory Lemma 2.6 is true w e do not need this assumption here. Pro of of B(i). B ( i ) follow s using Lemma 2.4, 2.5 and 2.6 and the follo wing Prop osition 3.1. Pro of of B(ii). At first recall that virtually p olycyclic groups ar e residually finite. And the F I C w F L is true for virtually p olycyclic groups b y [[17], Theorem 1.1 and (iv) of Theorem 1 .3]. Also no te that b y the same result IA( K ) is true in the L -theory case for a n y virtually p olycyclic g r oup K . This completes the pro o f using B ( i ).  Prop osition 3.1. Assume the same hyp o theses as i n B ( i ) of T he o- r em 1.3 r eplacing L by an arbitr ary e q uivariant homolo gy the ory and in addition assume that w t T V C , P V C and L V C ar e satisfie d. Then the F I C w F V C is true for π 1 ( G ) . Pr o of. W e need to apply (3) of Lemma 2.3 t o the homomorphism f : π 1 ( G ) → Q . Let C b e an infinite cyclic subgroup of Q . Then f − 1 ( C ) is isomorphic to the fundamen tal group of a gra ph of groups whose edge groups are finite and v ertex gro ups are subgroups of groups of the fo rm K ⋊ Z where K is finitely generated and residually finite and b y IA( K ) K ⋊ Z satisfies the F I C w F V C . The pro of will be completed b y [[16], (1) of Proposition 2.2] once w e sho w that K ⋊ Z is residually finite. W e apply [[16], Lemma 4.2]. That is, we ha v e to sho w that given an y finite index subgroup K ′ of K there is a (finite index) subgroup K ′′ of K ′ whic h is normal in K ⋊ Z and the quotient ( K ⋊ Z ) /K ′′ is residually finite. Since K is finitely generated w e can find a finite index c haracteristic subgroup K ′′ (and hence normal in K ⋊ Z ) of K con tained in K ′ . Then ( K ⋊ Z ) /K ′′ is residually finite since it is virtually cyclic b y [[16], Lemma 6.1]. This completes the pro of of the Prop osition.  Pr o of of The o r em 1.4. Pro ofs of ( A ) and ( B ) follow using the following. • F initely generated nilpo ten t g roups are virtually p o lycyclic. • L emmas 2.4, 2.5 and 2.6. • [[16], (3) of Prop osition 2.2], which sa ys that the statemen ts ( A ) and ( B ) are true for general equiv aria n t homology theories H ? ∗ if H ? ∗ is con tin uous and P V C and w t T V C are satisfied. In the pro o f of [[16], (3) of Prop osition 2 .2 ] w e needed the fact that L V C is satisfied whic h is THE ISOMORPHISM CONJECTURE IN L -THEOR Y 15 implied b y the h yp o thesis that H ? ∗ is con tin uous (see [[16], Prop o sition 5.1]). This completes the argumen t using the previous item. The pro of of ( C ) follo ws f rom t he fo llowing. A t first assume that the graph of groups is finite whic h we can b y Lemma 2.6. • π 1 ( G ) ≃ π 1 ( H ) where H is a graph of groups whose edge groups are finite and eac h v ertex group is either virtually cyclic or fundamen tal group of a tree of infinite virtually cyclic ab elian groups. See [[1 6], Lemma 3.1 ]. • The v ertex groups of H are residually finite. See [[16], Lemma 4.4]. • The v ertex groups satisfy F I C w F L . Use (1) and [[16], Lemma 3.5] whic h implies that the graph of groups G has the in tersection prop erty . F or t he pro of of ( D ) we need t he follo wing. • L emmas 2.4, 2.5 and 2.6. • [[16], 2( i ) o f Prop osition 2.3 ]. Here note that fo r the pro of of [[16], 2( i ) of Prop osition 2.3] w e needed that the F I C w F V C is true for Z n ⋊ Z for all n , whic h is the case for the F I C w F L b y [[17], Theorem 1.1 and ( iv ) of Theorem 1.3]. This completes the pro of.  Pr o of of The o r em 1.5. Let G b e a graph of g r oups. If G is a tree then there is nothing to pro v e. So a ssume that it is not a tree. Then there is a surjectiv e homomorphism f : π 1 ( G ) → F where F is a coun table free group. And the ke rnel of f is a tree of groups (the univ ersal co v ering graph of groups of G ). No w using the h ypothesis, Lemma 2.8 and (2 ) and (3) of Lemma 2.3 w e complete the pro o f of ( A ). F or the pro of of ( B ) w e j ust need to note that the univ ersal co v ering graph of groups of G is a tree of groups whose class of v ertex and edge groups is same as that of G .  Pr o of of The o r em 1.6. By ( B ) o f Theorem 1.5 w e can assume that the graph of groups is a tree of finitely generated ab elian groups. Next, b y [[16], Lemma 3.3] there is a surjectiv e homomorphism p : π 1 ( G ) → H 1 ( π 1 ( G ) , Z ) so that the restriction of p to any ve rtex gr o up has trivial k ernel. This implies that the kerne l of p acts on a tree with tr ivial stabilizers and hence it is a free group. No w using (2) and (3) of Lemma 2.3 and Lemma 2.8 w e comple te the pro of.  Pr o of of The o r em 1.7. The pro of of ( A ) follo ws fro m Lemmas 2.4, 2.5, 2.6 and [[16], (1) of Prop osition 2.2]. The pro of of ( B ) is routine using ( A ) a nd (2) and (3) of Lemma 2.3. The only fact we need to men tion is that a virtually residually finite group is residually finite.  16 S.K. ROUSHON 4. S ome spe cial cases In this section w e deduce some results for the follo wing simple cases of graphs of groups. This is con trary to the situation of ascending HNN extension f o r whic h the Fib ered Isomorphism Conjecture is still not pro v ed. The simplest case is the groups Z ∗ Z where the t w o inclusions Z → Z are iden tit y and multiplication by 2. Note here that Z ∗ Z = Z [ 1 2 ] ⋊ Z . See R emark 1.1. Prop osition 4.1. L et G a nd A b e two gr oups. L et i j : A → G b e two inje ctive hom omorphism fo r j = 1 , 2 . Assume that ther e ex ist an automorphism α : G → G with the pr op erty that α ( i 1 ( a )) = i 2 ( a ) for al l a ∈ A . T hen the F I C w F L is satisfie d for the H N N -extensio n G ∗ A (define d by the two homomorphism i 1 and i 2 ) pr ovide d G also satisfies the F I C w F L . Prop osition 4.2. L et G 1 and G 2 b e two gr oups. L et A b e a gr oup with two inje c tive homomorphi s m i j : A → G j for j = 1 , 2 . Assume that ther e is an isom orphism α : G 1 → G 2 with the pr op erty that α ( i 1 ( a )) = i 2 ( a ) for e ach a ∈ A . Th e n the F I C w F L is satisfie d for the gener al i z e d fr e e p r o duct G 1 ∗ A G 2 (define d by the two homom o rphism i 1 and i 2 ) pr ovide d G 1 (or G 2 ) also satisfies the F I C w F L . The f o llo wing is an immediate corollary of Proposition 4.2. Corollary 4.1. L et M and P b e two c omp act manifold with nonempty c onne cte d π 1 -inje c tive b oundary and let f : M → P b e a homo topy e quivalen c e so that f | ∂ M : ∂ M → ∂ P is a ho m e omorp hism. Then the F I C w F L is true for π 1 ( M ∪ ∂ P ) if the F I C w F L is true for π 1 ( M ) . Her e M ∪ ∂ P is the union of M and P glue d along the b oundary via the map f . Pr o of of Pr op osition 4.1. At first note that there is an ob vious surjec- tiv e ho mo mo r phism f : G ∗ A → G ⋊ α h t i . Using (2) of Lemma 2.3 it follo ws that the F I C w F L is true for G ⋊ h t i fo r any action of h t i ov er G . No w note that the group G ∗ A acts on a tree with v ertex gro ups conjugates of G and edge gro ups conjugates o f A . And also the r estric- tions of f to the v ertex g r oups are injectiv e. Therefore ( B ) of Theorem 1.7 completes the pro o f.  Pr o of of Pr op osition 4.2. Let us consider the free pro duct G = G 1 ∗ G 2 . Then there are t w o inclusions j 1 and j 2 from A to G defined b y i 1 and i 2 . And there is an isomorphism ˜ α : G → G defined b y α so that ˜ α ( j 1 ( a )) = j 2 ( a ). Next no te that there is an embedding G 1 ∗ A G 2 → G ∗ A where G 1 ∗ A G 2 is defined with resp ect to i 1 and i 2 and G ∗ A is THE ISOMORPHISM CONJECTURE IN L -THEOR Y 17 defined with resp ect to j 1 and j 2 . Hence b y L emma 2.2 it is enough to prov e the F I C w F L for G ∗ A . Since by Lemmas 2.4, 2.5 and 2.7 the F I C w F L is true for G we are done using Propo sition 4.1 .  Remark 4.1. The Prop ositions 4.1 and 4.2 can b e prov en for arbitra r y homology theories and with resp ect to the class F I N of finite groups if we add the extra assumptions that w t T F I N and L F I N are satisfied. W e hav e already men tioned in the in tro duction that w t T F I N in t he K -theory case is still not know n. 5. Some consequences The follo wing a r e some of t he we ll-kno wn consequences of the Iso- morphism Conjecture. Corollary 5.1. If Γ is a torsion fr e e gr oup fo r which the Fib er e d Iso- morphism Con je ctur e in pseudoisotopy the ory is true, then the f o l lowing holds. The Whitehe ad gr o up W h (Γ) , the low er K -gr oups K − i ( Z Γ) for i ≥ 1 and the r e duc e d pr oje ctive class gr oup ˜ K 0 ( Z Γ) vanish. Corollary 5.2. In addition to the hyp othes i s o f the pr e vious c or ol la ry, if the Isomo rphism C onje ctur e in L h−∞i -the ory is a lso true for the gr o up Γ then the fol lowing holds. The fol lowing assembly map is an isomorphism fo r al l n and for j = h −∞i , h and s . H n ( B Γ; L j ( Z )) → L j n ( Z Γ) . Note that the ab o v e t w o Corollaries giv e further evidence to the Whitehead Conjecture and the inte gral Novik ov Conjecture resp ec- tiv ely . The Whitehead Conjecture sa ys t hat the Whitehead group of an y to rsion free group v a nishes. And the in tegra l No vik o v Conjecture sa ys that the ab ov e assem bly map is split injectiv e fo r torsion free groups. Corollaries 5.1 and 5.2 together imply the fo llowing. Corollary 5.3. ( Generalized Borel Conjecture ) L et M b e a c lose d aspheric al manifold with π 1 ( M ) isom o rphic to G wher e G satisfies the Fib e r e d I somorphism Conje ctur e for the pseudoisotopy and the L -the o ry c ases. Then M × D k satisfies the Bor el Conje ctur e for dim ( M ) + k ≥ 5 . That is, if f : N → M × D k is a homo topy e quivalenc e fr om another c omp act manifold so that f | ∂ N : ∂ N → M × S k − 1 is a home omorph ism, then f is homotopic , r elative to b oundary, to a home omorphism. 18 S.K. ROUSHON Finally w e recall that, in our earlier w orks together with the presen t article w e hav e prov ed the Fib ered Isomorphism Conjecture b oth for the pseudoisotopy and for the L −h∞i -theory for a la rge class o f groups. Belo w we sk etch the pro ofs of the ab ov e corollaries. The arguments for the pro of s of Corollaries 5 .1 and 5.2 and Theorem 5.3 are already there in the litera t ure. W e briefly recall the pro ofs and then refer to the original sources. Pr o of of Cor ol lary 5.1. This is a consequenc e of the Fib ered Isomor- phism Conjecture in stable top ological pseudoisotop y theory . See [[6], 1.6.5] fo r details. Also see [[8], Theorem D].  Pr o of of Cor ol lary 5.2. The Isomorphism Conjecture in L h−∞i - theory for torsion free groups implies the isomorphism of the assem bly map H n ( B Γ; L h j i ( Z )) → L h j i n ( Z Γ) for j = −∞ . See [[6], 1.6.1] for details. No w recall the following Rothen b erg exact sequ ence. · · · → L h i +1 i n ( R ) → L h i i n ( R ) → ˆ H n ( Z / 2; ˜ K i ( R )) → L h i +1 i n − 1 ( R ) → L h i i n − 1 ( R ) → · · · . Where R = Z Γ and i ≤ 1. Recall that L h 1 i n = L h n and L h−∞i n is the limit of L h i i n . Now using Corollary 5.1 and b y a Five Lemma a rgumen t w e get the isomorphism of the assem bly map H n ( B Γ; L h ( Z )) → L h n ( Z Γ) . Using a similar Rothen b erg exact sequence whic h connects the surgery groups with h and s decoratio ns and t he T ate cohomology whic h ap- p ears is with co efficien t in the Whitehead gro up, one can show the follo wing isomorphism. H n ( B Γ; L s ( Z )) → L s n ( Z Γ) . See [[10], Section 1.5] for details and for o ther related features.  Pr o of of Cor ol lary 5.3. Let us first recall the surgery exact sequenc e. This sequence is for simple homotopy t ypes and f or the surgery groups with the decoration ‘ s ’. Since the Whitehead group of the group G in the presen t situation v anishes, there is no difference b et w een ‘ s ’ and ‘ h ’ and therefore w e do not use an y decoration. · · · → H n ( X ; L 0 ) → L n ( π 1 ( X )) → S n ( X ) → H n − 1 ( X ; L 0 ) → · · · . Where S ∗ ( − ) is the total surgery obstruction groups o f Ranick i a nd L 0 is a 1-connectiv e Ω-sp ectrum with 0-space homotopically equiv alen t THE ISOMORPHISM CONJECTURE IN L -THEOR Y 19 to G/T O P . If X is a compact n -dimensional manif o ld ( n ≥ 5) then the following part of the ab o v e surgery exact sequenc e · · · → S n +2 ( X ) → H n +1 ( X ; L 0 ) → L n +1 ( π 1 ( X )) → S n +1 ( X ) → H n ( X ; L 0 ) → L n ( π 1 ( X )) is iden tified with the original surgery exact sequence · · · → S T op ( X × D 1 , ∂ ( X × D 1 )) → [ X × D 1 , ∂ ( X × D 1 ); G/T O P , ∗ ] → L n +1 ( π 1 ( X )) → S T op ( X ) → [ X ; G/T O P ] → L n ( π 1 ( X )) . In particular, for an n -dimensional closed manifold X there is t he follo wing iden tification. S n + k +1 ( X ) = S T op ( X × D k , ∂ ( X × D k )) . Here S T op ( P , ∂ P ) denotes the structure set of a compact manifold P . When W h ( π 1 ( P ) ) = (1) and dim( P ) ≥ 5 (which is the case in the presen t situation) the structure set can b e defined in the follo wing sim- pler wa y . S T op ( P , ∂ P ) is the set of all equiv alence classes of homotopy equiv alences f : ( N , ∂ N ) → ( P , ∂ P ) from compact manifolds ( N , ∂ N ) so that f | ∂ N : ∂ N → ∂ P is a homeomorphism. Here tw o such maps f i : ( N i , ∂ N i ) → ( P , ∂ P ) for i = 1 , 2 a r e said t o b e equiv alen t if t here is a homeomorphism h : ( N 1 , ∂ N 1 ) → ( N 2 , ∂ N 2 ) so that f 2 ◦ h is homo- topic to f 1 relativ e to b oundary , that is during the homoto py the map on the bo undary is constan t. Next, there is a homomorphism H k ( X ; L 0 ) → H k ( X ; L ( Z )) whic h is an isomorphism for k > n and is injective for k = n . No w using the f a ct that M is aspherical a nd applying Corollaries 5.1 and 5.2 we see that S T op ( M × D k ) con tains o nly one elemen t for n + k ≥ 5. This completes the pro of of Corollary 5.3. F or some more details with related references see [[10], Theorem 1.28] or [[6], 1.6.3].  Remark 5 .1. In view of the fo otnote in [[16 ], in tro duction] w e finally remark that [[6], Remark 2 .1.3] is used in this pap er in the follo wing statemen ts: B ( ii ) of Theorem 1.3; A a nd B of Theorem 1.4; D of Theorem 1.4 when the ve rtex g roups of any component subgraph ha s rank > 1. The w ork on completing the pro of of [[6], Remark 2.1 .3 ] is in [1]. Ac kno wledgeme n t. I w ould like to thank F.T. F arrell for some help- ful e-mail comm unications. Also I am gra teful t o W olfgang L ¨ uc k for p oin ting out a n error in a preprin t whic h initiated some of the results in this pap er. 20 S.K. ROUSHON Reference s [1] Arth ur Bartels, F. Thomas F ar rell a nd W olfga ng L ¨ uck, The F a rrell-Jo nes co n- jecture fo r co compact lattices in virtually connected Lie groups, in prepara tion. [2] Arth ur Ba rtels and W olfgang L ¨ uck, Isomorphis m Conjecture for homotopy K - theory and groups a cting o n trees, J . Pur e Appl. Algebr a , 205 (200 6), 66 0-696 . [3] , The Borel conjecture for h yper b o lic and CA T(0)-gr oups, arXiv:090 1.044 2 . [4] Arth ur Bartels, W olfgang L¨ uc k a nd Holger Reic h, On the F a rrell-Jo nes con- jecture and its applications, J. T op ol. , 1 (2008 ), no . 1 , 5 7-86. [5] F.T. F arr ell and W.C-. Hsiang , T op ologica l characteriza tion of flat a nd almo s t flat Riemannian manifolds M n ( n 6 = 3 , 4), Amer. Jour. of Math. , 105 (1983), 641-6 72. [6] F.T. F arr ell and L .E . Jone s , Is o morphism conjectures in alg ebraic K -theory , J. A mer. Math. So c. , 6 (199 3), 2 49-29 7. [7] F.T. F arrell a nd P .A. Linnell, K -Theory of solv able groups, Pr o c. L ondo n Math. So c. (3) , 87 (2003), 309-3 36. [8] F.T. F arr ell and S.K. Roushon, The Whitehead gr oups of braid gr oups v anish, Internat. Math. R es. Notic es , no. 1 0 (20 00), 515-52 6. [9] W olfgang L ¨ uck, The F ar r ell-Jones conjecture in algebraic K - and L -theo ry , Enseign. Math. (2 ) 54 (20 0 8), 14 0-141 . In ‘Guido’s bo o k of c o njectures’, a g ift to Guido Mislin on the o ccas io n of his retirement from E THZ, June 20 06. Collected by Indira Chatterji. Enseign. Math. (2 ) 54 (2008), no. 1-2, 3–189. [10] W olfgang L¨ uck and Holger Reich, The B aum-Connes and the F a rrell-Jo nes conjectures in K - and L -theory , In Handb o ok of K-t he ory V o lume 2 , edited b y E.M. F r iedlander, D.R. Grayson, 703- 842, Springer, 200 5. [11] R.C. Lyndon and P .E. Sch upp, Combinatorial gr oup the ory. Class ic s in Math- ematics, Springer , 197 7. [12] W. Magnus, On a theorem of Mar shall Hall, Ann. of Math. , 40 (19 39), 76 4-768 . [13] Peter Or lik and Hiroa ki T erao, Arr angements of Hyp erpla nes. Springer -V er lag: Berlin, New Y ork , 1 992. [14] A. Ranicki, Alg ebraic L -theor y I I I, Twisted Laur e n t extensio n in Algebr aic K - The ory , I I I: Hermitian K -theor y and Geo metric Application (Pro c. Conf., Seattle, W as hington, 197 2), Lecture Notes in Mathe. 343, Spring er-V erlag, Berlin, 1973, 412-4 63. [15] S.K. Rous hon, The F arr ell-Jones iso morphism conjecture for 3-manifold groups, J. K -t he ory , 1 (2008 ), 49-8 2. [16] , On the isomorphism conjecture for groups acting on trees, math.KT/051 0297 v5. [17] , The fib ered iso morphism conjecture in L -theo ry , T op olo gy Appl. , 157 (2010), 508- 515. doi:10.1 016/j.top ol.2 009.1 0 .012. [18] , Surger y groups of fundamental g roups of hyperplane arr angement complements, ar Xiv:0909.2 133. School of Ma thema tics, T a t a Institute, Ho mi Bha bha Ro ad, Mumbai 400005, India E-mail add r ess : rousho n@math .tifr.res.in URL : http:/ /www.m ath.tifr.res.in/~ rousho n/