Algebraic K -theory, assembly maps, controlled algebra, and trace methods
AALGEBRAIC K -THEORY, ASSEMBLY MAPS,CONTROLLED ALGEBRA, AND TRACE METHODS A PRIMER AND A SURVEY OF THE FARRELL-JONES CONJECTURE
HOLGER REICH AND MARCO VARISCO
Abstract.
We give a concise introduction to the Farrell-Jones Conjecturein algebraic K -theory and to some of its applications. We survey the currentstatus of the conjecture, and we illustrate the two main tools that are used toattack it: controlled algebra and trace methods. Contents
1. Introduction 22. Conjectures 32.1. Idempotents and projective modules 32.2. h-Cobordisms 62.3. Assembly maps 72.4. The Farrell-Jones Conjecture 92.5. Rational computations 122.6. Some related conjectures 143. State of the art 143.1. What we know already 143.2. What we don’t know yet 163.3. Injectivity results 164. Controlled algebra methods 174.1. Geometric modules 194.2. Contracting maps 204.3. The Farrell-Hsiang Criterion 224.4. Z is a Farrell-Hsiang group 254.5. The Farrell-Hsiang Criterion (continued) 275. Trace methods 295.1. A warm-up example 305.2. Bökstedt-Hsiang-Madsen’s Theorem 325.3. Generalizations 35References 38Index 44 Date : February 23, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Algebraic K -theory, Farrell-Jones Conjecture. a r X i v : . [ m a t h . K T ] F e b HOLGER REICH AND MARCO VARISCO Introduction
The classification of manifolds and the study of their automorphisms are centralproblems in mathematics. For manifolds of sufficiently high dimension, theseproblems can often be successfully solved using algebraic topological invariants inthe algebraic K -theory and L -theory of group rings.In an article published in 1993 [FJ93a], Tom Farrell and Lowell Jones formulateda series of Isomorphism Conjectures about the K and L -theory of group rings, whichbecame universally known as the Farrell-Jones Conjectures . On the one hand theseconjectures represented the culmination of decades of seminal work by Farrell, Jones,and Wu Chung Hsiang, e.g. [FH78], [FH81a], [FH81b], [Hsi84], [FJ86], [FJ89]. Onthe other hand they have motivated and continue to motivate an impressive bodyof research.In this article we focus only on the Farrell-Jones Conjecture for algebraic K -theory, and mention briefly some of its variants in Subsection 2.6. We give a conciseintroduction to this conjecture and to some of its applications, survey its currentstatus, and most importantly we explain the main ideas and tools that are used toattack the conjecture: controlled algebra and trace methods.Section 2 begins with some fundamental conjectures in algebra and geometrictopology, which can be reformulated in terms of K and K of group rings. Theseconjectures are all implied by the Farrell-Jones Conjecture, but they are moreaccessible and elementary; moreover, their importance and appeal do not requirealgebraic K -theory, but may serve as motivation to study it.In Subsections 2.3 and 2.4 we define assembly maps and use them to formulatethe Farrell-Jones Conjecture. Then we discuss how the Farrell-Jones Conjectureimplies all other conjectures discussed in this article.In Section 3 we collect most of what is known today about the Farrell-JonesConjecture in algebraic K -theory. We invite the reader to compare that section tothe corresponding Section 2.6 in the survey article [LR05] from 2005, to appreciatethe tremendous amount of activity and progress that has taken place since then.The last two sections focus on proofs. In Section 4 we introduce the basic conceptsof controlled algebra and see them at work. In particular, we give an almost completeproof of the Farrell-Jones Conjecture in the simplest nontrivial case, that of thefree abelian group on two generators. Many ingenious ideas, mainly going backto Farrell and Hsiang, enter the proof already in this seemingly basic case. Thissection is meant to be an accessible introduction to controlled algebra. We do noteven mention the very important flow techniques, and highly recommend ArthurBartels’s survey article [Bar16].In Section 5 we illustrate how trace methods are used to prove rational injectivityresults about assembly maps. We give a complete proof of an elementary butilluminating statement about K in Subsection 5.1, and then explain how thisidea can be generalized using more sophisticated tools like topological Hochschildhomology and topological cyclic homology. The complicated technical detailsunderlying the construction of these tools are beyond the scope of this article,and we refer the reader to [DGM13], [Hes05], and [Mad94] for more information.However, we carefully explain the structure of the proof of the algebraic K -theoryNovikov Conjecture due to Marcel Bökstedt, Hsiang, and Ib Madsen [BHM93]. Wefollow the point of view used by the authors in joint work with Wolfgang Lückand John Rognes [LRRV17a], leading to a generalization of this theorem for the -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 3 Farrell-Jones assembly map. In particular, we highlight the importance of a variantof topological cyclic homology, Bökstedt-Hsiang-Madsen’s functor C , which hasseemingly disappeared from the literature since [BHM93].We tried to make our exposition accessible to nonexperts, and no deeper knowledgeof algebraic K -theory is required. However, we expect our reader to have seen thebasic definitions and properties of K and K , and to be willing to accept theexistence of a spectrum-valued algebraic K -theory functor. Classical and lessclassical sources for the K -theoretic background include [Bas68], [Cor11], [DGM13],[Mil71], [Ros94], and [Wei13].There are other survey articles about the Farrell-Jones and related conjectures:[Bar16], [LR05], and [Mad94], which we already recommended, and also [Lüc10]and the voluminous book project [Lüc]. Our hope is that this contribution mayserve as a more concise and accessible starting point, preparing the reader for theseother more advanced surveys and for the original articles. Acknowledgments.
This work was supported by the Collaborative Research Cen-ter 647
Space – Time – Matter in Berlin and by a grant from the Simons Foundation(
Conjectures
In this section we discuss many conjectures related to group rings and theiralgebraic K -theory. These conjectures are all implied by the Farrell-Jones Conjecture,which we formulate in Subsection 2.4. All of these conjectures are known in manycases but open in general, as we review in Section 3. An element p in a ring is an idem-potent if p = p . The trivial examples are the elements and . Conjecture 1 (trivial idempotents) . Let k be a field of characteristic zero and let G be a torsion-free group. Then every idempotent in the group ring k [ G ] is trivial. The assumption that G is torsion-free is necessary: if g ∈ G is an element offinite order n , then n (cid:80) n − i =0 g i is a nontrivial idempotent in Q [ G ] .A counterexample to the conjecture above would be in particular a zero-divisorin k [ G ] , and hence a counterexample to Problem 6 in Irving Kaplansky’s famousproblem list [Kap57], which is reproduced in [Kap70].It is interesting to notice that the analog of Conjecture 1 for the integral groupring is true for all groups, even for groups with torsion. The proof that we givebelow uses operator algebras, as suggested in [Kap70, page 451], and therefore it isvery different from the rest of this paper, even though the idea of using traces playsa central role in Section 5. Theorem 2.
For any group G , every idempotent in the integral group ring Z [ G ] is trivial.Proof. The integral group ring embeds into the reduced complex group C ∗ -algebra C ∗ r G , and the map Z [ G ] −→ Z , (cid:80) a g g (cid:55)−→ a e extends to a positive faithful trace tr : C ∗ r G −→ C . Let p ∈ Z [ G ] be an idempotent, i.e., p = p . It is known that inthe C ∗ -algebra C ∗ r G every idempotent is similar to a projection, i.e., there exist q, u ∈ C ∗ r G such that q = q = q ∗ , u is invertible, and p = u − qu ; see for example[CMR07, Proposition 1.8, Lemma 1.18]. Therefore tr ( p ) = tr ( q ) . Applying the HOLGER REICH AND MARCO VARISCO trace to q + (1 − q ) = q ∗ q + (1 − q ) ∗ (1 − q ) and using positivity one sees thatthe trace of q lies in [0 , . The trace of p is clearly an integer. Therefore tr ( q ) = 0 or tr ( q ) = 1 . By faithfulness of the trace this implies that q = 0 or q = 1 , and thenthe same holds for p = u − qu . (cid:3) The module Rp for an idempotent p = p in the ring R is an example of a finitelygenerated projective left R -module. In view of the conjecture and the result aboveit seems natural to ask whether all finitely generated projective modules over grouprings of torsion-free groups are necessarily free. Again, the assumption that G is torsion-free is necessary: if g ∈ G is an element of finite order n , then for thenon-trivial idempotent p = n (cid:80) n − i =0 g i ∈ Q [ G ] the module Q [ G ] p is projective butnot free. Examples 3. (i) Over fields and over principal ideal domains, hence in partic-ular over the polynomial and Laurent polynomial rings k [ t ] and k [ t ± ] withcoefficients in a field k , all projective modules are free.(ii) The question whether finitely generated projective modules over the polynomialring k [ t , . . . , t n ] for n ≥ are necessarily free was raised by Jean-Pierre Serrein [Ser55], and was answered affirmatively only 21 years later independently byDan Quillen and Andrei Suslin. The wonderful book [Lam06] gives a detailedaccount of this exciting story.The polynomial ring R [ t , . . . , t n ] is the monoid algebra R [ A ] of the free abelianmonoid A generated by { t , . . . , t n } . The statement in (ii) was generalized as followsto monoid algebras.(iii) If R is a principal ideal domain, then every finitely generated projectivemodule over the monoid algebra R [ A ] is free provided that A is a semi-normal,abelian, cancellative monoid without nontrivial units [Gub88], [Swa92]. Freeabelian groups are examples of monoids satisfying these conditions.(iv) If R is a principal ideal domain and F a finitely generated free group, thenevery finitely generated projective module over the group ring R [ F ] is free[Bas64].At this point one could over-optimistically conjecture that every finitely generatedprojective Q [ G ] -module is free if G is a torsion-free group. However:(v) Martin Dunwoody constructed in [Dun72] a torsion-free group G and a finitelygenerated projective Z [ G ] -module P which is not free but has the propertythat P ⊕ Z [ G ] ∼ = Z [ G ] ⊕ Z [ G ] . There are also finitely generated projectivemodules over Q [ G ] with analogous properties.A weakening of the question above is whether all finitely generated projective R [ G ] -modules are induced from finitely generated projective R -modules when G istorsion-free. Recall that K ( R ) is defined as the group completion of the monoid ofisomorphism classes of finitely generated projective R -modules. The surjectivity ofthe natural map K ( R ) −→ K ( R [ G ]) induced by [ M ] (cid:55)−→ (cid:2) R [ G ] ⊗ R M (cid:3) studies the stable version of this question: isevery finitely generated projective R [ G ] -module P stably induced? I.e., is there an n ≥ such that P ⊕ R [ G ] n is induced from a finitely generated projective R -module?Notice that this is true for Dunwoody’s example (v) above. The stable version of -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 5 Serre’s Conjecture (ii) above is a lot easier to prove and was established much earlierin [Ser58, Proposition 10].This discussion leads to the following conjecture. In order to formulate it, we needto recall some notions from the theory of rings. A ring R is called left Noetherianif submodules of finitely generated left modules are always finitely generated, andit is said to have finite left global dimension if every left module has a projectiveresolution of finite length. If both properties hold, then R is called left regular. Inthe sequel we only consider left modules and therefore simply say regular instead ofleft regular. The ring of integers Z , all PIDs, and all fields are examples of regularrings. Conjecture 4.
Let R be a regular ring, and assume that the orders of all finitesubgroups of G are invertible in R . Then the map colim H ∈ obj Sub G ( F in ) K ( R [ H ]) ∼ = −→ K ( R [ G ]) is an isomorphism. In particular, if G is torsion-free, then for any regular ring R there is an isomorphism K ( R ) ∼ = −→ K ( R [ G ]) . Here the colimit is taken over the finite subgroup category
Sub G ( F in ) , whoseobjects are the finite subgroups H of G and whose morphisms are defined asfollows. Given finite subgroups H and K of G , let conhom G ( H, K ) be the setof all group homomorphisms H −→ K given by conjugation by an element of G .The group inn( K ) of inner automorphisms of K acts on conhom G ( H, K ) by post-composition. The set of morphisms in Sub G ( F in ) from H to K is then defined asthe quotient conhom G ( H, K ) / inn( K ) . Since inner conjugation induces the identityon K ( R [ − ]) , this is indeed a well defined functor on Sub G ( F in ) . In the specialcase when G is abelian, the category Sub G ( F in ) is just the poset of finite subgroupsof G ordered by inclusion. Proposition 5.
Conjecture 4 implies Conjecture 1.Proof.
Let k be a field of characteristic zero and let G be a torsion-free group.Let (cid:15) : k [ G ] −→ k denote the augmentation and write (cid:15) ∗ M = k ⊗ k [ G ] M . If p ∈ k [ G ] is an idempotent, then k [ G ] ∼ = k [ G ] p ⊕ k [ G ](1 − p ) and k ∼ = (cid:15) ∗ k [ G ] ∼ = (cid:15) ∗ k [ G ] p ⊕ (cid:15) ∗ k [ G ](1 − p ) . Since k is a field, either (cid:15) ∗ k [ G ] p or (cid:15) ∗ k [ G ](1 − p ) is the zero module. Replacing p by − p if necessary, let us assume that (cid:15) ∗ k [ G ] p is zero. The assumption Z ∼ = K ( k ) ∼ = K ( k [ G ]) implies that there exist n and m such that k [ G ] p ⊕ k [ G ] n ∼ = k [ G ] m . Applying (cid:15) ∗ we see that n = m , and from this we conclude that k [ G ] p is zero asfollows. Recall that a ring R is called stably finite if M ⊕ R n ∼ = R n always impliesthat M is zero; see [Lam99, Section 1B]. Kaplansky showed that, if k is a field ofcharacteristic , then any group ring k [ G ] is stably finite; compare [Mon69]. (cid:3) HOLGER REICH AND MARCO VARISCO
Recall that a smooth cobordism over a closed n -dimensionalsmooth manifold M consists of another closed n -dimensional smooth manifold N andan ( n + 1) -dimensional compact smooth manifold W with boundary ∂W togetherwith a diffeomorphism ( f, g ) : M (cid:113) N ∼ = −→ ∂W . This is called an h -cobordism ifboth incl ◦ f and incl ◦ g are homotopy equivalences, where incl denotes the inclusionof ∂W in W . Two cobordisms W and W (cid:48) over M are called isomorphic if thereexists a diffeomorphism F : W ∼ = −→ W (cid:48) such that F | ∂W ◦ f = f (cid:48) . A cobordismover M is called trivial if it is isomorphic to the cylinder M × [0 , (and this inparticular implies that M and N are diffeomorphic). Conjecture 6 (trivial h -cobordisms) . Let M be a closed, connected, smooth mani-fold of dimension at least and with torsion-free fundamental group. Then every h -cobordism over M is trivial. Surprisingly, this conjecture can be reinterpreted in terms of algebraic K -theory.In fact, the celebrated s -Cobordism Theorem of Stephen Smale, Barry Mazur,John Stallings, and Dennis Barden (e.g., see [Mil65], [KL05]), states that there is abijection { h -cobordisms over M } / iso ∼ = Wh ( π ( M )) between the set of isomorphism classes of smooth cobordisms over M and theWhitehead group Wh ( π ( M )) of the fundamental group of M , whose definition wenow review.Recall that, given a ring R , invertible matrices with coefficients in R representclasses in K ( R ) . Given any group G , the elements ± g ∈ Z [ G ] are invertible forany g ∈ G , and hence represent elements in K ( Z [ G ]) . By definition the Whiteheadgroup Wh ( G ) is the quotient of K ( Z [ G ]) by the image of the map that sends ( ± , g ) to the element represented by ± g in K ( Z [ G ]) . This map factors over {± } ⊕ G ab ,where G ab is the abelianization of G , and the induced map {± }⊕ G ab −→ K ( Z [ G ]) is in fact injective; see for example [LR05, Lemma 2]. So there is a short exactsequence(7) −→ {± } ⊕ G ab −→ K ( Z [ G ]) −→ Wh ( G ) −→ . For Whitehead groups there is the following well-known folklore conjecture.The cases of the infinite cyclic group [Hig40], of finitely generated free abeliangroups [BHS64], and of finitely generated free groups [Sta65] provided early evidencefor this conjecture.
Conjecture 8. If G is a torsion-free group, then Wh ( G ) = 0 . By the s -Cobordism Theorem recalled above, the connection between the lasttwo conjectures is as follows. Proposition 9.
Let M be a closed, connected, smooth manifold of dimension atleast and with torsion-free fundamental group. Then Conjecture 6 for M isequivalent to Conjecture 8 for G = π ( M ) . For groups with torsion, the situation is much more complicated. For example,if C n is a finite cyclic group of order n (cid:54)∈ { , , , , } , then Wh ( C n ) (cid:54) = 0 , and infact even Wh ( C n ) ⊗ Z Q (cid:54) = 0 . The analog of Conjecture 8 for arbitrary groups is thefollowing. -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 7 Conjecture 10.
For any group G the map (11) colim H ∈ obj Sub G ( F in ) Wh ( H ) ⊗ Z Q −→ Wh ( G ) ⊗ Z Q is injective. We highlight two differences with the corresponding Conjecture 4 for K . First,Conjecture 10 is only a rational statement, i.e., after applying − ⊗ Z Q . Second, it isonly an injectivity statement. In order to obtain a rational isomorphism conjecturefor Wh ( G ) one needs to enlarge the source of the map (11). This requires someadditional explanations and is postponed to Conjecture 24 below. The Farrell-Jones Conjecture, which we formulate in thenext subsection, generalizes Conjectures 4, 8, and 10 from statements about theabelian groups K and Wh to statements about the non-connective algebraic K -theory spectra K ( R [ G ]) of group rings, for arbitrary coefficient rings and arbitrarygroups. In order to formulate the Farrell-Jones Conjecture, we need to first introducethe fundamental concept of assembly maps.Fixing a ring R , algebraic K -theory defines a functor K ( R [ − ]) from groups tospectra. In fact, it is very easy to promote this to a functor K ( R [ − ]) : Groupoids −→ Sp from the category of small groupoids (i.e., small categories whose morphisms areall isomorphisms) to the category of spectra. Moreover, this functor preservesequivalences, in the sense that it sends equivalences of groupoids to π ∗ -isomorphisms(i.e., weak equivalences) of spectra. For any such functor we now proceed to constructassembly maps, following the approach of [DL98]. It is not enough to work in thestable homotopy category of spectra, but any point-set level model would work.Let T : Groupoids −→ Sp be a functor that preserves equivalences. Given agroup G , consider the functor G ∫ − : Sets G −→ Groupoids that sends a G -set S toits action groupoid G ∫ S , with obj G ∫ S = S and mor G ∫ S ( s, s (cid:48) ) = { g ∈ G | gs = s (cid:48) } .Restricting to the orbit category Or G , i.e., the full subcategory of Sets G withobjects G/H as H varies among the subgroups of G , we obtain the horizontalcomposition in the following diagram. Or G Sets G Groupoids SpTop
Gι G ∫− T Lan ι T ( G ∫− ) Now we take the left Kan extension [Mac71, Section X.3] of T ( G ∫ − ) along thefull and faithful inclusion functor ι : Or G (cid:44) → Top G of Or G into the category of all G -spaces. The left Kan extension evaluated at a G -space X can be constructed asthe coend [Mac71, Sections IX.6 and X.4] (cid:0) Lan ι T ( G ∫ − ) (cid:1) ( X ) = X + ∧ Or G T ( G ∫ − ) of the functor ( Or G ) op × Or G −→ Sp , ( G/H, G/K ) (cid:55)−→ map( G/H, X ) G + ∧ T ( G ∫ ( G/K )) ∼ = X H + ∧ T ( G ∫ ( G/K )) . HOLGER REICH AND MARCO VARISCO
There are natural isomorphisms
G/H + ∧ Or G T ( G ∫ − ) ∼ = T ( G ∫ ( G/H )) , and the factthat T preserves equivalences implies that these spectra are π ∗ -isomorphic to T ( H ) .Notice that for pt = G/G we even have an isomorphism pt + ∧ Or G T ( G ∫ − ) ∼ = T ( G ) .To define the assembly map we apply this construction to the following G -spaces.Consider a family F of subgroups of G (i.e., a collection of subgroups closed underpassage to subgroups and conjugates) and consider a universal G -space EG ( F ) . Thisis a G -CW complex characterized up to G -homotopy equivalence by the propertythat, for any subgroup H ≤ G , the H -fixed point space (cid:0) EG ( F ) (cid:1) H is (cid:40) empty if H (cid:54)∈ F ;contractible if H ∈ F .The assembly map is by definition the map asbl F : EG ( F ) + ∧ Or G T ( G ∫ − ) −→ T ( G ) induced by the projection EG ( F ) −→ pt (where, in the target, we use the isomor-phism pt + ∧ Or G T ( G ∫ − ) ∼ = T ( G ) ). Remark 12. (i) In the special case of the trivial family F = 1 , a universal space EG (1) is bydefinition a free and non-equivariantly contractible G -CW complex, i.e., theuniversal cover of a classifying space BG . In this case, there is an identification EG (1) + ∧ Or G T ( G ∫ − ) ∼ = BG + ∧ T (1) and therefore we obtain the so-called classical assembly map asbl : BG + ∧ T (1) −→ T ( G ) . (ii) Any G -CW complex whose isotropy groups all lie in the family F has amap to EG ( F ) , and this map is unique up to G -homotopy. This applies inparticular to EG ( F (cid:48) ) when F (cid:48) ⊆ F , and we refer to the induced map asbl F (cid:48) ⊆F : EG ( F (cid:48) ) + ∧ Or G K ( R [ G ∫ − ]) −→ EG ( F ) + ∧ Or G K ( R [ G ∫ − ]) as the relative assembly map.(iii) The source of the assembly map is a model for hocolim G/H ∈ obj Or G s.t. H ∈F T ( G ∫ ( G/H )) , the homotopy colimit of the restriction of T ( G ∫ − ) to the full subcategoryof Or G of objects G/H with H ∈ F ; compare [DL98, Section 5.2].(iv) Taking the homotopy groups of X + ∧ Or G T ( G ∫ − ) defines a G -equivarianthomology theory for G -CW complexes X . This is an equivariant generaliza-tion of the well-known statement that π ∗ ( X + ∧ E ) gives a non-equivarianthomology theory for any spectrum E . The Atiyah-Hirzebruch spectral se-quence converging to π s + t ( X + ∧ E ) with E s,t = H s ( X ; π t E ) also generalizesto a spectral sequence converging to π s + t ( X + ∧ Or G T ( G ∫ − )) with E s,t = H Gs ( X ; π t T ( G ∫ − )) , the Bredon homology of X with coefficients in π t T ( G ∫ − ) : Or G −→ Ab ;compare [DL98, Theorem 4.7]. Using this we see that, if asbl F is a π ∗ -isomorphism, then in general all π t ( T ( H )) with H ∈ F and −∞ < t ≤ n contribute to π n ( T ( G )) . -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 9 We conclude with a historical comment. The classical assembly map asbl from Remark 12(i) for algebraic K -theory was originally introduced in Jean-LouisLoday’s thesis [Lod76, Chapitre IV] using pairings in algebraic K -theory and themultiplication map G × GL ( R ) −→ GL ( R [ G ]) . Friedhelm Waldhausen [Wal78a, Section 15] characterized this map as a universalapproximation by a homology theory evaluated on a classifying space. This pointof view was nicely explained by Michael Weiss and Bruce Williams in [WW95].In their original work [FJ93a], Farrell and Jones used the language developed byFrank Quinn [Qui82, Appendix]. Later, Jim Davis and Wolfgang Lück [DL98]gave an equivariant version of the point of view of [WW95], clarifying and unifyingthe underlying principles. Their approach leads to the concise description of theassembly map given above. The different approaches are compared and shown toagree in [HP04].
We begin by formulating the Farrell-JonesConjecture in the special case of torsion-free groups and regular rings.
Farrell-Jones Conjecture 13 (special case) . For any torsion-free group G andfor any regular ring R the classical assembly map asbl : BG + ∧ K ( R ) −→ K ( R [ G ]) is a π ∗ -isomorphism. On π the classical assembly map produces the map K ( R ) −→ K ( R [ G ]) inducedby the inclusion R −→ R [ G ] . So we see that the Farrell-Jones Conjecture 13 impliesthe torsion-free case of Conjecture 4.On π , in the special case when R = Z , we have(14) π (cid:0) BG + ∧ K ( Z ) (cid:1) ∼ = H (cid:0) BG ; K ( Z ) (cid:1) ⊕ H (cid:0) BG ; K ( Z ) (cid:1) ∼ = {± } ⊕ G ab . The first isomorphism comes from the Atiyah-Hirzebruch spectral sequence, whichis concentrated in the first quadrant because regular rings have vanishing negative K -theory. The second isomorphism comes from the computations K ( Z ) ∼ = {± } and K ( Z ) ∼ = Z . Under the isomorphism (14), it can be shown [Wal78a, Asser-tion 15.8] that the classical assembly map produces on π the left-hand map in (7),whose cokernel is by definition the Whitehead group Wh ( G ) . So we see that theFarrell-Jones Conjecture 13 implies Conjecture 8.From these identifications and computations of K ( Z [ G ]) and W h ( G ) for finitegroups we see that π (asbl ) and π (asbl ) may not be surjective for groups withtorsion, even when R = Z . The classical assembly map may also fail to be injectiveon homotopy groups if we drop the assumption torsion-free. This happens forexample for π (asbl ) if R = F is a finite field of characteristic prime to and G isthe non-cyclic group with elements [UW17].The regularity assumption cannot be dropped either. For example, considerthe case when G = C ∞ is the infinite cyclic group. Then of course BC ∞ = S and R [ C ∞ ] = R [ t, t − ] , and it can be shown that on π n the classical assembly mapproduces the left-hand map in the short exact sequence −→ K n ( R ) ⊕ K n − ( R ) −→ K n ( R [ t, t − ]) −→ N K n ( R ) ⊕ N K n ( R ) −→ given by the Fundamental Theorem of algebraic K -theory; see for example [BHS64]in low dimensions, [Swa95, Section 10], and [Wal78a, Theorem 18.1]. Recall that the groups N K n ( R ) are defined as the cokernel of the split injection K n ( R ) −→ K n ( R [ t ]) induced by the natural map R −→ R [ t ] . It is known that N K n ( R ) = 0 for each n if R is regular [Swa95, Theorem 10.1(1) and 10.3], but N K n ( R ) can be nontrivialfor arbitrary rings. So we see that the classical assembly map for the infinite cyclicgroup is a π ∗ -isomorphism if the ring R is regular, but otherwise it may fail to besurjective on homotopy groups.For arbitrary groups and rings, the generalization of Conjecture 13 is the following. Farrell-Jones Conjecture 15.
For any group G and for any ring R the Farrell-Jones assembly map asbl VC yc : EG ( VC yc ) + ∧ Or G K ( R [ G ∫ − ]) −→ K ( R [ G ]) is a π ∗ -isomorphism. Here VC yc denotes the family of virtually cyclic subgroups of G . A group iscalled virtually cyclic if it contains a cyclic subgroup of finite index.The Farrell-Jones Conjectures 13 and 15 are related as follows. Proposition 16. If G is a torsion-free group and R is a regular ring, then theFarrell-Jones Conjectures 13 and 15 are equivalent.Proof. This is an application of the following principle, which is proved in [LR05,Theorem 65].
Transitivity Principle 17.
Let F and F (cid:48) be families of subgroups of G with F ⊆ F (cid:48) . Assume that for each H ∈ F (cid:48) the assembly map EH ( F| H ) + ∧ Or H K ( R [ H ∫ − ]) −→ K ( R [ H ]) is a π ∗ -isomorphism, where F| H = { K ≤ H | K ∈ F } . Then the relative as-sembly map explained in Remark 12(ii), i.e., the left vertical map in the followingcommutative triangle, is a π ∗ -isomorphism. EG ( F ) + ∧ Or G K ( R [ G ∫ − ]) K ( R [ G ]) EG ( F (cid:48) ) + ∧ Or G K ( R [ G ∫ − ]) asbl F asbl F⊆F(cid:48) asbl
F(cid:48)
Therefore, asbl F is a π ∗ -isomorphism if and only if asbl F (cid:48) is a π ∗ -isomorphism. We now apply the transitivity principle in the case F = 1 and F (cid:48) = VC yc . Anynontrivial torsion-free virtually cyclic group is infinite cyclic. Recall that asbl canbe identified with the classical assembly map in Conjecture 13. So it is enough toshow that the classical assembly map is a π ∗ -isomorphism for the infinite cyclicgroup C . The fact that this is true in the case of regular rings is explained above,before the statement of Conjecture 15. (cid:3) The next result shows, as promised, that the Farrell-Jones Conjecture implies allthe other conjectures introduced in the first two subsections; the case of Conjecture 10is considered right after Conjecture 24 below.
Proposition 18.
The Farrell-Jones Conjecture 15 implies Conjectures 4 and 8,and so also Conjectures 1 and 6 by Propositions 5 and 9. -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 11
Proof.
The case of Conjecture 8 and the torsionfree case of Conjecture 4 is explainedabove, directly after the statement of Conjecture 13. The general case of Conjecture 4follows from the following isomorphisms. π (cid:18) EG ( VC yc ) + ∧ Or G K ( R [ G ∫ − ]) (cid:19) (cid:192) ∼ = π (cid:18) EG ( F in ) + ∧ Or G K ( R [ G ∫ − ]) (cid:19) (cid:193) ∼ = π (cid:18) EG ( F in ) + ∧ Or G ( F in ) K ( R [ G ∫ − ]) (cid:19) (cid:194) ∼ = H (cid:18) C ( EG ( F in )) ⊗ Z Or G ( F in ) K ( R [ G ∫ − ]) (cid:19) (cid:195) ∼ = Z ⊗ Z Or G ( F in ) K ( R [ G ∫ − ]) (cid:196) ∼ = colim Or G ( F in ) K ( R [ G ∫ − ]) (cid:197) ∼ = colim Sub G ( F in ) K ( R [ − ]) Theorem 19(ii) yields the isomorphism (cid:192) . Since EG ( F in ) H = ∅ if H is not finite,the isomorphism (cid:193) follows immediately by inspecting the construction of the coend.The assumptions that R is regular and that the order of every finite subgroup H of G is invertible in R imply that also R [ H ] is regular. For regular rings the negative K -groups vanish [Ros94, 3.3.1], and therefore the equivariant Atiyah-Hirzebruchspectral sequence explained in Remark 12(iv) is concentrated in the first quadrant.This gives the isomorphism (cid:194) . The singular or cellular chain complex C ( EG ( F in )) ,considered as a contravariant functor G/H (cid:55)−→ C ( EG ( F in ) H ) , resolves the constantfunctor Z , therefore (cid:195) follows from right exactness of − ⊗ Z Or G ( F in ) M for anyfixed M : Or G ( F in ) −→ Ab . The coend with the constant functor Z is one possibleconstruction of the colimit in abelian groups, hence (cid:196) . Since K n ( R [ G ∫ G/H ]) ∼ = K n ( R [ H ]) and since inner automorphisms induce the identity on K -theory, thefunctor K n ( R [ G ∫ − ]) factors over Or G ( F in ) −→ Sub G ( F in ) , the functor sending G/H → G/K , gH (cid:55)→ gaH to the class of H → K , h (cid:55)→ a − ha . The isomorphism (cid:197) then follows by standard properties of colimits. (cid:3) The next result deals with the passage from finite to virtually cyclic subgroupsin the source of the Farrell-Jones assembly map.
Theorem 19 (finite to virtually cyclic) . (i) The relative assembly map asbl F in ⊆VC yc : EG ( F in ) + ∧ Or G K ( R [ G ∫ − ]) −→ EG ( VC yc ) + ∧ Or G K ( R [ G ∫ − ]) is always split injective.(ii) If R is regular and the order of every finite subgroup of G is invertible in R ,then asbl F in ⊆VC yc is a π ∗ -isomorphism.(iii) If R is regular then asbl F in ⊆VC yc is a π Q ∗ -isomorphism, i.e., it induces iso-morphisms on π n ( − ) ⊗ Z Q for all n ∈ Z . Proof.
Part (i) is the main result of [Bar03a]. Part (ii) is shown in [LR05, Proposi-tion 70]. Part (iii) is proved in [LS16, Theorem 0.2] and generalizes [Gru08, Corollaryon page 165]. (cid:3)
After tensoring with the rational numbers, theFarrell-Jones Conjecture 15 for regular rings can be reformulated in a more concreteand computational fashion as follows.Assume that R is a regular ring. Recall from Theorem 19(iii) that the relativeassembly map asbl F in ⊆VC yc induces isomorphisms(20) π n (cid:16) EG ( F in ) + ∧ Or G K ( R [ G ∫ − ]) (cid:17) ⊗ Z Q ∼ = −→ π n (cid:16) EG ( VC yc ) + ∧ Or G K ( R [ G ∫ − ]) (cid:17) ⊗ Z Q . The theory of equivariant Chern characters developed by Lück in [Lüc02] yields thefollowing isomorphisms:(21) (cid:77) ( C ) ∈ ( FC yc ) (cid:77) s + t = n H s ( BZ G C ; Q ) ⊗ Q [ W G C ] Θ C (cid:16) K t ( R [ C ]) ⊗ Z Q (cid:17) π n (cid:16) EG ( FC yc ) + ∧ Or G K ( R [ G ∫ − ]) (cid:17) ⊗ Z Q π n (cid:16) EG ( F in ) + ∧ Or G K ( R [ G ∫ − ]) (cid:17) ⊗ Z Q . ∼ = ∼ = Before we explain the notation, notice the analogy with the well-known isomorphism (cid:77) s + t = n H s ( BG ; Q ) ⊗ Z (cid:16) K t ( R ) ⊗ Z Q (cid:17) ∼ = −→ π n (cid:0) BG + ∧ K ( R ) (cid:1) ⊗ Z Q , whose source corresponds to the summand in (21) indexed by C = 1 .Given a subgroup H of G , we denote by N G H the normalizer and by Z G H the centralizer of H in G , and we define the Weyl group as the quotient W G H = N G H/ ( Z G H · H ) . Notice that the Weyl group W G H of a finite subgroup H isalways finite, since it embeds into the outer automorphism group of H . We write FC yc for the family of finite cyclic subgroups of G , and ( FC yc ) for the set ofconjugacy classes of finite cyclic subgroups. Furthermore, Θ C is an idempotentendomorphism of K t ( R [ C ]) ⊗ Z Q , which corresponds to a specific idempotent inthe rationalized Burnside ring of C , and whose image is a direct summand of K t ( R [ C ]) ⊗ Z Q isomorphic to(22) coker (cid:32) (cid:77) D (cid:8) C ind CD : (cid:77) D (cid:8) C K t ( R [ D ]) ⊗ Z Q −→ K t ( R [ C ]) ⊗ Z Q (cid:33) . The Weyl group acts via conjugation on C and hence on Θ C ( K t ( R [ C ]) ⊗ Z Q ) . TheWeyl group action on the homology groups in the source of (21) comes from thefact that EN G C/Z G C is a model for BZ G C . -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 13 Farrell-Jones Conjecture 23 (rationalized version) . For any group G and forany regular ring R the composition of the Farrell-Jones assembly map and theisomorphisms (21) and (20) (cid:77) ( C ) ∈ ( FC yc ) (cid:77) s + t = n H s ( BZ G C ; Q ) ⊗ Q [ W G C ] Θ C (cid:16) K t ( R [ C ]) ⊗ Z Q (cid:17) −→ K n ( R [ G ]) ⊗ Z Q is an isomorphism for each n ∈ Z . Analogously one obtains the following conjecture for Whitehead groups, which isthe correct generalization of Conjecture 10 mentioned at the end of Subsection 2.2.
Conjecture 24.
For any group G there is an isomorphism (cid:77) ( C ) ∈ ( FC yc ) (cid:32) Q ⊗ Q [ W G C ] Θ C (cid:16) Wh ( C ) ⊗ Z Q (cid:17) ⊕ H ( BZ G C ; Q ) ⊗ Q [ W G C ] Θ C (cid:16) K − ( Z [ C ]) ⊗ Z Q (cid:17)(cid:33) Wh ( G ) ⊗ Z Q . ∼ = Conjecture 24 implies Conjecture 10, because in fact colim H ∈ obj Sub G ( F in ) Wh ( H ) ⊗ Z Q ∼ = (cid:77) ( C ) ∈ ( FC yc ) Q ⊗ Q [ W G C ] Θ C (cid:16) Wh ( C ) ⊗ Z Q (cid:17) and the map (11) coincides with the restriction to this summand of the map inConjecture 24. Remark 25.
For finite groups H we have that Wh ( H ) ⊗ Z Q ∼ = K ( Z [ H ]) ⊗ Z Q bythe exact sequence (7). The only difference between the sources of the maps inConjectures 23 and 24 is the absence from 24 of the summands with ( s, t ) = (1 , .For finite groups H the natural map Q ∼ = K ( Z ) ⊗ Z Q −→ K ( Z [ H ]) ⊗ Z Q is anisomorphism, and hence it follows from (22) that the only non-vanishing summandamong these is H ( BG ; Q ) ∼ = G ab ⊗ Z Q corresponding to C = 1 . This is consistentwith the exact sequence (7).Finally, we note that in the special case when R = Z the dimensions of the Q -vector spaces in (22) for any t and any finite cyclic group C can be explicitlycomputed as follows. Theorem 26.
Let C be a cyclic group of order c . Then dim Q Θ C (cid:16) K t ( Z [ C ]) ⊗ Z Q (cid:17) = s ( c ) − if t = − ; ϕ ( c ) / − if t = 1 and c > ; if t > , t ≡ , and c = 2 ; ϕ ( c ) / if t > , t ≡ , and c > ; otherwise.Here ϕ ( c ) = { x ∈ C | x generates C } is Euler’s ϕ -function, c = (cid:81) si =1 p e i i is theprime factorization of c , and s ( c ) = (cid:80) si =1 ϕ ( n/p e i i ) /f p i , where f p i is the smallestnumber such that p f pi i ≡ n/p e i . This result is proved in [Pat14, Theorem on page 9], and more details will appearin [PRV].
We now survey very briefly some other conjec-tures that are analogous to Conjecture 15. For details and further explanations werecommend [FRR95], [KL05], [Lüc], [LR05], and [MV03].In [FJ93a], Farrell and Jones formulated Conjecture 15 not only for algebraic K -theory, but also for L -theory; more precisely, for L (cid:104)−∞(cid:105) ( R [ G ]) , the quadraticalgebraic L -theory spectrum of R [ G ] with decoration −∞ , for any ring with involu-tion R . The corresponding assembly map is constructed completely analogously, byapplying the machinery of Subsection 2.3 to the functor L (cid:104)−∞(cid:105) ( R [ − ]) . In the specialcase of torsion-free groups G , this conjecture is equivalent to the statement that theclassical assembly map BG + ∧ L (cid:104)−∞(cid:105) ( R ) −→ L (cid:104)−∞(cid:105) ( R [ G ]) is a π ∗ -isomorphism, forany ring R , not necessarily regular.If G is a torsion-free group and the Farrell-Jones Conjectures hold for both K ( Z [ G ]) and L (cid:104)−∞(cid:105) ( Z [ G ]) , then the Borel Conjecture is true for manifolds withfundamental group G and dimension at least . The Borel Conjecture states that, if M and N are closed connected aspherical manifolds with isomorphic fundamentalgroups, then M and N are homeomorphic, and every homotopy equivalence between M and N is homotopic to a homeomorphism. In short, the Borel Conjecture saysthat closed aspherical manifolds are topologically rigid. Recall that a connectedCW complex X is aspherical if its universal cover is contractible, or equivalently if π n ( X ) = 0 for all n > .We also mention that the Farrell-Jones Conjecture in algebraic L -theory impliesthe Novikov Conjecture about the homotopy invariance of higher signatures.Furthermore, Farrell and Jones also formulated an analog of Conjecture 15 forthe stable pseudo-isotopy functor, or equivalently for Waldhausen’s A -theory, alsoknown as algebraic K -theory of spaces. We refer to [ELP +
16] for a modern approachto this conjecture and in particular for its many applications to automorphisms ofmanifolds.Finally, the analog of the Farrell-Jones Conjecture 15 for the complex topolog-ical K -theory of the reduced complex group C ∗ -algebra of G is equivalent to thefamous Baum-Connes Conjecture, formulated by Paul Baum, Alain Connes, andNigel Higson in [BCH94]. For the Baum-Connes Conjecture, the relative assemblymap asbl F in ⊆VC yc is always a π ∗ -isomorphism; compare and contrast with Theo-rem 19. Also the Baum-Connes Conjecture implies the Novikov Conjecture. Formore information on the relation between the Baum-Connes Conjecture and theFarrell-Jones Conjecture in L -theory we refer to [LN17] and [Ros95].3. State of the art
We now overview what we know and don’t know about the Farrell-Jones Conjec-ture 15, to the best of our knowledge in January 2017. We aim to give immediatelyaccessible statements, which may not always reflect the most general available results.We restrict our attention to algebraic K -theory and ignore the related conjecturesmentioned in the previous subsection. The following theorem is the result of the effortof many mathematicians over a long period of time. The methods of controlledalgebra and topology that underlie this theorem (and that we illustrate in the nextsection) were pioneered by Steve Ferry [Fer77] and Frank Quinn [Qui79], and werethen applied with enourmous success by Farrell-Hsiang [FH78], [FH81b], [FH83]and Farrell-Jones [FJ86], [FJ89], [FJ93b], [FJ93a]. Many ideas in the proofs of -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 15 the following results originate in these articles. The formulation of the theorembelow is meant to be a snapshot of the best results available today, as opposed toa comprehensive historical overview of the many important intermediate resultspredating the works quoted here.
Theorem 27.
Let G be the smallest class of groups that satisfies the following twoconditions.(1) The class G contains:(a) hyperbolic groups [BLR08] ;(b) finite-dimensional CAT(0)-groups [BL12a] , [Weg12] ;(c) virtually solvable groups [FW14] , [Weg15] ;(d) Baumslag-Solitar groups and graphs of abelian groups [FW14] , [GMR15] ;(e) lattices in virtually connected Lie groups [BFL14] , [KLR16] ;(f ) arithmetic and S -arithmetic groups [BLRR14] , [Rüp16] ;(g) fundamental groups of connected manifolds of dimension at most [Rou08] ;(h) Coxeter groups;(i) Artin braid groups [AFR00] ;(j) mapping class groups of oriented surfaces of finite type [BB16] .(2) The class G is closed under:(A) subgroups [BR07a] ;(B) overgroups of finite index [BLRR14, Section 6] ;(C) finite products;(D) finite coproducts;(E) directed colimits [BEL08] ;(F) graph products [GR13] ;(G) if −→ N −→ G p −→ Q −→ is a group extension such that Q ∈ G and p − ( C ) ∈ G for each infinite cyclic subgroup C ≤ Q , then G ∈ G ;(H) if G is a countable group that is relatively hyperbolic to subgroups P , . . . , P n and each P i ∈ G , then G ∈ G [Bar17] .Then the Farrell-Jones Conjecture 15 holds for any ring R and for any group G ∈ G .Proof. In order to have the inheritance properties formulated in (2) one needsto work with a slight generalization of the Farrell-Jones Conjecture. First, oneneeds to allow coefficients in arbitrary additive categories with G -actions [BR07a];then, one says that the conjecture with finite wreath products is true for G if theconjecture holds not only for G , but also for all wreath products G (cid:111) F of G withfinite groups F [Rou07], [BLRR14, Section 6]. The Farrell-Jones Conjecture withcoefficients and finite wreath products is true for all groups listed under (1) and hasall the inheritance properties listed under (2).Some of the earlier references given above omit the discussion of the version withfinite wreath products; consult [BLRR14, Section 6] and [GR13, Proposition 1.1]for the corresponding extensions.We discuss the statements (h), (C), (D) and (G), for which no reference wasprovided above. Coxeter groups (h) are known to fall under (b) by a result ofMoussong; compare [Dav08, Theorem 12.3.3]. For (C) use [BR07a, Corollary 4.3]applied to the projection to the factors, the Transitivity Principle 17, and the factthat the Farrell-Jones Conjecture is known for finite products of virtually cyclicgroups. The extension to the version with finite wreath products uses the fact that ( G × G ) (cid:111) F is a subgroup of ( G (cid:111) F ) × ( G (cid:111) F ) . Finite coproducts are treated similarly using property (G) and the natural map from the coproduct to the product;compare [GR13, Proposition 1.1]. Statement (G) itself is simply a combination of[BR07a, Corollary 4.3] and the Transitivity Principle 17. (cid:3) At the time of writing the Farrell-Jones Conjec-ture 15 seems to be open for the following classes of groups:(i) Thompson’s groups;(ii) outer automorphism groups of free groups;(iii) linear groups;(iv) (elementary) amenable groups;(v) infinite products of groups (satisfying the Farrell-Jones Conjecture).However, for some of these groups there are partial injectivity results, as we explainin Remark 29 below.
The next theorem gives two examples of injectivityresults for assembly maps in algebraic K -theory. Part (i) is proved using the tracemethods explained in Section 5 below, where more rational injectivity results aredescribed. Part (ii) is based on a completely different approach using controlledalgebra, the descent method due to Gunnar Carlsson and Erik Pedersen [CP95]. Forthis method to work, the group has to satisfy some mild metric conditions, whichare not needed for the weaker statement in part (i). One such condition goes backto [FH81a]. The condition of finite asymptotic dimension appeared in the contextof algebraic K -theory in [Bar03b] and [CG04, CG05], and was later generalized tofinite decomposition complexity in [RTY14]. The extension to non-classical assemblymaps appeared in [BR07b, BR17] and [Kas15]. The statement in part (ii) belowis from [KNR18] and further improves and combines these developments. We alsomention [FW91] for yet another approach to injectivity results.Recall from Theorem 19 that the relative assembly map π n (cid:16) EG ( F in ) + ∧ Or G K ( Z [ G ∫ − ]) (cid:17) −→ π n (cid:16) EG ( VC yc ) + ∧ Or G K ( Z [ G ∫ − ]) (cid:17) is always split injective, and it becomes an isomorphism after applying − ⊗ Z Q if R is regular, e.g., if R = Z . Therefore the results below would follow if we knew theFarrell-Jones Conjecture 15. Theorem 28.
Assume that there exists a finite-dimensional EG ( F in ) , and thatthere exists an upper bound on the orders of the finite subgroups of G .(i) If R = Z , then there exists an integer L > such that for every n ≥ L therationalized assembly map π n (cid:16) EG ( F in ) + ∧ Or G K ( Z [ G ∫ − ]) (cid:17) ⊗ Z Q −→ K n ( Z [ G ]) ⊗ Z Q is injective.(ii) Assume furthermore that G has regular finite decomposition complexity. Thenfor any ring R the assembly map EG ( F in ) + ∧ Or G K ( R [ G ∫ − ]) −→ K ( R [ G ]) is split injective on π ∗ . -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 17 Proof. (i) is a consequence of Theorem 69 below, or rather of its more general versionin [LRRV17a, Main Technical Theorem 1.16]; see Remark 70(iv) and [LRRV17a,Theorem 1.15], where the result is only stated for cocompact EG ( F in ) , but theproof given on page 1015 only uses finite-dimensionality and the existence of abound on the order of the finite cyclic subgroups. (ii) is [KNR18, Theorem 1.3]. (cid:3) Remark 29.
Theorem 28 applies to groups for which no isomorphism results wereknown at the time of writing:(i) The existence of an upper bound on the orders of the finite subgroups of G follows from the existence of a cocompact EG ( F in ) . For example, this is thecase for outer automorphism groups of free groups, to which Theorem 28(i)then applies.(ii) Regular finite decomposition complexity is a property shared by all groupsthat are either (a) of finite asymptotic dimension, (b) elementary amenable,(c) linear, or (d) subgroups of virtually connected Lie groups.4. Controlled algebra methods
As noted in the previous section, most proofs of the Farrell-Jones Conjecture 15use the ideas and technology of controlled algebra, which are the focus of this section.The ultimate goal is to explain the Farrell-Hsiang Criterion for assembly maps tobe π ∗ -isomorphisms. The criterion goes back to [FH78] and has been successfullyapplied in many cases, e.g. [FH81b], [FH83], [Qui12], and plays an important rolein the proof of Theorem 27(1)(e) [BFL14]. The formulation that we give here inTheorem 57 is due to [BL12b].Our goal is to keep the exposition as concrete as possible, and to work out themain details of the proof of the following result, establishing the first nontrivial caseof the Farrell-Jones Conjecture. Theorem 30.
The Farrell-Jones Conjecture 15 holds for finitely generated freeabelian groups, i.e., for any n ≥ and for any ring R , the assembly map E Z n ( C yc ) + ∧ Or Z n K ( R [ Z n ∫ − ]) −→ K ( R [ Z n ]) is a π ∗ -isomorphism. Before we get to the proof, we want to show how Theorem 30 leads to a simpleformula for the Whitehead groups of Z n ; the article [LR14] contains many similarbut way more general explicit computations. The Whitehead groups of G over R aredefined as Wh Rk ( G ) = π k ( Wh R ( G )) , where Wh R ( G ) is the homotopy cofiber of theclassical assembly map asbl : BG + ∧ K ( R ) −→ K ( R [ G ]) appearing in Conjecture 13.Of course, Wh Z ( G ) = Wh ( G ) . Corollary 31.
For any n ≥ and k ∈ Z there are isomorphisms Wh Rk ( Z n ) ∼ = (cid:77) C ∈M ax C yc Wh Rk ( C ) ∼ = (cid:77) C ∈M ax C yc N K k ( R ) ⊕ N K k ( R ) , where M ax C yc denotes the set of maximal cyclic subgroups of Z n . Observe that the set of maximal cyclic subgroups of Z n can be identified with P n − ( Q ) , the set of all -dimensional subspaces of the Q -vector space Q n . Proof.
There is a Z n -equivariant homotopy pushout square (cid:96) C ∈M ax C yc Z n × C EC E Z n (cid:96) C ∈M ax C yc Z n × C pt E Z n ( C yc ) . Applying ( ? ) + ∧ Or ( Z n ) K ( R [ Z n ∫ − ]) preserves homotopy pushout squares, and theinduced left vertical map can be identified with a wedge sum of copies of the classicalassembly map asbl for C , using induction isomorphisms. The homotopy cofibrationsequence BC + ∧ K ( R ) ∼ = EC + ∧ Or ( C ) K ( R [ C ∫ − ]) asbl −−−→ K ( R [ C ]) −→ Wh R ( C ) is known to split, and W h Rn ( C ) ∼ = N K n ( R ) ⊕ N K n ( R ) ; compare [Swa95, Section 10]and [Wal78a, Theorem 18.1]. Therefore we obtain the following homotopy pushoutsquare. pt E Z n + ∧ Or Z n K ( R [ Z n ∫ − ]) (cid:95) C ∈M ax C yc Wh R ( C ) E Z n ( C yc ) + ∧ Or Z n K ( R [ Z n ∫ − ]) Theorem 30 identifies the bottom right corner with K ( R [ Z n ]) , and therefore thehomotopy cofiber of the right vertical map agrees with the homotopy cofiber of theclassical assembly map for Z n , completing the proof. (cid:3) Working with the Farrell-Jones Conjecture with coefficients mentioned in theproof of Theorem 27, we can use induction and reduce the proof of Theorem 30 to thecase n = 2 , by applying the inheritance property formulated in Theorem 27(2)(G) toa surjective homomorphism Z n −→ Z . Notice that for Z itself Theorem 27(2)(G)is useless.However, even in the case n = 2 the full proof of Theorem 30 involves manytechnicalities that obscure the underlying ideas. For this reason, we concentrate onthe following partial result. Proposition 32.
The assembly map (33) π (cid:16) E Z ( C yc ) + ∧ Or G K ( R [ Z ∫ − ]) (cid:17) −→ K ( R [ Z ]) is surjective for any ring R . In the rest of this section we give a complete proof of this proposition moduloTheorem 37, which we use as a black box. The proof is completed right after thestatement of Claim 50. -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 19
The main characters of controlled algebra are definednext.
Definition 34 (geometric modules) . Given a ring R and G -space X , the cate-gory C ( X ) = C G ( X ; R ) of geometric R [ G ] -modules over X is defined as follows.The objects of C ( X ) are cofinite free G -sets S together with a G -map ϕ : S −→ X .Notice that, given a cofinite free G -set S , the R -module R [ S ] is in a natural way afinitely generated free R [ G ] -module. The morphisms in C ( X ) from ϕ : S −→ X to ϕ (cid:48) : S (cid:48) −→ X are simply the R [ G ] -linear maps R [ S ] −→ R [ S (cid:48) ] .The category C ( X ) is additive and depends functorially on X , in the sense thata G -map f : X −→ X (cid:48) induces an additive functor f ∗ : C ( X ) −→ C ( X (cid:48) ) whichsends the object ϕ to f ◦ ϕ . Let F ( R [ G ]) be the category of finitely generated free R [ G ] -modules. The functor U : C ( X ) −→ F ( R [ G ]) (where U stands for underlying)that sends ϕ : S −→ X to R [ S ] is obviously an equivalence of additive categories,since ϕ does not enter the definition of the morphisms in C ( X ) . Therefore we obtaina π ∗ -isomorphism(35) K ( C ( X )) K ( R [ G ]) . U (cid:39) However, the advantage of C ( X ) is that morphisms have a geometric shadow in X ,and if X is equipped with a metric we can talk about their size. Definition 36 (support and size) . Let α : R [ S ] −→ R [ S (cid:48) ] be a morphism in C ( X ) from ϕ : S −→ X to ϕ (cid:48) : S (cid:48) −→ X . Let ( α s (cid:48) s ) ( s (cid:48) ,s ) ∈ S (cid:48) × S be the associated matrix.Define the support of α to be supp α = (cid:8) (cid:0) ϕ (cid:48) ( s (cid:48) ) , ϕ ( s ) (cid:1) ∈ X × X (cid:12)(cid:12) α s (cid:48) s (cid:54) = 0 (cid:9) ⊆ X × X . If X is equipped with a G -invariant metric d , define the size of α to be size α = sup (cid:8) d (cid:0) ϕ (cid:48) ( s (cid:48) ) , ϕ ( s ) (cid:1) (cid:12)(cid:12) α s (cid:48) s (cid:54) = 0 (cid:9) . Figure 1.
Support of a morphism with G = Z acting via shift on a band.Note that the supremum is really a maximum, since α is G -equivariant, d is G -invariant, and S is cofinite. As we will see, sometimes it is convenient to workwith extended metrics, i.e., metrics for which d ( x, x (cid:48) ) = ∞ is allowed. Being of finitesize is then a severe restriction on α . In the support picture no arrow is allowedbetween points at distance ∞ ; compare Figure 2 on page 23.The main idea now is that assembly maps can be described as forget control maps. Proving that an element is in the image of an assembly map can be achievedby proving that it has a representative of small size. Before making this precise weintroduce some more conventions and definitions. Recall that a point a in a simplicial complex Z can be written uniquely in theform a = (cid:88) v ∈ V a v v , where V is the set of vertices of the underlying abstract simplicial complex, a v ∈ [0 , ,and (cid:80) v ∈ V a v = 1 . The point a lies in the interior of the realization ∆ v of the uniqueabstract simplex given by { v | a v (cid:54) = 0 } . The l -metric on Z is defined as d ( a, b ) = (cid:88) v ∈ V | a v − b v | . Observe that the distance between points is always ≤ , and that every simplicialautomorphism is an isometry with respect to the l -metric. Theorem 37 (small elements are in the image) . For any integer n > there isan ε = ε ( n ) > such that for every G -simplicial complex Z of dimension n thefollowing is true. Let x ∈ K ( R [ G ]) and consider the assembly map asbl Z inducedby Z −→ pt . K ( C ( Z )) [ α ] π (cid:16) Z + ∧ Or G K ( R [ G ∫ − ]) (cid:17) K ( R [ G ]) x ∼ = U (cid:51) asbl Z (cid:51) Then x ∈ im(asbl Z ) if there exists an automorphism α in C ( Z ) with U ([ α ]) = x and size( α ) ≤ ε and size( α − ) ≤ ε . Corollary 38.
Retain the notation and assumptions of Theorem 37. If all isotropygroups of Z belong to the family F , then x is also in the image of the assembly map (39) π (cid:16) EG ( F ) ∧ Or G K ( R [ G ∫ − ]) (cid:17) asbl F −−−→ K ( R [ G ]) . Proof.
The universal property of EG ( F ) in Remark 12(ii) gives a G -equivariantmap Z −→ EG ( F ) . Hence the assembly map asbl Z , which is induced by Z −→ pt ,factors over the assembly map asbl F , which is induced by EG ( F ) −→ pt . (cid:3) The sufficient condition for surjectivity on π from the preceding two results isgeneralized in Theorem 55 below to a necessary and sufficient condition for assemblymaps to be π ∗ -isomorphisms. In Remark 56 we explain how and where in theliterature Theorem 37 is proved. In view of Theorem 37 and Corollary 38, a possiblestrategy to prove surjectivity of asbl F is to look for contracting maps. This leads tothe following criterion. Criterion 40.
Fix G , R , F , and a word metric d G for G . Suppose that thereis an N > such that for any arbitrarily large D > there exist a simplicialcomplex Z D with a simplicial G -action and a G -equivariant map f D : G/ −→ Z D satisfying the following conditions: -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 21 (i) dim Z D ≤ N ;(ii) all isotropy groups of Z D lie in F ;(iii) the map f D is D -contracting with respect to the l -metric in the target andthe word metric in the source, i.e., for all g, g (cid:48) ∈ G we have d ( f D ( g ) , f D ( g (cid:48) )) ≤ D d G ( g, g (cid:48) ) . Then the map (39) is surjective.
The projection map to a point always satisfies (i) and (iii) but not (ii). The N -skeleton of a simplicial model for EG ( F ) always satisfies (i) and (ii). But howcan we produce contracting maps f D that satisfy all three conditions? In Remark 41below we explain why the assumptions of the criterion are too strong to be useful.Nevertheless, we spell out the proof of the criterion as a warm-up exercise. Proof.
Set (cid:15) = min { (cid:15) ( n ) | n ≤ N } , where (cid:15) ( n ) comes from Theorem 37. Givenany x ∈ K ( R [ G ]) consider the following diagram. K ( C ( Z D )) K ( C ( G/ α ] K ( R [ G ]) x U ∼ = U ∼ = f D ∗ (cid:51)(cid:51) Choose an automorphism α in C ( G/ whose class [ α ] ∈ K ( C ( G/ maps to x under U . Determine the sizes of α and α − , and then choose D so large that size f D ∗ ( α ) ≤ D size α < (cid:15) and analogously for α − . Then Corollary 38 implies that x is in the image of theassembly map asbl F in (39). (cid:3) Remark 41.
The case of Proposition 32 is when G = Z and F = C yc . Unfortu-nately, the conditions of Criterion 40 cannot possibly be satisfied in this case. Toexplain why, we need the following lemma. Lemma 42.
Let s be a simplicial automorphism of a simplicial complex Z with dim Z ≤ N . If x = (cid:80) x v v ∈ Z is such that d ( x, sx ) < N + 1) , then a barycenter of a face of the simplex ∆ x spanned by V x = { v | x v (cid:54) = 0 } is fixedunder s .Proof. For a vertex v with x v (cid:54) = 0 set A v = { v, sv, s v, . . . } . If A v ⊂ V x then s permutes the finitely many elements in A v and in particular fixes the barycenter ofthe face spanned by A v .Suppose that for no vertex v with x v (cid:54) = 0 we have A v ⊂ V x . Then for all v ∈ V x there exists a smallest n ( v ) ≥ such that s n ( v ) v / ∈ V x and hence x s n ( v ) v = 0 . Since V x contains at most N + 1 vertices we know that n ( v ) ≤ N + 1 for all v ∈ V x . Write (cid:15) = N +1) ; then from d ( x, s − x ) = d ( s − x, s − x ) = · · · = d ( s − N x, s − ( N +1) x ) < (cid:15) and x s n ( v ) v = 0 we conclude that x s n ( v ) − v < (cid:15) , x s n ( v ) − v < (cid:15) , . . . , x v < n ( v ) (cid:15) ≤ ( N + 1) (cid:15) = 1 N + 1 . However, since (cid:80) v ∈ V x x v = 1 , the last inequality cannot be true for all verticesin V x . (cid:3) Now suppose we can arrange (i) and (iii) from Criterion 40. If S is a (very large)finite subset of G , then by (iii) there exists a G -equivariant map f : G/ −→ Z to a G -simplicial complex that is contracting enough in order to have d ( f (1) , f ( s )) < N +1) for all s ∈ S . The lemma implies that for each s ∈ S a barycenter of a faceof the simplex ∆ f (1) determined by f (1) is fixed under s . Let b ( N ) be the numberof vertices in the barycentric subdivision of an N -simplex. Then there exists asubset T ⊂ S with cardinality T ≥ Sb ( N ) and a point in ∆ f (1) fixed by all elementsof T . The subgroup generated by T must lie in F if we require (ii). Since S can bearbitrarily large it seems difficult to keep F small.In the case G = Z we can choose S = S l = (cid:26) ( x , x ) (cid:12)(cid:12)(cid:12)(cid:12) | x | ≤ l and | x | ≤ l (cid:27) . Then if l > b ( N ) we have T ≥ l b ( N ) > l , and since a subset of S l with more than l elements cannot be contained in a line, the set T generates a finite index subgroup.Hence we can never arrange F = C yc as desired, proving the claim in Remark 41. The trick to obtain sufficiently contractingmaps is to relax the requirement that the maps are G -equivariant, and instead onlyask for equivariance with respect to (finite index) subgroups. We first illustrate thisphenomenon in an example that is too simple to be useful. Example 43.
Consider the standard shift action of the infinite cyclic group G = Z on the real line: Z × R −→ R , ( z, x ) (cid:55)−→ z + x . This is a simplicial action if we consider R as -dimensional simplicial complex withset of vertices Z ⊂ R . The map f D : Z −→ R , z (cid:55)−→ D z is D -contracting but not Z -equivariant. It becomes Z -equivariant if we change theaction on R to the action given by Z × R −→ R , ( z, x ) (cid:55)−→ D z + x . However, this action is no longer simplicial. If we restrict the action to the subgroup D Z < Z or to any subgroup H with H ≤ D Z , then the H -action on res Z H R issimplicial, and f D : res Z H Z −→ res Z H R is a D -contracting H -equivariant map.Assume for a moment that for a subgroup H ≤ G of finite index we have an H -equivariant map f D : res GH G/ −→ E H -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 23 to an H -simplicial complex E H that is D -contracting with respect to a word metricin the source and the l -metric in the target. Let us see what happens when weinduce up to G .If ( X, d ) is a metric space with an isometric H -action, then ind GH X = G × H X has an isometric G -action with respect to the extended metric d ([ g, x ] , [ g (cid:48) , x (cid:48) ]) = (cid:40) d X ( x, g − g (cid:48) x ) if g − g (cid:48) ∈ H ; ∞ if g − g (cid:48) (cid:54)∈ H .Applying this to f D we obtain a map ind GH f D : ind GH res GH G/ −→ ind GH E H which is still D -contracting. However, observe that D ∞ = ∞ , and that a pair ofpoints at distance ∞ in the source is mapped to a pair of points still at distance ∞ inthe target. Hence the map can be used to diminish the size of a morphism betweengeometric modules only if the morphism over ind GH res GH G/ is of finite size, i.e.,only if it has no components that connect points at distance ∞ .The usual induction homomorphism ind GH : K ( R [ H ]) −→ K ( R [ G ]) given by thefunctor R [ G ] ⊗ R [ H ] − can be easily lifted to the categories of geometric modules,i.e., for any metric space X the functor ind GH : C ( X ) −→ C (ind GH X ) , ( ϕ : S −→ X ) (cid:55)−→ (ind GH ϕ : ind GH S −→ ind GH X ) induces the upper horizontal map in the following commutative diagram. K ( C ( X )) K ( C (ind GH X )) K ( R [ H ]) K ( R [ G ]) ind GH ∼ = U ∼ = U ind GH If X is a metric space in the usual sense (where ∞ is not allowed), then morphismsin the image of ind GH have the desired property: the size of ind GH α is finite eventhough ind GH X is a metric space in the extended sense. Moreover size ind GH α = size α . Therefore, using the map ind GH f D we can hope to show that, maybe not arbitraryelements, but at least elements of the form ind GH [ β ] belong to the image of asbl F . Figure 2.
Support of α and ind GH α in an index situation.The reason why this is useful is the following theorem of Swan. Recall thata finite group E is called hyperelementary if it fits into a short exact sequence −→ C −→ E −→ P −→ where C is cyclic and the order of P is a prime power. Theorem 44 (Swan induction) . Let F be a finite group, pr : G −→ F a surjectivehomomorphism, and H pr = (cid:8) pr − ( E ) (cid:12)(cid:12) E is a hyperelementary subgroup of F (cid:9) . Then for every H ∈ H pr there exist Z [ H ] -modules M + H and M − H that are finitelygenerated free as Z -modules, and such that, for each n ∈ Z and each x ∈ K n ( R [ G ]) ,we have (45) x = (cid:88) H ∈H p ind GH (cid:0) [ M + H ] · res GH x (cid:1) − ind GH (cid:0) [ M − H ] · res GH x (cid:1) . Here, for y ∈ K n ( R [ H ]) and a Z [ H ] -module M which is finitely generated free asa Z -module, we write [ M ] · y = l M ( y ) for the image of y under the map induced in K -theory by the functor l M that sends the R [ H ] -module P to M ⊗ Z P equippedwith the diagonal H -action. Proof.
The Swan group Sw ( H ; Z ) is by definition the K -group of Z H -modules thatare finitely generated free as Z -modules. The relation is the usual additivity relationfor (not necessarily split) short exact sequences. Tensor products over Z equippedwith the diagonal H -actions induce the structure of a unital commutative ringon Sw ( H ; Z ) , and also define an action of Sw ( H ; Z ) on K n ( R [ H ]) . Swan showedin [Swa60] that for a finite group F there exist Z [ E ] -modules N + E and N − E , where E runs through all hyperelementary subgroups of F , such that in Sw ( F ; Z ) we have(46) Z ] = (cid:88) E ind FE [ N + E ] − ind FE [ N − E ] . The natural isomorphisms ind GH (cid:16) M ⊗ Z res GH P (cid:17) ∼ = → (cid:0) ind GH M (cid:1) ⊗ Z P and ind G pr − ( E ) res pr N ∼ = → res pr ind FE N , given by g ⊗ m ⊗ p (cid:55)→ g ⊗ m ⊗ gp and g ⊗ n (cid:55)→ pr( g ) ⊗ n , respectively, yield thefollowing identity in K n ( R [ G ]) for H = pr − ( E ) : (cid:0) res pr ind FE [ N ] (cid:1) · x = (cid:0) ind GH [res pr N ] (cid:1) · x = ind GH (cid:0) [res pr N ] · res GH x (cid:1) . Using this and res pr pr Z ] = [ Z ] = 1 one derives the statement in the theoremwith M pr − ( E ) = res pr N E from (46). (cid:3) If we want to use H -equivariant contracting maps, as explained above, to showthat each of the summands in (45) is in the image of asbl F , we need to controlthe size of a geometric representative of [ M ± H ] · res GH x in terms of the size of arepresentative of x .This is indeed easy. Similarly to induction, also the functors restriction res GH and l M = M ⊗ Z − can be lifted to categories of geometric modules. For restriction simplysend the object given by φ : S −→ Z to res GH φ : res GH S −→ res GH Z . For l M observethat if B is a finite Z -basis for the Z [ H ] -module M , then there are isomorphisms of Z [ H ] -modules(47) Z [ B ] ⊗ Z R [ (cid:96) H/ ∼ = −→ Z [ B ] ⊗ Z R [ (cid:96) H/ ∼ = −→ R [ B × (cid:96) H/ . Here the first isomorphism is given by m ⊗ h (cid:55)→ h − m ⊗ h , where in the sourceone uses the diagonal H -action, and in the target the H -action on the right tensorfactor. The second isomorphism is the obvious one. One constructs the desiredfunctor by working only with objects of the form φ : (cid:96) H/ −→ Z and sending such -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 25 a φ to φ ◦ pr , where pr : B × (cid:96) H/ −→ (cid:96) H/ is the projection onto the secondfactor. The behaviour on morphisms is determined by the isomorphism (47): onedefines l M α between the objects on the right in (47) in such a way that on the leftit corresponds to id ⊗ α . One then checks easily that size res GH α = size α and size l M α = size α . In summary, given a finite index subgroup H ≤ G , a Z [ H ] -module M that isfinitely generated free as a Z -module, and an H -equivariant D -contracting map f D : res GH G/ −→ Z to an H -simplicial complex, we have a commutative diagram(48) [ α ] (cid:104) (ind GH f D ) ∗ (ind GH l M res GH α ) (cid:105) K ( C (ind GH Z )) K ( C ( G/ K ( C (res GH G/ K ( C (res GH G/ K ( C (ind GH res GH G/ K ( R [ G ]) K ( R [ H ]) K ( R [ H ]) K ( R [ G ]) ∈ ∈ res GH ∼ = U l M ∼ = U ind GH ∼ = U ∼ = U (ind GH f D ) ∗ res GH l M ind GH and the estimate(49) size (cid:16) (ind GH f D ) ∗ (ind GH l M res GH α ) (cid:17) ≤ D size (cid:16) ind GH l M res GH α (cid:17) = 1 D size α < ∞ . In order to prove surjectivity of asbl F it remains to find suitable finite quotients pr : G −→ F and suitable H -equivariant contracting maps for each H ∈ H pr . Thisleads to the criterion formulated in Theorem 57 below for arbitrary groups G .Groups that meet this criterion have been named Farrell-Hsiang groups in [BL12b]. Z is a Farrell-Hsiang group. Now we concentrate on the concrete situationwhere G = Z , and explain how the criterion is met in this special case. Claim 50 ( Z is a Farrell-Hsiang group with respect to C yc ) . Fix a word metric d Z × Z on Z × Z . Consider R as a simplicial complex with vertices Z ⊂ R and with thecorresponding (cid:96) -metric d . For any arbitrarily large D > there exists a surjectivehomomorphism pr D : Z × Z −→ F to a finite group F with the following property.For each H ∈ H pr D = (cid:8) pr − D ( E ) (cid:12)(cid:12) E is a hyperelementary subgroup of F (cid:9) there exist:(i) a simplicial H -action on R with only cyclic isotropy,(ii) a map f H : res H ( Z × Z ) −→ R that is H -equivariant and D -contracting, i.e., (51) d ( f H ( g ) , f H ( g (cid:48) )) ≤ D d Z × Z ( g, g (cid:48) ) for all g, g (cid:48) ∈ Z × Z . We first show that this implies Proposition 32.
Proof of Proposition 32.
The simplicial complex R is -dimensional. Let (cid:15) = (cid:15) (1) be as in Theorem 37. Given x ∈ K ( R [ G ]) choose an automorphism α in C ( G/ such that [ α ] maps to x under the forgetful map U : K ( C ( G/ −→ K ( R [ G ]) .Choose D > so large that D max { size( α ) , size( α − ) } ≤ (cid:15) . Use Claim 50 in orderto find a finite quotient pr D : Z × Z −→ F and H -equivariant D -contracting maps f H : res GH G/ −→ R for every H ∈ H pr D . For each H ∈ H pr D , let M = M ± H be as in Theorem 44, and send [ α ] through theupper row in diagram (48). Use estimate (49) to conclude that size (cid:16) (ind GH f H ) ∗ (ind GH l M res GH α ) (cid:17) ≤ (cid:15) . By Corollary 38 and the commutativity of (48), we see that ind GH l M res GH x is in theimage of the map (33). Because of the decomposition (45) in Theorem 44, also x isin the image. (cid:3) Proof of Claim 50.
We begin with some simplifications. With respect to the stan-dard generating set { ( ± , , (0 , ± } , the word metric is Lipschitz equivalent tothe Euclidean metric on Z × Z ⊂ R × R . On R the simplicial (cid:96) metric and theEuclidean metric satisfy d ( x, y ) ≤ C d
Eucl ( x, y ) for some fixed constant C . Therefore it is enough to establish (51) with respect tothe Euclidean metrics on Z × Z ⊂ R × R and on R , instead of the word and (cid:96) -metrics.Moreover, it is enough to consider only maximal hyperelementary subgroups of F ,because then for any H (cid:48) < H we can take f H (cid:48) = res H (cid:48) f H .Let us start to look for suitable finite quotients F of Z × Z . If F itself werehyperelementary, then we would have to find a contracting map f Z × Z to a ( Z × Z ) -simplicial complex with cyclic isotropy that is ( Z × Z ) -equivariant. But in Remark 41we saw that this is impossible.Every finite quotient F of Z × Z is isomorphic to Z /a × Z /ab , which is hyper-elementary if and only if a is a prime power. Hence a simple choice of F which isnot itself hyperelementary is Z /pq × Z /pq for distinct primes p and q . In order toachieve the contracting property we will later choose the primes to be very large.Let pr pq : Z × Z −→ Z /pq × Z /pq be the projection. A maximal hyperelementarysubgroup E of Z /pq × Z /pq has order pq or p q . By symmetry it is enough toconsider the case where the order of E is pq . Let H = pr − pq ( E ) . Now we need toconstruct f H .For every v ∈ Z × Z with v (cid:54) = 0 , consider the map f v : Z × Z (cid:96) v = (cid:104) v, −(cid:105) −−−−−−→ Z − /p −−−−−−→ R , w (cid:55)−→ p (cid:104) v, w (cid:105) where (cid:104)− , −(cid:105) is the standard inner product on R . If we equip R with the ( Z × Z ) -action given by ( Z × Z ) × R −→ R , ( w, x ) (cid:55)−→ x + p (cid:104) v, w (cid:105) then f v is ( Z × Z ) -equivariant. More importantly, we have that:(A) f v is p/ (cid:107) v (cid:107) -contracting, i.e., | f v ( w ) − f v ( w (cid:48) ) | ≤ (cid:107) v (cid:107) p (cid:107) w − w (cid:48) (cid:107) . This followsimmediately from the linearity of f v and the Cauchy-Schwarz inequality.(B) The isotropy group at every point of R is ker( (cid:96) v ) = { w ∈ Z × Z | (cid:104) v, w (cid:105) = 0 } ,and hence cyclic since we assumed that v (cid:54) = 0 .(C) The action restricts to a simplicial H -action if (cid:96) v ( H ) ⊆ p Z . -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 27 Let us reformulate the last condition. Consider the following commutative diagram.(52) H = pr − pq ( E ) Z × Z Z E Z /pq × Z /pq pr( E ) = F p · u F p × F p F p(cid:96) v pr pq pr (cid:96) v = (cid:96) v Here u ∈ F p × F p is a generator of the F p -vector space pr( E ) < F p × F p . Observethat pr( E ) (cid:54) = F p × F p because the order of E is pq . Then the last condition aboveis equivalent to saying that the composition in diagram (52) from H to F p is trivial,i.e., that (cid:96) v ( u ) = 0 .Hence, if we can find a vector v ∈ Z × Z such that(53) < (cid:107) v (cid:107) ≤ √ p and (cid:96) v ( u ) = 0 , then from (A) we get that f v is a ( p/ √ p = √ p/ -contracting H -equivariant mapto R , where R is equipped with a simplicial H -action by (C) and has cyclic isotropyby (B).The existence of such a vector v is established by the following counting argument.Consider the set S = (cid:110) v = ( x , x ) ∈ Z × Z (cid:12)(cid:12)(cid:12) | x | ≤ (cid:112) p and | x | ≤ (cid:112) p (cid:111) . This set has more than p elements, and therefore the map S −→ F p , v (cid:55)−→ (cid:96) v ( u ) is not injective, where u was defined right after diagram (52). If v and v are twodistinct vectors in S with (cid:96) v ( u ) = (cid:96) v ( u ) , then v = v − v is a vector which satisfiesthe equality in (53). For the inequality in (53) we estimate (cid:107) v (cid:107) ≤ (cid:107) v (cid:107) + (cid:107) v (cid:107) ≤ √ (cid:112) p = 4 √ p . So we define f H = f v for such a v and finish the argument using Euclid’s Theorem:since there are infinitely many primes, for any given D > we can find distinctprimes p and q such that both √ p/ ≥ D and √ q/ ≥ D , and hence for every H ∈ H pr pq = (cid:8) pr − pq ( E ) (cid:12)(cid:12) hyperelementary E < Z /pq × Z /pq (cid:9) the map f H is D -contracting. (cid:3) We now indicate how theideas developed in this section can be used to prove isomorphism results in alldimensions instead of just surjectivity results for K . In [BLR08] the authorsintroduce, for an arbitrary G -space X , the additive categories T G ( X ) , O G ( X ) , and D G ( X ) , and establish in [BLR08, Lemma 3.6] a homotopy fibration sequence(54) K ( T G ( X )) −→ K ( O G ( X )) −→ K ( D G ( X )) . The category T G ( X ) is a variant of the category denoted C ( X ) in this section. Thefunctor X (cid:55)−→ K ( D G ( X )) is a G -equivariant homology theory on G -CW complexes [BFJR04, Section 5], and the value at G/H is π ∗ -isomorphic to Σ K ( R [ H ]) [BFJR04,Section 6]. Therefore the general principles in [WW95] and [DL98] identify the map K ( D G ( EG ( F ))) −→ K ( D G (pt)) with the (suspended) assembly map asbl F .A variant of the category O G ( X ) can be defined as follows. Objects are G -equivariant maps ϕ : S −→ X × [1 , ∞ ) , where now the free G -set S is allowed to becocountable instead of only cofinite. Moreover we require that ϕ − ( X × [1 , N ]) iscofinite for every N .A morphism α from ϕ to ϕ (cid:48) is again an R [ G ] -linear map α : R [ S ] −→ R [ S (cid:48) ] , butnow there is a severe restriction on the support of a morphism: towards ∞ thearrows representing non-vanishing components must become smaller and smaller.Notice though that X is only a topological and not a metric space, and “small”has no immediate meaning. We refer to [BFJR04, Definition 2.7] for the precisedefinition of this condition, which is known as equivariant continous control atinfinity . Figure 3.
A morphism in the obstruction category O G ( X ) .The following result explains the choice of notation: the category O G ( X ) is the obstruction category . Theorem 55.
The assembly map EG ( F ) + ∧ Or G K ( R [ G ∫ − ]) −→ K ( R [ G ]) is a π ∗ -isomorphism if and only if K ∗ ( O G ( EG ( F ))) = 0 .Proof. The map EG ( F ) −→ pt and the homotopy fibration sequence (54) inducethe following commutative diagram with exact rows. · · · K n ( O G ( EG ( F ))) K n ( D G ( EG ( F ))) K n − ( T G ( EG ( F ))) · · ·· · · K n ( O G (pt)) = 0 K n ( D G (pt)) K n − ( T G (pt)) · · · (cid:195)(cid:193) (cid:192) ∼ = (cid:194) ∼ = The map (cid:192) is an isomorphism, because source and target are both isomorphic to K n − ( R [ G ]) via the forgetful map (35). Using the shift map [1 , ∞ ) −→ [1 , ∞ ) , x (cid:55)−→ x + 1 , it is not difficult to prove that O G (pt) admits an Eilenberg swindle,and so K ∗ ( O G (pt)) = 0 . Therefore also the map (cid:194) is an isomorphism. Since themap (cid:193) is identified with the assembly map, the result follows. (cid:3) -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 29 Remark 56 (Proof of Theorem 37) . Consider the ladder diagram in the previousproof, but replace EG ( F ) with a simplicial complex Z . The maps (cid:192) and (cid:194) arestill isomorphisms. The maps (cid:193) and (cid:195) for n = 2 are both models for the assemblymap asbl Z in Theorem 37. Exactness implies that [ α ] ∈ K ( T G ( Z )) is in the imageof the assembly map if it maps to ∈ K ( O G ( Z )) . The statement of Theorem 37 isnow a special case of [BL12a, Theorem 5.3(i)].With some additional work, the program carried out above to decompose anarbitrary K -element into summands with sufficiently small representatives can begeneralized to show that the K -theory of the obstruction category in Theorem 55vanishes. This leads to the following theorem, which is the main result of [BL12b]. Theorem 57 (Farrell-Hsiang Criterion) . Let F be a family of subgroups of G . Fixa word metric on G . Assume that there exists an N > such that for any arbitrarilylarge D > there exists a surjective homomorphism pr D : G −→ F to a finitegroup F with the following property. For each H ∈ H pr D = (cid:8) pr − D ( E ) (cid:12)(cid:12) hyperelementary E ≤ F (cid:9) there exist:(i) an H -simplicial complex Z H of dimension at most N and whose isotropygroups are all contained in F ;(ii) a map f H : res H G −→ Z H that is H -equivariant and D -contracting, i.e., d ( f H ( g ) , f H ( g (cid:48) )) ≤ D d G ( g, g (cid:48) ) for all g, g (cid:48) ∈ G .Then the assembly map EG ( F ) + ∧ Or G K ( R [ G ∫ − ]) −→ K ( R [ G ]) is a π ∗ -isomorphism. Trace methods
Trace maps are maps from algebraic K -theory to other theories like Hochschildhomology, topological Hochschild homology, and their variants, which are usuallyeasier to compute than K -theory. These trace maps have been used successfullyto prove injectivity results about assembly maps in algebraic K -theory. In fact,the most sophisticated trace invariant, topological cyclic homology, was inventedby Bökstedt, Hsiang, and Madsen specifically to attack the rational injectivity ofthe classical assembly map for K ( Z [ G ]) , as explained in Subsection 5.2 below. Injoint work with Lück and Rognes, we applied similar techniques to the Farrell-Jonesassembly map, and in particular we obtained the following partial verification ofConjecture 10; see [LRRV17a, Theorem 1.1]. Theorem 58.
Assume that, for every finite cyclic subgroup C of a group G , thefirst and second integral group homology H ( BZ G C ; Z ) and H ( BZ G C ; Z ) of thecentralizer Z G C of C in G are finitely generated abelian groups. Then G satisfiesConjecture 10, i.e., the map colim H ∈ obj Sub G ( F in ) Wh ( H ) ⊗ Z Q −→ Wh ( G ) ⊗ Z Q is injective. In this section we want to explain the ideas and the structure of the proofs ofBökstedt-Hsiang-Madsen’s Theorem 66 and its generalization, suppressing some ofthe technical details. We first consider a K -analog of Theorem 58 and explain infull detail its proof, which is an illuminating example of the trace methods. Let k be any field of characteristic zero. Then for any group G the map colim H ∈ obj Sub G ( F in ) K ( k [ H ]) ⊗ Z Q −→ K ( k [ G ]) ⊗ Z Q is injective. This is closely related to Conjecture 4 for R = k , but observe that, even though K ( k [ H ]) is a finitely generated free abelian group for each finite group H , thecolimit in the source of the map in Conjecture 4 may contain torsion [KM91].Therefore Proposition 59 does not imply the injectivity of the map in Conjecture 4.The key ingredient in the proof of Proposition 59 is the trace map tr : K ( R ) −→ R/ [ R, R ] , where [ R, R ] denotes the subgroup of the additive group of R generated by commu-tators. The trace map is defined as follows. The projection R −→ R/ [ R, R ] extendsto a map tr : M n ( R ) −→ R/ [ R, R ] , a = ( a ji ) (cid:55)−→ tr( A ) = n (cid:88) i =1 [ a ii ] , which is easily seen to be the universal additive map out of M n ( R ) with the traceproperty: tr( ab ) = tr( ba ) . If p is an idempotent matrix in M n ( R ) , then tr( p ) only depends on the isomorphism class of the projective R -module R n p . Since thetrace sends the block sum of matrices to the sum of the traces, it induces a grouphomomorphism(60) tr : K ( R ) −→ R/ [ R, R ] , [( p ji )] (cid:55)−→ (cid:88) i [ p ii ] . Now consider the case of group algebras. We denote by conj G the set of conjugacyclasses of elements of G . The map R [ G ] −→ R [conj G ] induced by the projectionsends (cid:2) R [ G ] , R [ G ] (cid:3) to zero, and it induces an isomorphism R [ G ] / (cid:2) R [ G ] , R [ G ] (cid:3) ∼ = R [conj G ] . The composition of the trace map tr from (60) with this isomorphism gives a map tr : K ( R [ G ]) −→ R [conj G ] , which is known as the Hattori-Stallings rank. In the special case of group algebrasof finite groups with coefficients in fields of characteristic zero we have the followingresult. -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 31 Lemma 61.
Suppose that the group G is finite and that R = k is a field ofcharacteristic zero. Let R k ( G ) be the representation ring of G over k , and considerthe map χ : R k ( G ) −→ k [conj G ] , ρ (cid:55)−→ (cid:0) χ ρ : g (cid:55)→ tr k ( ρ ( g )) (cid:1) that sends each representation to its character. Then there is a commutative diagram K ( k [ G ]) k [conj G ] [ g ] R k ( G ) k [conj G ] (cid:0) Z G (cid:104) g (cid:105) (cid:1) [ g − ] tr ∼ = ∼ = χ whose vertical maps are isomorphisms. In other words, the Hattori-Stallings rank can be identified up to isomorphismwith the character map χ . Notice, though, that unlike χ the Hattori-Stallings rankis natural in G . Proof of Lemma 61.
Since G is finite and k has characteristic zero, a finitely gener-ated projective k [ G ] -module V is the same as a finite-dimensional k -vector space V equipped with a linear G -action ρ : G −→ GL ( V ) . This explains the left ver-tical isomorphism in the diagram above. It is well known that every irreduciblerepresentation is contained as a direct summand in the regular representation k [ G ] .Therefore we can assume that the idempotent p = p = (cid:80) k ∈ G p k k lies in k [ G ] .Let (cid:104)− , −(cid:105) be the k -bilinear form on k [ G ] that is determined on group elements by (cid:104) g, h (cid:105) = δ gh . Then χ ρ ( g ) = tr k ( k [ G ] p → k [ G ] p, x (cid:55)→ gx ) = tr k ( k [ G ] → k [ G ] , x (cid:55)→ gxp ) == (cid:88) h ∈ G (cid:104) h, ghp (cid:105) = (cid:88) h ∈ G (cid:88) k ∈ G p k (cid:104) h, ghk (cid:105) = (cid:88) h ∈ G p h − g − h = (cid:88) x ∈ [ g − ] (cid:0) Z G (cid:104) g − (cid:105) (cid:1) p x . For the last equality observe that the stabilizer of g ∈ G under the action of G onitself via conjugation is the centralizer Z G (cid:104) g (cid:105) . For the Hattori-Stallings rank wehave tr( p )([ g ]) = (cid:80) x ∈ [ g ] p x . (cid:3) We are now ready to prove Proposition 59.
Proof of Proposition 59.
It suffices to prove the injectivity of the map in Proposi-tion 59 with − ⊗ Z Q replaced by − ⊗ Z k . We explain the proof in the case k = C .Consider the following commutative diagram. colim H ∈ obj Sub G ( F in ) K ( C [ H ]) ⊗ Z C K ( C [ G ]) ⊗ Z C colim H ∈ obj Sub G ( F in ) C [conj H ] C [conj G ] C (cid:104) colim H ∈ obj Sub G ( F in ) conj H (cid:105) ∼ = (cid:192) ∼ = (cid:193) (cid:194) The vertical maps are induced by the C -linear extension of the Hattori-Stallingsrank. For each finite group H this extension is an isomorphism by Lemma 61 and [Ser78, Corollary 1 in §12.4], and so the map (cid:192) is an isomorphism. The map (cid:193) is an isomorphism because the functor C [ − ] is left adjoint and hence preservescolimits. Since conjugation with elements in G represents morphisms in Sub G ( F in ) ,the map (cid:194) is easily seen to be injective already before applying C [ − ] .The proof for an arbitrary field k of characteristic zero is completely analogous,but the set conj G needs to be replaced by the set conj k G of k -conjugacy classes, acertain quotient of conj G . (cid:3) Notice that for each finite group H the Hattori-Stallings rank itself (before k -linear extension) is always injective. But we cannot leverage this fact to proveintegral injectivity results because colimits need not preserve injectivity. The map tr in (60) is just the first(or rather the zeroth) and the easiest trace invariant of the algebraic K -theory of R .We now briefly overview how it can be generalized, starting with the Dennis tracewith values in Hochschild homology.Consider the simplicial abelian group(62) R ⊗ R ⊗ R R ⊗ R R · · · whose face maps are d i ( r ⊗ · · · ⊗ r n ) = (cid:40) r ⊗ · · · ⊗ r i r i +1 ⊗ · · · ⊗ r n if i < n ; r n r ⊗ r ⊗ · · · ⊗ r n − if i = n .The geometric realization of the simplicial abelian group (62) is the zeroth space ofan Ω -spectrum denoted HH ( R ) = HH ( R | Z ) , whose homotopy groups HH ∗ ( R ) = π ∗ HH ( R ) are the Hochschild homology groups of R .In particular, we see that HH ( R ) is the cokernel of the map r ⊗ s (cid:55)−→ rs − sr , andhence HH ( R ) ∼ = R/ [ R, R ] . The trace map tr : K ( R ) −→ HH ( R ) in (60) lifts to a map of spectra trd : K ≥ ( R ) −→ HH ( R ) called the Dennis trace , such that π trd = tr . We use K ≥ to denote connectivealgebraic K -theory, the ( − -connected cover of the functor K we used throughout.Following ideas of Goodwillie and Waldhausen, Bökstedt [Bök86] introduced afar-reaching generalization of HH ( R ) , called topological Hochschild homology anddenoted THH ( R ) . We omit the technical details of the definitions, and we ratherexplain the underlying ideas and structures.The key idea in the definition of topological Hochschild homology is to passfrom the ring R to its Eilenberg-Mac Lane ring spectrum H R , and to replace thetensor products (over the initial ring Z ) with smash products (over the initial ringspectrum S ). In order to make this precise, one needs to work within a symmetricmonoidal model category of spectra (e.g., symmetric spectra), or with ad hoc point-set level constructions (as Bökstedt did, long before symmetric spectra and thelike were discovered). Once these technical difficulties are overcome, one obtains asimplicial spectrum -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 33 (63) H R ∧ H R ∧ H R H R ∧ H R H R , · · · whose geometric realization is
THH ( R ) = HH ( H R | S ) . Notice that of course thisdefinition applies not only to Eilenberg-Mac Lane ring spectra H R but to arbitraryring spectra A .Bökstedt also lifted the Dennis trace to topological Hochschild homology for anyconnective ring spectrum A : THH ( A ) K ≥ ( A ) HH ( π A ) . trdtrb Cyclic permutation of the tensor factors in (62) or smash factors in (63) makesthose simplicial objects into cyclic objects, thus inducing a natural S -action on theirgeometric realizations; see for example [Jon87, Section 3] and [Dri04]. Bökstedt,Hsiang, and Madsen [BHM93] discovered that topological Hochschild homology haseven more structure, which Hochschild homology lacks. Fix a prime p . As n varies,the fixed points of the induced C p n -actions are related by maps(64) THH ( A ) C pn THH ( A ) C pn − , RF called Restriction and Frobenius. The map F is simply the inclusion of fixed points,whereas the definition of the map R is much more delicate and specific to theconstruction of THH . The homotopy equalizer of (64) is denoted TC n +1 ( A ; p ) .One important property of the maps R and F is that they commute, and thereforethey induce a map TC n +1 ( A ; p ) −→ TC n ( A ; p ) . The topological cyclic homology of A at the prime p is then defined as the homotopy limit TC ( A ; p ) = holim n TC n ( A ; p ) . Bökstedt, Hsiang, and Madsen lifted the Bökstedt trace to topological cyclichomology, thus obtaining the following commutative diagram for any connectivering spectrum A :(65) TC ( A ; p ) THH ( A ) K ≥ ( A ) HH ( π A ) . trdtrbtrc The map trc is called the cyclotomic trace map .They then used this technology to prove the following striking theorem, whichis often referred to as the algebraic K -theory Novikov Conjecture; see [BHM93,Theorem 9.13] and [Mad94, Theorem 4.5.4]. Theorem 66 (Bökstedt-Hsiang-Madsen) . Let G be a group. Assume that thefollowing condition holds. [ A ] For every s ≥ , the integral group homology H s ( BG ; Z ) is a finitely generatedabelian group.Then the classical assembly map asbl : BG + ∧ K ( Z ) −→ K ( Z [ G ]) is π Q ∗ -injective, i.e., π n (asbl ) ⊗ Z Q is injective for all n ∈ Z . We now explain the structure of the proof of Theorem 66, following the approachof [LRRV17a]. As mentioned above, the idea is to use the cyclotomic trace map.However, it is not enough to work with topological cyclic homology, and one needsa variant of it that we proceed to explain. Instead of taking the homotopy equalizerof R and F in (64), we may consider just the homotopy fiber of R and define C n +1 ( A ; p ) = hofib (cid:16) THH ( A ) C pn R −→ THH ( A ) C pn − (cid:17) . The map F induces a map C n +1 ( A ; p ) −→ C n ( A ; p ) , and we define C ( A ; p ) = holim n C n ( A ; p ) . A fundamental property, also established in [BHM93], is that C n +1 ( A ; p ) can beidentified with THH ( A ) hC pn , up to a zigzag of π ∗ -isomorphisms. In [LRRV17a,Section 8] we provided a natural zigzag of π ∗ -isomorphisms between THH ( A ) hC pn and C n +1 ( A ; p ) , natural even before passing to the stable homotopy category ofspectra. The key tool here is the natural Adams isomorphism for equivariantorthogonal spectra developed in [RV16].In the special case when A = S [ G ] is a spherical group ring, then the maps R split, and these splittings can be used to construct a map(67) TC ( S [ G ]; p ) −→ C ( S [ G ]; p ) . The crucial advantage of using C instead of TC is that more general rationalinjectivity statements can be proved for the assembly maps for C ; compare Remark 73below.In order to prove Theorem 66 one studies the following commutative diagram.(68) BG + ∧ K ( Z ) K ( Z [ G ]) BG + ∧ K ≥ ( Z ) K ≥ ( Z [ G ]) BG + ∧ K ≥ ( S ) K ≥ ( S [ G ]) BG + ∧ TC ( S ; p ) TC ( S [ G ]; p ) BG + ∧ (cid:0) THH ( S ) × C ( S ; p ) (cid:1) THH ( S [ G ]) × C ( S [ G ]; p ) asbl (cid:192) (cid:202)(cid:193)(cid:194) (cid:203)(cid:204)(cid:195) (cid:205)(cid:196) -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 35 The horizontal maps are all classical assembly maps, and we want to prove that theone at the top of the diagram is π Q ∗ -injective. The maps (cid:192) and (cid:202) are induced bythe natural maps from connective to non-connective algebraic K -theory. Since Z isregular, (cid:192) is a π ∗ -isomorphism. The maps (cid:193) and (cid:203) come from the linearization(or Hurewicz) map S −→ Z , and they are both π Q ∗ -isomorphisms by a result ofWaldhausen [Wal78b, Proposition 2.2]. The maps (cid:194) and (cid:204) are given by thecyclotomic trace map, and (cid:195) and (cid:205) by the natural maps in (65) and (67).So, in order to prove that the top horizontal map in diagram (68) is π Q ∗ -injective,it is enough to show that:(a) The assembly map (cid:196) is π Q ∗ -injective.(b) The composition (cid:195) ◦ (cid:194) is π Q ∗ -injective.The assumption [ A ] is then shown to imply (a), and in fact not just for S but forarbitrary connective ring spectra A . This is the special case F = 1 of Theorems 71and 72 below. The difficult part in proving (b) is the analysis of the map (cid:194) . TheAtiyah-Hirzebruch spectral sequences collapse rationally, and therefore it is enoughto study the rational injectivity of trc : K ≥ ( S ) −→ TC ( S ; p ) . To this end, considerthe following commutative diagram. K ≥ ( Z p ) K ≥ ( S ) K ≥ ( Z ) K ≥ ( Z p ) ∧ p K ≥ ( Z ) ∧ p TC ( S ; p ) TC ( Z p ; p ) ∧ p (cid:198)(cid:193) trc (cid:197)(cid:199) trc ∧ p (cid:200) Here ( − ) ∧ p denotes the p -completion of spectra and Z p are the p -adic numbers. Themap trc ∧ p is a π n -isomorphism for each n ≥ by a result of Hesselholt and Mad-sen [HM97, Theorem D]. We already mentioned above that (cid:193) is a π Q ∗ -isomorphism.It remains to discuss the diamond. Since the groups K n ( Z ) are known to befinitely generated, (cid:199) is π Q ∗ -injective. The question whether (cid:200) is π Q ∗ -injective is openin general. It can be reformulated in terms of similar maps in étale K -theory, étalecohomology, or Galois cohomology, as surveyed in [LRRV17a, Section 18]. Luckilythe equivalent conjecture in Galois cohomology is known to be true if p is a regularprime by results in [Sch79]; see [LRRV17a, Proposition 2.9]. Recall that a prime p is regular if it does not divide the order of the ideal class group of Q ( ζ p ) . Sinceregular primes exist we obtain the following statement and we are done. [ B ] There exists a prime p such that (cid:200) ◦ (cid:199) is π Q ∗ -injective.We remark that little is known about the rationalized homotopy groups of K ≥ ( Z p ) without p -completion; compare [Wei05, Warning 60].This concludes our explanation of the proof of Theorem 66. The following result generalizes Theorem 66 from the clas-sical to the Farrell-Jones assembly map, and is a special case of [LRRV17a, MainTechnical Theorem 1.16].
Theorem 69.
Let G be a group and let F ⊆ FC yc be a family of finite cyclicsubgroups of G . Assume that the following two conditions hold. [ A F ] For every C ∈ F and every s ≥ , the integral group homology H s ( BZ G C ; Z ) of the centralizer of C in G is a finitely generated abelian group. [ B F ] For every C ∈ F and every t ≥ , the natural homomorphism K t ( Z [ ζ c ]) ⊗ Z Q −→ (cid:81) p prime K t (cid:16) Z p ⊗ Z Z [ ζ c ]; Z p (cid:17) ⊗ Z Q is injective, where c is the order of C , ζ c is any primitive c -th root of unity,and K t ( R ; Z p ) = π t ( K ( R ) ∧ p ) .Then the assembly map asbl F : EG ( F ) + ∧ Or G K ≥ ( Z [ G ∫ − ]) −→ K ≥ ( Z [ G ]) is π Q ∗ -injective. Remark 70.
Several comments are in order.(i) When F = 1 is the trivial family, Theorems 66 and 69 coincide. This is becauseassumption [ A ] of Theorem 69 is literally the same as assumption [ A ] ofTheorem 66, and assumption [ B ] follows at once from the corresponding truestatement explained at the end of the previous subsection.(ii) When F = FC yc , then the rationalized assembly map for connective algebraic K -theory studied in Theorem 69 can be rewritten as in Conjecture 23, becausethe isomorphisms (20) and (21) hold for both connective and non-connectivealgebraic K -theory. The only difference is that the summands indexed by t = − in the source of the map in Conjecture 23 are now missing. Notice thatthe negative K -groups K t ( Z [ C ]) are known to vanish for any t < − if C isfinite or even virtually cyclic [FJ95].(iii) As noted above, assumption [ A F ] implies and is the obvious generalization ofassumption [ A ] . For any F ⊆ FC yc , assumption [ A F ] is satisfied if there is auniversal space EG ( F in ) of finite type, i.e., whose skeleta are all cocompact.Hyperbolic groups, finite-dimensional CAT(0)-groups, cocompact lattices invirtually connected Lie groups, arithmetic groups in semisimple connected lin-ear Q -algebraic groups, mapping class groups, and outer automorphism groupsof free groups are all examples of groups that even have a finite-dimensionaland cocompact EG ( F in ) . Among these groups, outer automorphism groupsof free groups do not appear in Theorem 27, and for them Theorem 69 givesthe first result about the Farrell-Jones Conjecture. An interesting exampleof a group that satisfies [ A FC yc ] without having an EG ( F in ) of finite typeis given by Thompson’s group T of orientation preserving, piecewise linear,dyadic homeomorphisms of the circle; see [GV17].(iv) Conjecturally assumption [ B F ] of Theorem 69 is always satisfied; in fact, it isimplied by a weak version of the Leopoldt-Schneider Conjecture for cyclotomicfields, as explained carefully in [LRRV17a, Sections 2 and 18]. When t = 0 or t = 1 , i.e., for K and K , the map in [ B F ] is injective for arbitrary c bydirect computation; compare [LRRV17a, Proposition 2.4]. For any fixed c itis known that injectivity may fail for at most finitely many values of t . Thesetwo facts allow to deduce Theorems 58 and 28(i) from Theorem 69, or ratherfrom its more general version in [LRRV17a, Main Technical Theorem 1.16],as explained in loc. cit., Section 17 and page 1015. Notice that, on the other -THEORY, ASSEMBLY MAPS, CONTROLLED ALGEBRA, AND TRACE METHODS 37 hand, Theorem 66 cannot be used to deduce information about the Whiteheadgroup Wh ( G ) , which is the cokernel of the map induced on π by the classicalassembly map asbl .The proof of Theorem 69 follows the same strategy as the proof of Theorem 66outlined above. We consider the analog of diagram (68) for the generalized as-sembly map asbl F ; compare [LRRV17a, “main diagram” (3.1)]. The key resultsabout assembly maps are summarized in the following two theorems [LRRV17a,Theorem 1.19, parts (i) and (ii)]. We point out that all the following results holdfor arbitrary connective ring spectra A . Theorem 71.
For any group G and for any family F of subgroups of G , theassembly map asbl F : EG ( F ) + ∧ Or G THH ( A [ G ∫ − ]) −→ THH ( A [ G ]) induces split monomorphisms on π ∗ , and it is a π ∗ -isomorphism if and only if F contains all cyclic subgroups of G , i.e., F ⊇ C yc . Theorem 72.
Let G be a group and let F ⊆ FC yc be a family of finite cyclicsubgroups of G . Assume that the following condition holds. [ A F ] For every C ∈ F and every s ≥ , the integral group homology H s ( BZ G C ; Z ) of the centralizer of C in G is a finitely generated abelian group.Then the assembly map asbl F : EG ( F ) + ∧ Or G C ( A [ G ∫ − ; p ]) −→ C ( A [ G ]; p ) is π Q ∗ -injective. Remark 73.
In order to establish an analog of Theorem 72 for the assembly map(74) asbl F : EG ( F ) + ∧ Or G TC ( A [ G ∫ − ; p ]) −→ TC ( A [ G ]; p ) in topological cyclic homology, we need to assume not only condition [ A F ] , but alsothe following two conditions: [ A (cid:48)F ] the family F contains only finitely many conjugacy classes of subgroups; [ A (cid:48)(cid:48)F ] for every g ∈ G , (cid:104) g (cid:105) ∈ F if and only if (cid:104) g p (cid:105) ∈ F .The fact that the assembly map (74) is π Q ∗ -injective under assumptions [ A F ] , [ A (cid:48)F ] ,and [ A (cid:48)(cid:48)F ] is a special case of [LRRV17b, Theorem 1.8]. Notice the following facts.(i) When F = 1 , assumption [ A (cid:48) ] is vacuously true, but [ A (cid:48)(cid:48) ] is not satisfied if G has p -torsion. This is the reason why, in the proof of Bökstedt-Hsiang-Madsen’s Theorem 66, we need to work with C and not just TC .(ii) As pointed out in Remark 70(iii), Thompson’s group T satisfies [ A FC yc ] andobviously also [ A (cid:48)(cid:48)FC yc ] . However, T contains finite cyclic subgroups of anygiven order, and therefore does not satisfy [ A (cid:48)FC yc ] . It is an interesting openquestion whether the assembly map (74) is π Q ∗ -injective for G = T .(iii) Without homological finiteness assumptions on G , the assembly map (74) isnot rationally injective in general. For example, if G = Q and F = 1 = FC yc ,then (74) is essentially trivial after applying π ∗ ( − ) ⊗ Z Q . This is explainedin [LRRV17a, Remark 3.7]. Of course, the group G = Q does not satisfy [ A ] . Finally, we mention the following two additional results about assembly maps fortopological cyclic homology, which we proved in [LRRV17b, Theorems 1.1, 1.4(ii),and 1.5].One should view Theorem 75 as a cyclic induction theorem for the topologicalcyclic homology of any finite group, with coefficients in any connective ring spectrum.It allows to reduce the computation of TC of any finite group to the case of thefinite cyclic subgroups; this is carried out explicitly in [LRRV17b, Proposition 1.2]for the basic case of the symmetric group on three elements.Theorem 76 studies the analog for TC of the Farrell-Jones Conjecture 15. For alarge class of groups (for which Conjecture 15 is already known; see Theorem 27),we prove that asbl VC yc is injective, but surprisingly not surjective. Theorem 75.
For any finite group G the assembly map asbl C yc : EG ( C yc ) + ∧ Or G TC ( A [ G ∫ − ]; p ) −→ TC ( A [ G ]; p ) is a π ∗ -isomorphism. Theorem 76.
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E-mail address : [email protected] URL : mi.fu-berlin.de/math/groups/top/members/Professoren/reich.html Department of Mathematics and Statistics, University at Albany, SUNY, USA
E-mail address : [email protected] URL : albany.edu/~mv312143/albany.edu/~mv312143/