aa r X i v : . [ m a t h . K T ] J a n ASSEMBLY MAPS
WOLFGANG L ¨UCK
Abstract.
We introduce and analyze the concept of an assembly map fromthe original homotopy theoretic point of view. We give also interpretations interms of surgery theory, controlled topology and index theory. The motivationis that prominent conjectures of Farrell-Jones and Baum-Connes about K - and L -theory of group rings and group C ∗ -algebras predict that certain assemblymaps are weak homotopy equivalences. Introduction
The homotopy theoretic description of assembly maps.
The quickestand probably for a homotopy theorist most convenient approach to assembly mapsis via homotopy colimits as explained in Subsection 6.3. Let F be a family of sub-groups of G , i.e., a collection of subgroups closed under conjugation and passingto subgroups. Let Or ( G ) be the orbit category and Or F ( G ) be the full subcat-egory consisting of objects G/H satisfying H ∈ F . Consider a covariant functor E G : Or ( G ) → Spectra to the category of spectra. We get from the inclusion Or F ( G ) → Or ( G ) and the fact that G/G is a terminal object in Or ( G ) a map(0.1) hocolim Or F ( G ) E G | Or F ( G ) → hocolim Or ( G ) E G = E G ( G/G ) . It is called assembly map since we are trying to assemble the values of E G onhomogeneous spaces G/H for H ∈ F to get E ( G/G ).On homotopy groups this assembly map can also be described as the map(0.2) H Gn (pr; E G ) : H Gn ( E F ( G ); E G ) → H Gn ( G/G ; E G ) = π n ( E G ( G/G ))induced by the projection pr : E F ( G ) → G/G of the classifying G -space E F ( G ) forthe family F , see Section 5, to G/G , where H Gn ( − ; E G ) is the G -homology theoryin the sense of Definition 2.1 associated to E G , see Lemma 2.5.In all interesting situations one can take a global point of view. Namely, onestarts with a covariant functor respecting equivalences E : Groupoids → Spectra and defines for a group G the functor E G to be the composite of E with the functor Or ( G ) → Groupoids given by the transport groupoid of a G -set, see Subsection 6.4.0.2. Isomorphism Conjectures.
The Meta Isomorphism Conjecture for G , F and E G , see Section 6, says that the assembly map of (0.1) is a weak homotopyequivalence, or, equivalently, that the map (0.2) is bijective for all n ∈ Z .If we take for E an appropriate functor modelling the algebraic K -theory orthe algebraic L -theory with decoration h−∞i of the group ring RG and for F thefamily of virtually cyclic subgroups, we obtain the Farrell-Jones Conjecture 7.3. Itassembles K n ( RG ) and L h−∞i n ( RG ) in terms of K n ( RH ) and L h−∞i n ( RH ), where H runs through the virtually cyclic subgroups of G . Date : January, 2019.2010
Mathematics Subject Classification. primary: 18F25, secondary: 46L80, 55P91, 57N99.
Key words and phrases. assembly maps, equivariant homology and homotopy theory, Farrell-Jones Conjecture, Baum-Connes Conjecture.
If we take for E an appropriate functor modelling the topological K -theory ofthe reduced group C ∗ -algebra C ∗ r ( G ) and for F the family of finite subgroups, weobtain the Baum-Connes Conjecture 8.2. It assembles K top n ( C ∗ r ( G )) in terms of K top n ( C ∗ r ( H )), where H runs through the finite subgroups of G .The Farrell-Jones Conjecture 7.3 and the Baum-Connes Conjecture 8.2 are verypowerful conjectures and are the main motivation for the study of assembly maps. Asurvey of a lot of striking applications such as the ones to the conjectures of Bass,Borel, Gromov-Lawson-Rosenberg, Kadison, Kaplansky, and Novikov is given inSubsections 7.2 and 8.4. The Farrell-Jones Conjecture 7.3 and the Baum-ConnesConjecture 8.2 are known to be true for a surprisingly large class of groups, asexplained in Subsections 7.7 and 8.5. All of this is an impressive example howhomotopy theoretic methods can be used for problems in other fields such as algebra,geometry, manifold theory and operator algebras.0.3. Other interpretations of assembly maps.
The homotopy theoretic ap-proach is the best for structural purposes. The applications and the proofs of theFarrell-Jones Conjecture 7.3 and the Baum-Connes Conjecture 8.2 require sophisti-cated analytic, topological and geometric interpretations of the homotopy theoreticassembly maps, for instance in terms of surgery theory, see Subsection 7.3, as forgetcontrol maps, see Subsection 7.4, and in terms of index theory, see Subsection 8.3.This presents an intriguing interaction between homotopy theory, geometry andoperator theory.0.4.
The universal property of the assembly map.
In Section 4 we character-ize the assembly map in the sense that it is the universal approximation from the leftby an excisive functor of a given homotopy invariant functor G - CW → Spectra .This is the key ingredient in the difficult identification of the various assembly mapsmentioned in Subsection 0.3 above. It reflects the fact that in all of the Isomor-phism Conjectures the hard and interesting object is the target and the source isgiven by the G -homology of the classifying G -spaces for a specific family of sub-groups. The source is more accessible than the target since one can apply standardmethods from algebraic topology such as spectral sequences and equivariant Cherncharacters.0.5. Relative assembly maps.
Relative assembly maps are studied in Section 11.They address the problem to make the families appearing in the various Isomor-phism Conjectures as small as possible.0.6.
Further aspects of assembly maps.
The homotopy theoretic approach toassembly allows to relate assembly maps for various theories, such as the algebraic K -theory and A -theory via linearization, see Subsection 7.6, algebraic K -theory ofgroups rings and topological K -theory of reduced group C ∗ -algebras, see Subsec-tion 8.6, and algebraic K -theory of groups rings and the topological cyclic homologyof the spherical group ring via cyclotomic traces, see Subsection 9.3. How assemblymaps can be used for computations is illustrated in Section 12 which is based onthe global point of view described in Section 10. Finally we formulate the challengeof extending equivariant homotopy theory for finite groups to infinite groups inSection 13.The idea of the geometric assembly map is due to Quinn [94, 95] and its algebraiccounterpart was introduced by Ranicki [97].0.7. Conventions.
Throughout this paper G denotes a (discrete) group. Ringmeans associative ring with unit. All spectra are non-connective. SSEMBLY MAPS 3
Acknowledgments.
This paper is dedicated to Andrew Ranicki. It is finan-cially supported by the ERC Advanced Grant “KL2MG-interactions” (no. 662400)of the author granted by the European Research Council, and by the Cluster ofExcellence “Hausdorff Center for Mathematics” at Bonn.The paper is organized as follows:
Contents
0. Introduction 10.1. The homotopy theoretic description of assembly maps 10.2. Isomorphism Conjectures 10.3. Other interpretations of assembly maps 20.4. The universal property of the assembly map 20.5. Relative assembly maps 20.6. Further aspects of assembly maps 20.7. Conventions 20.8. Acknowledgments 31. Some basic categories 41.1. G - CW -complexes 41.2. The orbit category 41.3. Spectra 42. G -homology theories and Or ( G )-spectra 53. Approximation by an excisive functor 74. The universal property 85. Classifying spaces for families of subgroups 96. The Meta-Isomorphism Conjecture 96.1. The Meta-Isomorphism Conjecture for G -homology theories 106.2. The Meta-Isomorphism Conjecture on the level of spectra 106.3. The assembly map in terms of homotopy colimits 106.4. Spectra over Groupoids K -and L -theory 117.2. Applications of the Farrell-Jones Conjecture 127.3. The interpretation of the Farrell-Jones assembly map for L -theory interms of surgery theory 137.4. The interpretation of the Farrell-Jones assembly map in terms ofcontrolled topology 157.5. The Farrell-Jones Conjecture for Waldhausen’s A -theory 167.6. Relating the assembly maps for K -theory and for A -theory 167.7. The status of the Farrell-Jones Conjecture 178. The Baum-Connes Conjecture 188.1. The Baum-Connes Conjecture 188.2. The Baum-Connes Conjecture with coefficients 188.3. The interpretation of the Baum Connes assembly map in terms ofindex theory 198.4. Applications of the Baum-Connes Conjecture 198.5. The status of the Baum-Connes Conjecture 218.6. Relating the assembly maps of Farrell-Jones to the one of Baum-Connes 219. Topological cyclic homology 239.1. Topological Hochschild homology 239.2. Topological cyclic homology 239.3. Relating the assembly maps of Farrell-Jones to the one for topologicalcyclic homology via the cyclotomic trace 24 WOLFGANG L¨UCK
10. The global point of view 2411. Relative assembly maps 2512. Computationally tools 2713. The challenge of extending equivariant homotopy theory to infinitegroups 28References 291.
Some basic categories
In this section we recall some well-known basic categories.1.1. G - CW -complexes.Definition 1.1 ( G - CW -complex) . A G - CW -complex X is a G -space together witha G -invariant filtration ∅ = X − ⊆ X ⊂ X ⊆ . . . ⊆ X n ⊆ . . . ⊆ [ n ≥ X n = X such that X carries the colimit topology with respect to this filtration (i.e., a set C ⊆ X is closed if and only if C ∩ X n is closed in X n for all n ≥
0) and X n isobtained from X n − for each n ≥ n -dimensional cells,i.e., there exists a G -pushout ` i ∈ I n G/H i × S n − ` i ∈ In q ni / / (cid:15) (cid:15) X n − (cid:15) (cid:15) ` i ∈ I n G/H i × D n ` i ∈ In Q ni / / X n A map f : X → Y between G - CW -complexes is called cellular if f ( X n ) ⊆ Y n holds for all n ≥
0. We denote by G - CW the category of G - CW -complexes withcellular G -maps as morphisms and by G - CW the corresponding category of G - CW -pairs. For basic information about G - CW -complexes we refer for instanceto [65, Chapter 1 and 2].1.2. The orbit category.
The orbit category Or ( G ) has as objects homogeneousspaces G/H and as morphisms G -maps. It can be viewed as the category of 0-dimensional G - CW -complexes, whose G -quotient space is connected. In particularwe can think of Or ( G ) as a full subcategory of G - CW .1.3. Spectra.
In this paper we can work with the most elementary category
Spectra of spectra. A spectrum E = { ( E ( n ) , σ ( n )) | n ∈ Z } is a sequence of pointed spaces { E ( n ) | n ∈ Z } together with pointed maps called structure maps σ ( n ) : E ( n ) ∧ S −→ E ( n + 1). A map of spectra f : E → E ′ is a sequence of maps f ( n ) : E ( n ) → E ′ ( n ) which are compatible with the structure maps σ ( n ), i.e., we have f ( n + 1) ◦ σ ( n ) = σ ′ ( n ) ◦ ( f ( n ) ∧ id S ) for all n ∈ Z . Maps of spectra are sometimes calledfunctions in the literature, they should not be confused with the notion of a mapof spectra in the stable category, see [1, III.2.].The homotopy groups of a spectrum are defined by π i ( E ) := colim k →∞ π i + k ( E ( k )) , (1.2)where the i th structure map of the system π i + k ( E ( k )) is given by the composite π i + k ( E ( k )) S −→ π i + k +1 ( E ( k ) ∧ S ) σ ( k ) ∗ −−−→ π i + k +1 ( E ( k + 1)) SSEMBLY MAPS 5 of the suspension homomorphism S and the homomorphism induced by the struc-ture map. A weak equivalence of spectra is a map f : E → F of spectra inducing anisomorphism on all homotopy groups.2. G -homology theories and Or ( G ) -spectra Let Λ be a commutative ring. Next we recall the obvious generalization of thenotion of a (generalized) homology theory to a G -homology theory. Definition 2.1 ( G -homology theory) . A G -homology theory H G ∗ with values in Λ -modules is a collection of covariant functors H Gn from the category G - CW of G - CW -pairs to the category of Λ-modules indexed by n ∈ Z together with naturaltransformations ∂ Gn ( X, A ) : H Gn ( X, A ) → H Gn − ( A ) := H Gn − ( A, ∅ )for n ∈ Z such that the following axioms are satisfied: • G -homotopy invariance If f and f are G -homotopic G -maps of G - CW -pairs ( X, A ) → ( Y, B ),then H Gn ( f ) = H Gn ( f ) for n ∈ Z ; • Long exact sequence of a pair
Given a pair (
X, A ) of G - CW -complexes, there is a long exact sequence . . . H Gn +1 ( j ) −−−−−→ H Gn +1 ( X, A ) ∂ Gn +1 −−−→ H Gn ( A ) H Gn ( i ) −−−−→ H Gn ( X ) H Gn ( j ) −−−−→ H Gn ( X, A ) ∂ Gn −−→ . . . , where i : A → X and j : X → ( X, A ) are the inclusions; • Excision
Let (
X, A ) be a G - CW -pair and let f : A → B be a cellular G -map of G - CW -complexes. Equip ( X ∪ f B, B ) with the induced structure of a G - CW -pair. Then the canonical map ( F, f ) : (
X, A ) → ( X ∪ f B, B ) inducesan isomorphism H Gn ( F, f ) : H Gn ( X, A ) ∼ = −→ H Gn ( X ∪ f B, B ); • Disjoint union axiom
Let { X i | i ∈ I } be a family of G - CW -complexes. Denote by j i : X i → ` i ∈ I X i the canonical inclusion. Then the map M i ∈ I H Gn ( j i ) : M i ∈ I H Gn ( X i ) ∼ = −→ H Gn a i ∈ I X i ! is bijective.If E is a spectrum, then one gets a (non-equivariant) homology theory H ∗ ( − ; E )by defining H n ( X, A ; E ) = π n (cid:0) ( X + ∪ A + cone( A + )) ∧ E (cid:1) for a CW -pair ( X, A ) and n ∈ Z , where X + is obtained from X by adding a disjointbase point and cone denotes the (reduced) mapping cone. Its main property is H n ( {•} ; E ) = π n ( E ). This extends to G -homology theories as follows. Since thebuilding blocks of G -spaces are homogeneous spaces, we will have to consider acovariant Or ( G )-spectrum, i.e., a covariant functor E G : Or ( G ) → Spectra , insteadof a spectrum.
WOLFGANG L¨UCK
Definition 2.2 (Excisive) . We call a covariant functor E : G - CW → Spectra homotopy invariant if it sends G -homotopy equivalences to weak homotopy equiv-alences of spectra.The functor E is excisive if it has the following four properties: • It is homotopy invariant; • The spectrum E ( ∅ ) is weakly contractible; • It respects homotopy pushouts up to weak homotopy equivalence, i.e., ifthe G - CW -complex X is the union of G - CW -subcomplexes X and X with intersection X , then the canonical map from the homotopy pushoutof E ( X ) ←− E ( X ) −→ E ( X ) to E ( X ) is a weak homotopy equivalenceof spectra; • It respects disjoint unions up to weak homotopy, i.e., the natural map W i ∈ I E ( X i ) → E ( ` i ∈ I X i ) is a weak homotopy equivalence for all indexsets I .One easily checks Lemma 2.3.
Suppose that the covariant functor E : G - CW → Spectra is excisive.Then we obtain a G -homology theory with values in Z -modules by assigning to G - CW -pair ( X, A ) and n ∈ Z the abelian group π n ( E ( X, A )) . A G -space X defines a contravariant Or ( G )-space O G ( X ) by sending G/H tomap G ( G/H, X ) = X H . Given a contravariant pointed Or ( G )-space Y and a covari-ant pointed Or ( G )-space Z , there is the pointed space Y ∧ Or ( G ) Z . Its constructionis explained for instance in [25, Section 1]. This construction is natural in Y and Z . Its main property is that one obtains for every pointed space X an adjunctionhomeomorphismmap( Y ∧ Or ( G ) Z, X ) ∼ = −→ mor Or ( G ) ( Y, map( Z, X ))where the source is the pointed mapping space and the target is the topologicalspace of natural transformations from Y to the contravariant pointed Or ( G )-spacemap( Z, X ) sending
G/H to the pointed mapping space map( Z ( G/H ) , X ). If E G is a covariant Or ( G )-spectrum, then one obtains a spectrum Y ∧ Or ( G ) E G . Hencewe can extend a covariant functor E G : Or ( G ) → Spectra to a covariant functor(2.4) ( E G ) % : G - CW → Spectra , ( X, A ) O G ( X + ∪ A + cone( A + )) ∧ Or ( G ) E G . The easy proofs of the following two results are left to the reader.
Lemma 2.5. If E G is a covariant Or ( G ) -spectrum, then ( E G ) % is excisive and weobtain a G -homology theory H G ∗ ( − ; E G ) by H Gn ( X, A ; E G ) = π n (( E G ) % ( X, A )) = π n (cid:0) O G ( X + ∪ A + cone( A + )) ∧ Or ( G ) E G (cid:1) satisfying H Gn ( G/H ; E G ) = π n ( E G ( G/H )) for n ∈ Z and H ⊆ G . Lemma 2.6.
Let t : E → F be a natural transformation of covariant functors G - CW → Spectra . Suppose that E and F are excisive and t ( G/H ) is a weakhomotopy equivalence for any homogeneous G -space G/H .Then t ( X, A ) : E ( X, A ) → F ( X, A ) is a weak homotopy equivalence for every G - CW -pair ( X, A ) . SSEMBLY MAPS 7 Approximation by an excisive functor
The following result follows from [25, Theorem 6.3]. Its non-equivariant versionis due to Weiss-Williams [120].
Theorem 3.1 (Approximation by an excisive functor) . Let E : G - CW → Spectra be a covariant functor which is homotopy invariant. Let E | : Or ( G ) → Spectra beits composite with the obvious inclusion Or ( G ) → G - CW .Then there exists a covariant functor E % : G - CW → Spectra and natural transformations A E : E % → E ; B E : E % → E | % , satisfying:(1) The functor E % is excisive;(2) The map A E ( G/H ) : E % ( G/H ) → E ( G/H ) is a weak homotopy equiva-lence for every homogeneous space G/H ;(3) The map B E ( X, A ) : E % ( X, A ) → E | % ( X, A ) is a weak homotopy equiva-lence for every G - CW -pair ( X, A ) ;(4) The functor E is excisive if and only if A E ( X, A ) is a weak homotopyequivalence for every G - CW -pair ( X, A ) ;(5) The transformations A E and B E are functorial in E . Although one does not need to understand the explicite construction of E % , A E and B E and the proof of Theorem 3.1 for the applications of Theorem 3.1 and forthe reminder of this paper, we make some comments about it for the interestedreader.As an illustration we firstly present a naive suggestion in the non-equivariantcase, which turns out to require too restrictive assumptions on E and therefore willnot be the final solution, but conveys a first idea. Namely, we can define a map ofpointed sets X + ∧ E ( {•} ) → E ( X ) for a CW -complex X by sending an element inthe target represented by ( x, e ) for x ∈ X and e ∈ E ( {•} ) to E ( c x : {•} → X )( e ),where c x : {•} → X is the constant map with value x . The problem is that the onlyreasonable way of ensuring the continuity of this map is to require that E itself iscontinuous, i.e., the map map( X, Y ) → map( E ( X ) n , E ( Y ) n ) sending f to E ( f ) n has to be continuous for all n ∈ Z . But this assumption is not satisfied for thefunctors E which are of interest for us and will be considered below.The solution is to take homotopy invariance into account and to work simplicially.Let us consider the special case, where G is trivial and X is a simplicial complex.For any simplex σ of X we have the inclusion i [ σ ] : σ → X and can therefore definemaps A E [ σ ] n : σ + ∧ E ( σ ) n pr + ∧ id E ( σ ) n −−−−−−−−−→ {•} + ∧ E ( σ ) n = E ( σ ) n E ( i [ σ ]) n −−−−−→ E ( X ) n ; B E [ σ ] n : σ + ∧ E ( σ ) n id σ + ∧ E (pr) n −−−−−−−−−→ σ + ∧ E ( {•} ) n , where pr denotes the projection onto {•} . Now define a space E % ( X ) n by glueingthe spaces σ + ∧ E ( σ ) n for σ running over the simplices of X together accordingto the simplicial structure, more precisely, for an inclusion j : τ → σ of simpliceswe identify a point in τ + ∧ E ( τ ) n with its image in σ + ∧ E ( σ ) n under the obviousmap j + ∧ E ( j ) n . One easily checks that the various maps A E [ σ ] n and B E [ σ ] n fit WOLFGANG L¨UCK together to maps of pointed spaces A E ( X ) n : E % ( X ) n → E ( X ) n ; B E ( X ) n : E % ( X ) n → E | % ( X ) n := X + ∧ E ( {•} ) n , and thus to maps of spectra A E ( X ) : E % ( X ) → E ( X ) n ; B E ( X ) : E % ( X ) → E | % ( X ) := X + ∧ E ( {•} ) . Notice that each map E (pr) : E ( σ ) → E ( {•} ) is by assumption a weak homotopyequivalence. This implies that B E ( X ) : E % ( X ) → E | % ( X ) is a weak homotopyequivalence. Since the functor E % is excisive, the functor E % is excisive. If X = {•} ,the map A E ( {•} ) : E % ( {•} ) → E ( {•} ) is an isomorphism and in particular a weakhomotopy equivalence.Now we see, where the name assembly map comes from. In the case of a simplicialcomplex X we want to assemble E ( X ) by its values E ( σ ) for the various simplices of X , which leads to the definition of E % ( X ). Intuitively it is clear that E ( X ) carriesthe same information as E | % ( X ) if and only if E is excisive since the conditionexcisive allows to compute the values of E on X by its values on the simplicestaking into account how the simplices are glued together to yield X .Finally one wants a definition that is independent of the simplicial structure andactually applies to more general spaces X than simplicial complexes. Therefore oneuses simplicial sets and in particular the singular simplicial set S.X , Recall that
S.X is the functor from the category of finite ordered sets ∆ to the category of sets
Sets sending the finite ordered set [ p ] to the set map(∆ p , Y ) for ∆ p the standard p -simplex. For the equivariant version one has to bring the orbit category into play.So one considers for a G -space X the functor Or ( G ) × ∆ → Sets , ( G/H, [ p ]) map G ( G/H × ∆ p , X ) . In some sense on uses free resolution of contravariant functors Or ( G ) × ∆ → Spaces to get the right construction of E % and of the desired transformations A E and B E so that the claims appearing in Theorem 3.1 can be proved. Details can be foundin [25]. 4. The universal property
Next we explain why Theorem 3.1 characterizes the assembly map in the sensethat A E : E % −→ E is the universal approximation from the left by an excisive func-tor of a homotopy invariant functor E : G - CW → Spectra . Namely, let T : F → E be a transformation of covariant functors G - CW → Spectra such that F is exci-sive. Then for any G - F - CW -pair ( X, A ) the following diagram commutes F % ( X ) A F ( X ) ≃ / / T % ( X ) (cid:15) (cid:15) F ( X ) T ( X ) (cid:15) (cid:15) E % ( X ) A E ( X ) / / E ( X )and A F ( X ) is a weak homotopy equivalence by Theorem 3.1 (4). Hence T ( X )factorizes over A E ( X ) up to natural weak homotopy equivalence.Suppose additionally that T ( G/H ) is a weak homotopy equivalence for everysubgroup H ⊆ G . Then both T % ( X ) and A F ( X ) are weak homotopy equivalencesby Lemma 2.6 and Theorem 3.1 (4), and hence T ( X ) can be identified with A E ( X )up to natural weak homotopy equivalence. SSEMBLY MAPS 9
Recall that there is a natural weak equivalence B E ( X ) : E % ( X ) ≃ −→ E | % ( X ), sothat one may replace in the considerations above E % ( X ) by E | % ( X ), which dependson the values of E on homogeneous spaces only. This universal property will be thekey ingredient for the identification of various versions of assembly maps.5. Classifying spaces for families of subgroups
We recall the notion classifying space for a family which was introduced by tomDieck [108].
Definition 5.1 (Family of subgroups) . A family F of subgroups of a group G isa set of subgroups of G which is closed under conjugation with elements of G andunder passing to subgroups.Our main examples of families are the trivial family T R consisting of the trivialsubgroup, the family
ALL of all subgroups, and the families
FCY , CY , FIN ,and
VCY of finite cyclic subgroups, of cyclic subgroups, of finite subgroups, and ofvirtually cyclic subgroups.
Definition 5.2 (Classifying G -space for a family of subgroups) . Let F be a familyof subgroups of G . A model E F ( G ) for the classifying spaces for the family F ofsubgroups of G is a G - CW -complex E F ( G ) that has the following properties:(1) All isotropy groups of E F ( G ) belong to F ;(2) For any G - CW -complex Y , whose isotropy groups belong to F , there is upto G -homotopy precisely one G -map Y → X .We abbreviate EG := E FIN ( G ) and call it the universal G -space for proper G -actions . We also write EG := E VCY ( G ).Equivalently, E F ( G ) is a terminal object in the G -homotopy category of G - CW -complexes, whose isotropy groups belong to F . In particular two models for E F ( G )are G -homotopy equivalent and for two families F ⊆ F there is up to G -homotopyprecisely one G -map E F ( G ) → E F ( G ). There are functorial constructions for E F ( G ) generalizing the bar construction, see [25, Section 3 and Section 7]. Theorem 5.3 (Homotopy characterization of E F ( G )) . A G - CW -complex X is amodel for E F ( G ) if and only if for every subgroup H ⊆ G its H -fixed point set X H is weakly contractible if H ∈ F , and is empty if H / ∈ F . A model for E ALL ( G ) is G/G . A model for E T R ( G ) is the same as a modelfor EG i.e, the universal covering of BG , or, equivalently, the total space of theuniversal G -principal bundle. There are many interesting geometric models forclassifying spaces EG = E FIN ( G ), e.g., the Rips complex for a hyperbolic group,the Teichm¨uller space for a mapping class group, and so on. The question whetherthere are finite-dimensional models, models of finite type or finite models has beenstudied intensively during the last decades. For more information about classifyingspaces for families we refer for instance to [73].6. The Meta-Isomorphism Conjecture
In this section we formulate the Meta-Isomorphism Conjecture, from which allother Isomorphism Conjectures such as the one due to Farrell-Jones and Baum-Connes are obtained by specifying the parameters E and F . The Meta-Isomorphism Conjecture for G -homology theories. Let H G ∗ be a G -homology theory with values in Λ-modules for some commutative ring Λ.The projection pr : E F ( G ) → G/G induces for all integers n ∈ Z a homomorphismof Λ-modules(6.1) H Gn (pr) : H Gn ( E F ( G )) → H Gn ( G/G )which is called the assembly map . Conjecture 6.2 (Meta-Isomorphism Conjecture for G -homology theories) . Thegroup G satisfies the Meta-Isomorphism Conjecture with respect to the G -homologytheory H G ∗ and the family F of subgroups of G , if the assembly map H Gn (pr) : H Gn ( E F ( G )) → H Gn ( G/G ) of (6.1) is bijective for all n ∈ Z . If we choose F to be the family ALL of all subgroups, then
G/G is a model for E ALL ( G ) and the Meta-Isomorphism Conjecture 6.2 is obviously true. The pointis to find an as small as possible family F . The idea of the Meta-IsomorphismConjecture 6.2 is that one wants to compute H Gn ( G/G ), which is the unknown andthe interesting object, by assembling it from the values H Gn ( G/H ) for H ∈ F .6.2. The Meta-Isomorphism Conjecture on the level of spectra.
Often theconstruction of the assembly map is done already on the level of spectra or can belifted to this level. Consider a covariant functor E G : Or ( G ) → Spectra . Conjecture 6.3 (Meta-Isomorphism Conjecture for spectra) . The group G satisfiesthe Meta-Isomorphism Conjecture with respect to the covariant functor E G : Or ( G ) → Spectra and the family F of subgroups of G , if the projection pr : E F ( G ) → G/G induces a weak homotopy equivalence ( E G ) % (pr) : ( E G ) % ( E F ( G )) → ( E G ) % ( G/G ) = E G ( G/G ) . Notice that ( E G ) % (pr) : ( E G ) % ( E F ( G )) → ( E G ) % ( G/G ) = E G ( G/G ) is a weakhomotopy equivalence if and only if for every n ∈ Z the map H Gn (pr; E G ) : H Gn ( E F ( G ); E G ) → H Gn ( G/G ; E G )is a bijection, where H G ∗ ( − ; E G ) is the G -homology theory associated to E G , seeLemma 2.5. In other words, Conjecture 6.3 is equivalent to Conjecture 6.2 if wetake for H G ∗ the G -homology theory associated to E G .6.3. The assembly map in terms of homotopy colimits.
The assembly mapappearing in Conjecture 6.3 can be interpreted in terms of homotopy colimits asfollows. Let Or F ( G ) be the full subcategory of Or ( G ) consisting of those objects G/H for which H belongs to F . Let E G | Or F ( G ) be the restriction of E G to Or F ( G ).Then we get from the inclusion Or F ( G ) → Or ( G ) and the fact that G/G is aterminal object in Or ( G ) a maphocolim Or F ( G ) E G | Or F ( G ) → hocolim Or ( G ) E G = E G ( G/G ) . This map can be identified with ( E G ) % (pr) : ( E G ) % ( E F ( G )) → ( E G ) % ( G/G ),see [25, Section 5.2]. Again this explains the name assembly map: we try to putthe values of E G on homogeneous spaces G/H for H ∈ F together to get its valueat G/G . SSEMBLY MAPS 11
Spectra over
Groupoids . In all interesting cases we will obtain E G as fol-lows. Let Groupoids be the category of small groupoids. Consider a covariantfunctor E : Groupoids → Spectra which respects equivalences , i.e., it sends equivalences of groupoids to weak equiv-alences of spectra. Given a G -set S , its transport groupoid T G ( S ) has S as set ofobjects and the set of morphism from s to s is { g ∈ G | gs = s } . Compositioncomes from the multiplication in G . We get for every group G a functor E G : Or ( G ) → Spectra by composing E with the functor Or ( G ) → Groupoids , G/H
7→ T G ( G/H ).Notice that a group G can be viewed as a groupoid with one object and G asset of automorphisms of this object and hence we can consider E ( G ). We havethe obvious identifications E ( G ) = E G ( G/G ) = ( E G ) % ( G/G ). Moreover, for everysubgroup H ⊆ G there is an equivalence of groupoids H → T G ( G/H ) sending theunique object of H to the object eH , which induces a weak homotopy equivalence E ( H ) → E G ( G/H ).The various prominent Isomorphism Conjectures such as the one due to Farrell-Jones and Baum-Connes are now obtained by specifying E : Groupoids → Spectra ,the group G and the family F .7. The Farrell-Jones Conjectures
The Farrell-Jones Conjecture for K -and L -theory. Let R be a ring (withinvolution). There exist covariant functors respecting equivalences K R : Groupoids → Spectra ;(7.1) L h−∞i R : Groupoids → Spectra , (7.2)such that for every group G and all n ∈ Z we have π n ( K R ( G )) ∼ = K n ( RG ); π n ( L h−∞i R ( G )) ∼ = L h−∞i n ( RG ) . Here K n ( RG ) is the n -th algebraic K -group of the group ring RG and L h−∞i n ( RG )is the n th quadratic L -group with decoration h−∞i of the group ring RG equippedwith the involution sending P g ∈ G r g g to P g ∈ G r g g − .The details of this construction can be found in [25, Section 2]. If we now takethese functors and the family VCY of virtually cyclic subgroups, we obtain
Conjecture 7.3 (Farrell-Jones Conjecture) . A group G satisfies the K -theoreticor L -theoretic Farrell-Jones Conjecture if for every ring (with involution) R theassembly maps induced by the projection pr : EG → G/GH Gn (pr; K R ) : H Gn ( EG ; K R ) → H Gn ( G/G ; K R ) = K n ( RG ); H Gn (pr; L h−∞i R ) : H Gn ( EG ; L h−∞i R ) → H Gn ( G/G ; L h−∞i R ) = L h−∞i n ( RG ) , are bijective for all n ∈ Z . It is crucial that we use non-connective K -spectra and that the decoration forthe L -theory is h−∞i , see [38].The original version of the Farrell-Jones Conjecture appeared in [37, 1.6 onpage 257]. A detailed exposition on the Farrell-Jones Conjecture will be givenin [75], see also [78]. Applications of the Farrell-Jones Conjecture.
Here are some conse-quences of the Farrell-Jones Conjecture. For more information about these andother applications we refer for instance to [9, 75, 78].7.2.1.
Computations.
One can carry out explicite computations of K and L -groupsof group rings by applying methods from algebraic topology to the left side givenby a G -homology theory and by finding small models for the classifying spaces offamilies using the topology and geometry of groups, see Section 12.7.2.2. Vanishing of lower and middle K -groups. If G is a torsionfree group satis-fying the K -theoretic Farrell-Jones Conjecture 7.3, then K n ( Z G ) for n ≤ −
1, thereduced projective class group e K ( Z G ), and the Whitehead group Wh( G ) vanish.This has the following consequences. Every homotopy equivalence f : X → Y of connected CW -complexes with π ( Y ) ∼ = G is simple. Every h -cobordism overa closed manifold M of dimension ≥ G ∼ = π ( M ) is trivial. Every finitelygenerated projective Z G -module is stably free. Every finitely dominated connected CW -complex X with π ( X ) ∼ = G is homotopy equivalent to a finite CW -complex.7.2.3. Kaplansky’s Idempotent Conjecture.
If the torsionfree group G satisfies the K -theoretic Farrell-Jones Conjecture 7.3, then G satisfies the Idempotent Conjec-ture that for a commutative integral domain R the only idempotents of RG are 0and 1.7.2.4. Novikov Conjecture. If G satisfies the L -theoretic Farrell-Jones Conjecture 7.3,then G satisfies the Novikov Conjecture about the homotopy invariance of highersignatures. For more information about the Novikov Conjecture we refer for in-stance to [41, 42, 58].7.2.5.
Borel Conjecture. If G is a torsionfree group satisfying the K -theoretic andthe L -theoretic Farrell-Jones Conjecture 7.3, then G satisfies the Borel Conjecture in dimensions ≥
5, i.e., if M and N are closed aspherical manifolds of dimension ≥ π ( M ) ∼ = π ( N ) ∼ = G , then M and N are homeomorphic and every homotopyequivalence from M to N is homotopic to a homeomorphism.7.2.6. Bass Conjecture. If G satisfies the K -theoretic Farrell-Jones Conjecture 7.3,then G satisfies the Bass Conjecture , see [9, 13].7.2.7.
Automorphism groups.
The Farrell-Jones Conjecture 7.3 yields rational com-putations of the homotopy groups and homology groups of the automorphismsgroups of an aspherical closed manifold in the topological, PL and smooth cate-gory, see for instance instance [35], [36, Section 2] and [34, Lecture 5].For instance, if M is an aspherical orientable closed (smooth) manifold of di-mension >
10 with fundamental group G such that G satisfies the Farrell-JonesConjecture 7.3, then we get for 1 ≤ i ≤ (dim M − / π i (Top( M )) ⊗ Z Q = (cid:26) center( G ) ⊗ Z Q if i = 1;0 if i > , and π i (Diff( M )) ⊗ Z Q = center( G ) ⊗ Z Q if i = 1; L ∞ j =1 H ( i +1) − j ( M ; Q ) if i > M odd;0 if i > M even . For a survey on automorphisms of manifolds we refer to [121].
SSEMBLY MAPS 13
Boundary of hyperbolic groups.
In [11] a proof of a conjecture of Gromov isgiven in dimensions n ≥ S n as boundary is the fundamental group of an asphericalclosed topological manifold. This manifold is unique to homeomorphism. Thestable Cannon Conjecture is treated in [40].7.2.9. Poincar´e duality groups. If G is a Poincar´e duality group of dimension n ≥ s -cobordism.Whether it can be chosen to be an aspherical closed topological manifold, dependson its Quinn obstruction.7.2.10. Tautological classes and aspherical manifolds.
The vanishing of tautologicalclasses for many bundles with fibre an aspherical manifold is proved in [44].7.2.11.
Fibering manifolds.
The problem when a map from some closed connectedmanifold to an aspherical closed manifold approximately fibers, i.e., is homotopicto Manifold Approximate Fibration, is analyzed in [39].7.3.
The interpretation of the Farrell-Jones assembly map for L -theory interms of surgery theory. So far we have given a homotopy theoretic approachto the assembly map. This is the easiest approach and well-suited for structuralquestions such as comparing the assembly maps of various different theories, asexplained below. For concrete applications it is important to give geometric oranalytic interpretations. For instance, one key ingredient in the proof that the BorelConjecture follows from the Farrell-Jones Conjecture is a geometric interpretationof the assembly for the trivial family in terms of surgery theory, notably the surgeryexact sequence, which we briefly sketch next.
Definition 7.4 (The structure set) . Let N be a closed topological manifold ofdimension n . We call two simple homotopy equivalences f i : M i → N from closedtopological manifolds M i of dimension n to N for i = 0 , g : M → M such that f ◦ g is homotopic to f .The structure set S ( N ) of N is the set of equivalence classes of simple homotopyequivalences M → X from closed topological manifolds of dimension n to N . Thisset has a preferred base point, namely, the class of the identity id : N → N .One easily checks that the Borel Conjecture holds for G = π ( N ) for a closedaspherical manifold N if and only if S ( N ) consists of precisely one element, namely,the class of id N : N → N . The surgery exact sequence, which we will explain next,gives a way of calculating the structure set. Definition 7.5 (Normal map of degree one) . A normal map of degree one withtarget the connected closed manifold N of dimension n consists of: • A connected closed n -dimensional manifold M ; • A map of degree one f : M → N ; • A ( k + n )-dimensional vector bundle ξ over N ; • A bundle map f : T M ⊕ R k → ξ covering f .There is an obvious normal bordism relation and we denote by N ( N ) the setof bordism classes of normal maps with target N . One can assign to a normalmap f : M → N its surgery obstruction σ ( f ) ∈ L sn ( Z G ) taking values in the n thquadratic L -group with decoration s , where G = π ( N ) and n = dim( N ). If n ≥
5, the surgery obstruction vanishes if and only if one can find (by doingsurgery) a representative in the normal bordisms class, whose underlying map f is a simple homotopy equivalence. It yields a map σ : N ( N ) → L sn ( Z G ). There is a map η : S top n ( N ) → N ( N ) which assigns to the class of a simple homotopyequivalence f : M → N with a closed manifold M as source the normal map givenby f itself and the bundle data coming from T M and ξ = ( f − ) ∗ T M for somehomotopy inverse f − of f . We denote by N ( N × [0 , , N × ∂ [0 , M, ∂M ) → ( N × [0 , , N × ∂ [0 , σ : N ( N × [0 , , N × ∂ [0 , → L sn +1 ( Z G ). There is a also amap ∂ : L sn +1 ( Z G ) → S top n ( N ) which sends an element x ∈ L sn +1 ( Z G ) to the classof a simple homotopy equivalence f : M → N for which there exists a normal maprelative boundary of triads ( F, f , id N ) : ( W ; M, N ) → ( N × [0 , N × { } , N × { } )whose relative surgery obstruction is x . If n ≥
5, then one obtains a long exactsequence of abelian groups, the surgery exact sequence due to Browder, Novikov,Sullivan and Wall(7.6) N ( N × [0 , , N × ∂ [0 , σ n +1 −−−→ L sn +1 ( Z G ) ∂ −→ S ( N ) η −→ N ( N ) σ n −−→ L sn ( Z G ) . If we can show that σ n +1 is surjective and σ n is injective, then the Borel Conjectureholds for G = π ( N ), if N is an aspherical closed manifold of dimension n ≥ L be the L -theory spectrum. It has the property π n ( L ) = L h−∞i n ( Z ). Denoteby L h i its 1 -connective cover . It comes with a natural map of spectra L h i → L ,which induces on π i an isomorphism for i ≥
1, and we have π i ( L h i ) = 0 for i ≤ u n : N ( N ) ∼ = −→ H n ( N ; L h i ) = π n ( N + ∧ L h i ); u n +1 : N ( N × [0 , , N × { , } ) ∼ = −→ H n +1 ( N ; L h i ) = π n +1 (cid:0) N + ∧ L h i ) . An easy spectral sequence argument shows that the canonical map v n : H n ( N ; L h i ) → H n ( N ; L )is injective and the canonical map v n +1 : H n +1 ( N ; L h i ) → H n +1 ( N ; L )is bijective for n = dim( N ).For the remainder of this subsection we assume additionally that N is aspherical.There is a natural identification for m = n, n + 1, see Definition 10.1, w m : H Gm ( EG ; L h−∞i Z ) ∼ = −→ H m ( BG ; L ) = H m ( N ; L ) . The K -theoretic Farrell-Jones Conjecture applied to the torsionfree group G impliesthat K n ( Z G ) for n ≤ −
1, the reduced projective class group e K ( Z G ), and theWhitehead group Wh( G ) vanish. One concludes from the so called Rothenbergsequences , see [98, Theorem 17.2 on page 146], that for m = n, n + 1 the canonicalmap r m : L sm ( Z G ) ∼ = −→ L h−∞i m ( Z G )is bijective. The up to G -homotopy unique G -map i : EG = E T R ( G ) → EG inducesfor all m ∈ Z an isomorphism, see Theorem 11.2 (5), H Gm ( i ; L h−∞i Z ) : H Gm ( EG ; L h−∞i Z ) ∼ = −→ H Gm ( EG ; L h−∞i Z ) . The L -theoretic Farrell-Jones Conjecture predicts the bijectivity of the assemblymap H Gn (pr , L h−∞i Z ) : H Gn ( EG ; L h−∞i Z ) ∼ = −→ H Gn ( G/G ; L h−∞i Z ) = L h−∞i n ( Z G ) . SSEMBLY MAPS 15
The following diagram commutes N ( N ) σ n / / u n ∼ = (cid:15) (cid:15) L sn ( Z G ) H n ( N ; L h i ) v n (cid:15) (cid:15) H n ( N ; L ) H Gn ( EG ; L h−∞i Z ) w n ∼ = O O ∼ = H Gn ( i, L h−∞i Z ) / / H Gn ( EG ; L h−∞i Z ) ∼ = H Gn (pr , L h−∞i Z ) / / L h−∞i n ( Z G ) r n ∼ = O O If we replace everywhere n by n + 1 and in the upper left corner N ( N ) by N ( N × [0 , , N × ∂ [0 , σ n is injective and σ n +1 is bijective since v n is injective and v n +1 is bijective. Recall that this impliesthe vanishing of the structure set S ( N ). Hence the Farrell-Jones Conjecture 7.3implies the Borel Conjecture in dimensions ≥ L -groups and surgery theory and the arguments andfacts above we refer for instance to [18, 19, 24, 66, 97, 117].7.4. The interpretation of the Farrell-Jones assembly map in terms ofcontrolled topology.
We have defined the assembly map appearing in the Farrell-Jones Conjecture as a map induced by the projection EG → G/G for a G -homologytheory H G ∗ ( X ; E G ) or for the functor ( E G ) % : G - CW → Spectra . We have alsogiven a homotopy theoretic interpretation in terms of homotopy colimits and de-scribed its universal property to be the best approximation from the left by anexcisive functor. This interpretation is good for structural and computational as-pects but it turns out that it is not helpful for the proof that the assembly maps isa weak homotopy equivalence. There is no direct homotopy theoretic constructionof an inverse up to weak homotopy equivalence known to the author.For the actual proofs that the assembly maps are weak homotopy equivalences,the interpretation of the assembly map as a forget control map is crucial. Thisfundamental idea is due to Quinn.Roughly speaking, one attaches to a metric space certain categories, to thesecategories spectra and then takes their homotopy groups, where everything dependson a choice of certain control conditions which in some sense measure sizes of cycles.If one requires certain control conditions, one obtains the source of the assemblymap. If one requires no control conditions, one obtains the target of the assemblymap. The assembly map itself is forgetting the control condition.One of the basic features of a homology theory is excision. It often comes fromthe fact that a representing cycle can be arranged to have arbitrarily good control.An example is the technique of subdivision which allows to make the representingcycles for singular homology arbitrarily controlled, i.e., the diameter of the imageof a singular simplex appearing in a singular chain with non-zero coefficient canbe arranged to be arbitrarily small. This is the key ingredient in the proof thatsingular homology satisfies excision. In general one may say that requiring controlconditions amounts to implementing homological properties.With this interpretation it is clear what the main task in the proof of surjectivityof the assembly map is: achieve control , i.e., manipulate cycles without changing their homology class so that they become sufficiently controlled. There is a generalprinciple that a proof of surjectivity also gives injectivity. Namely, proving injec-tivity means that one must construct a cycle whose boundary is a given cycle, i.e.,one has to solve a surjectivity problem in a relative situation. The actual imple-mentation of this idea is rather technical. The proof that this forget control versionof the assembly map agrees up to weak homotopy equivalence with the homotopytheoretic one appearing in the Farrell-Jones Conjecture 7.3 is a direct applicationof Section 4. The same is true also for the version of the assembly map appearingin [37, 1.6 on page 257], as explained in [25, page 239].To achieve control one can now use geometric methods. The key ingredients arecontracting maps and open coverings, transfers, flow spaces and the geometry ofthe group G .For more information about the general strategy of proofs we refer for instanceto [2, 74, 75].7.5. The Farrell-Jones Conjecture for Waldhausen’s A -theory. Waldhausenhas defined for a CW -complex X its algebraic K -theory space A ( X ) in [113, Chap-ter 2]. As in the case of algebraic K -theory of rings it will be necessary to considera non-connective version. Vogell [110] has defined a delooping of A ( X ) yielding anon-connective spectrum A ( X ) for a CW -complex X . This construction actuallyyields a covariant functor from the category of topological spaces to the categoryof spectra. We can assign to a groupoid G its classifying space B G . Thus we obtaina covariant functor(7.7) A : Groupoids → Spectra , G 7→ A ( B G ) , denoted by A again. It respects equivalences, see [113, Proposition 2.1.7] and [110].If we now take this functor and the family VCY of virtually cyclic subgroups, weobtain the A -theoretic Farrell-Jones Conjecture Conjecture 7.8 ( A -theoretic Farrell-Jones Conjecture) . A group G satisfies the A -theoretic Farrell-Jones Conjecture if the assembly maps induced by the projection pr : EG → G/GH Gn (pr; A ) : H Gn ( EG ; A ) → H Gn ( G/G ; A ) = π n ( A ( BG )) is bijective for all n ∈ Z . The A -theoretic Farrell-Jones Conjecture 7.8 is an important ingredient in thecomputation of the group of selfhomeomorphisms of an aspherical closed manifoldin the stable range using the machinery of Weiss-Williams [121]. Moreover, it isrelated to Whitehead spaces, pseudo-isotopy spaces and spaces of h-cobordisms, seefor instance [31, 112, 113, 114, 115, 116, 121].7.6. Relating the assembly maps for K -theory and for A -theory. Let X be a connected CW -complex with fundamental group π = π ( X ). Essentially bypassing to the cellular Z π -chain complex of the universal covering one obtains anatural map of (non-connective) spectra, natural in X , called linearization map L ( X ) : A ( X ) → K ( Z π ( X )) . (7.9)The next result follows by combining [111, Section 4] and [112, Proposition 2.2and Proposition 2.3]. Theorem 7.10 (Connectivity of the linearization map) . Let X be a connected CW -complex. Then: SSEMBLY MAPS 17 (1) The linearization map L ( X ) of (7.9) is -connected, i.e., the map L n := π n ( L ( X )) : A n ( X ) → K n ( Z π ( X )) is bijective for n ≤ and surjective for n = 2 ;(2) Rationally the map L n is bijective for all n ∈ Z , provided that X is aspher-ical. Thus one obtain a transformation L : A → K Z of covariant functors Groupoids → Spectra , where K Z and A have been defined in (7.1) and (7.7). It induces a com-mutative diagram(7.11) H Gn ( EG ; A ) H Gn (pr; A ) / / H Gn (id EG ; L ) (cid:15) (cid:15) H Gn ( G/G ; A ) = π n ( A ( BG )) π n ( L ( BG )) (cid:15) (cid:15) H Gn ( EG ; K Z ) H Gn (pr; K R ) / / K n ( Z G )where the upper horizontal arrow is the assembly map appearing in A -theoreticFarrell-Jones Conjecture 7.8, the lower horizontal arrow is the assembly map ap-pearing in the K -theoretic Farrell-Jones Conjecture 7.3, and both vertical arrowsare bijective for n ≤
1, surjective for n = 2 and rationally bijective for all n ∈ Z .In particular the K -theoretic Farrell-Jones Conjecture 7.3 for R = Z and the A -theoretic Farrell-Jones Conjecture 7.8 are rationally equivalent.7.7. The status of the Farrell-Jones Conjecture.
There is a more generalversion of the Farrell-Jones Conjecture, the so called
Full Farrell-Jones Conjecture ,where one allows coefficients in additive categories and the passage to finite wreathproducts, It implies the Farrell-Jones Conjectures 7.3. For A -theory there is a socalled fiibered version which implies Conjecture 7.8. Let FJ be the class of groupsfor which the Full Farrell-Jones Conjecture and the fibered A -theoretic Farrell-Jones Conjecture holds. Notice that then any group in FJ satisfies in particularConjectures 7.3 and 7.8. Theorem 7.12 (The class FJ ) . (1) The following classes of groups belong to FJ :(a) Hyperbolic groups;(b) Finite dimensional CAT(0) -groups;(c) Virtually solvable groups;(d) (Not necessarily cocompact) lattices in second countable locally compactHausdorff groups with finitely many path components;(e) Fundamental groups of (not necessarily compact) connected manifolds(possibly with boundary) of dimension ≤ ;(f ) The groups GL n ( Q ) and GL n ( F ( t )) for F ( t ) the function field over afinite field F ;(g) S -arithmetic groups;(h) mapping class groups;(2) The class FJ has the following inheritance properties:(a) Passing to subgroups
Let H ⊆ G be an inclusion of groups. If G belongs to FJ , then H belongs to FJ ;(b) Passing to finite direct products
If the groups G and G belong to FJ , then also G × G belongs to FJ ; (c) Group extensions
Let → K → G → Q → be an extension of groups. Suppose thatfor any cyclic subgroup C ⊆ Q the group p − ( C ) belongs to FJ andthat the group Q belongs to FJ .Then G belongs to FJ ;(d) Directed colimits
Let { G i | i ∈ I } be a direct system of groups indexed by the directedset I (with arbitrary structure maps). Suppose that for each i ∈ I thegroup G i belongs to FJ .Then the colimit colim i ∈ I G i belongs to FJ ;(e) Passing to finite free products
If the groups G and G belong to FJ , then G ∗ G belongs to FJ ;(f ) Passing to overgroups of finite index
Let G be an overgroup of H with finite index [ G : H ] . If H belongs to FJ , then G belongs to FJ ;Proof. See [3, 4, 5, 7, 8, 10, 33, 53, 57, 102, 118, 119]. (cid:3)
It is not known whether all amenable groups belong to FJ .8. The Baum-Connes Conjecture
The Baum-Connes Conjecture.
Recall that the reduced group C ∗ -algebra C ∗ r ( G ) is a certain completion of the complex group ring C G . Namely, there is acanonical embedding of C G into the space B ( l ( G )) of bounded operators L ( G ) → L ( G ) equipped with the supremums norm given by the right regular representation,and C ∗ r ( G ) is the norm closure of C G in B ( L ( G )). There is a covariant functorrespecting equivalences(8.1) K top : Groupoids → Spectra , such that for every group G and all n ∈ Z we have π n ( K top ( G )) ∼ = K top n ( C ∗ r ( G )) , where K top n ( C ∗ r ( G )) is the topological K -theory of the reduced group C ∗ -algebra C ∗ r ( G ), see [51]. If we now take this functors and the family FIN of finite sub-groups, we obtain
Conjecture 8.2 (Baum-Connes Conjecture) . A group G satisfies the Baum-ConnesConjecture if the assembly maps induced by the projection pr : EG → G/GH Gn (pr; K top ) : H Gn ( EG ; K top ) → H Gn ( G/G ; K top ) = K top n ( C ∗ r ( G )) . is bijective for all n ∈ Z . The original version of the Baum-Connes Conjecture is stated in [14, Conjec-ture 3.15 on page 254]. There is also a version, where the ground field C is replacedby R . The complex version of the Baum-Connes Conjecture 8.2 implies automati-cally the real version, see [16, 105].8.2. The Baum-Connes Conjecture with coefficients.
There is also a moregeneral version of the Baum-Connes Conjecture 8.2, where one allows twisted coef-ficients. However, there are counterexamples to this more general version, see [47,Section 7]. There is a new formulation of the Baum-Connes Conjecture with coef-ficients in [15], where these counterexamples do not occur anymore. At the time ofwriting no counterexample to the Baum-Connes Conjecture 8.2 or to the versionof [15] is known to the author.
SSEMBLY MAPS 19
The interpretation of the Baum Connes assembly map in terms ofindex theory.
For applications of the Baum-Connes Conjecture 8.2 it is essentialthat the Baum-Connes assembly maps can be interpreted in terms of indices ofequivariant operators with values in C ∗ -algebras. Namely, one assigns to a Kas-parov cycle representing an element in the equivariant KK -group KK Gn ( C ( X ) , C )in the sense of Kasparov [54, 55, 56] its C ∗ -valued index in K n ( C ∗ r ( G )) in the senseof Mishchenko-Fomenko [86], thus defining a map KK Gn ( C ( X ) , C ) → K top n ( C ∗ r ( G )) , provided that X is proper and cocompact and C ( X ) is the C ∗ -algebra (possiblywithout unit) of continuous function X → C vanishing at infinity. This is theapproach appearing in [14].The other equivalent approach is based on the Kasparov product. Given aproper cocompact G - CW -complex X , one can assign to it an element [ p X ] ∈ KK G ( C , C ( X ) ⋊ r G ), where C ( X ) ⋊ r G denotes the reduced crossed product C ∗ -algebra associated to the G - C ∗ -algebra C ( X ). Now define the Baum-Connesassembly map by the composition of a descent map and a map coming from theKasparov product KK Gn ( C ( X ) , C ) j Gr −−→ KK n ( C ( X ) ⋊ r G, C ∗ r ( G )) [ p X ] b ⊗ C X ) ⋊ rG − −−−−−−−−−−−→ KK n ( C , C ∗ r ( G )) = K top n ( C ∗ r ( G )) . This extends to arbitrary proper G - CW -complexes X by defining the source by K Gn ( C ( X ) , C ) := colim C ⊆ X K Gn ( C ( C ) , C ) , where C runs through the finite G - CW -subcomplexes of Y directed by inclusion.Hence we can take X = EG above without assuming any finiteness conditions on EG . For some information about these two approaches and their identification, atleast for torsionfree G , we refer to [61].One can identify the original assembly map of [14] with the assembly map ap-pearing in Conjecture 8.2 using Section 8.2 and the fact thatcolim C ⊆ X H Gn ( C ; K top ) ∼ = −→ H Gn ( X ; K top )is an isomorphism. This is explained in [25, page 247-248], Unfortunately, theproof is based on an unfinished preprint by Carlsson-Pedersen-Roe [20], where theassembly map appearing in [14, Conjecture 3.15 on page 254] is implemented onthe spectrum level. Another proof of the identification is given in [43, Corollary 8.4]and [88, Theorem 1.3].8.4. Applications of the Baum-Connes Conjecture.
Computations.
One can carry out explicite computations of topological K -groups of group C ∗ -algebras and related C ∗ -algebras by applying methods fromalgebraic topology to the left side given by a G -homology theory and by findingsmall models for the classifying spaces of families using the topology and geometryof groups. This leads to classification results about certain C ∗ -algebras, see forinstance [27, 32, 63, 64].8.4.2. (Modified) Trace Conjecture. The Baum-Connes Conjecture 8.2 implies the
Trace Conjecture for torsionfree groups that for a torsionfree group G the image oftr C ∗ r ( G ) : K ( C ∗ r ( G )) → R consists of the integers. If one drops the condition torsionfree, there is the socalled Modified Trace Conjecture , which is implied by Baum-Connes Conjecture 8.2,see [69].
Kadison Conjecture.
The Baum-Connes Conjecture 8.2 implies the
KadisonConjecture that for a torsionfree group G the only idempotent elements in C ∗ r ( G )are 0 and 1.8.4.4. Novikov Conjecture.
The Baum-Connes Conjecture 8.2 implies the
NovikovConjecture .8.4.5.
The Zero-in-the-spectrum Conjecture.
The
Zero-in-the-spectrum Conjecture says, if f M is the universal covering of an aspherical closed Riemannian manifold M ,then zero is in the spectrum of the minimal closure of the p th Laplacian on f M forsome p ∈ { , , . . . , dim M } . It is a consequence of the Strong Novikov Conjectureand hence of the Baum-Connes Conjecture 8.2, see [68, Chapter 12].8.4.6. The Stable Gromov-Lawson-Rosenberg Conjecture.
Let Ω
Spin n ( BG ) be thebordism group of closed Spin-manifolds M of dimension n with a reference mapto BG . Given an element [ u : M → BG ] ∈ Ω Spin n ( BG ), we can take the C ∗ r ( G ; R )-valued index of the equivariant Dirac operator associated to the G -covering M → M determined by u . Thus we get a homomorphismind C ∗ r ( G ; R ) : Ω Spin n ( BG ) → K top n ( C ∗ r ( G ; R )) . (8.3)A Bott manifold is any simply connected closed Spin-manifold B of dimension 8whose b A -genus b A ( B ) is 1. We fix such a choice, the particular choice does notmatter for the sequel. Notice that ind C ∗ r ( { } ; R ) ( B ) ∈ K top8 ( R ) ∼ = Z is a genera-tor and the product with this element induces the Bott periodicity isomorphisms K top n ( C ∗ r ( G ; R )) ∼ = −→ K top n +8 ( C ∗ r ( G ; R )). In particularind C ∗ r ( G ; R ) ( M ) = ind C ∗ r ( G ; R ) ( M × B ) , (8.4)if we identify K top n ( C ∗ r ( G ; R )) = K top n +8 ( C ∗ r ( G ; R )) via Bott periodicity. Conjecture 8.5 (Stable Gromov-Lawson-Rosenberg Conjecture) . Let M be a closedconnected Spin -manifold of dimension n ≥ . Let u M : M → Bπ ( M ) be the clas-sifying map of its universal covering. Then M × B k carries for some integer k ≥ a Riemannian metric with positive scalar curvature if and only if ind C ∗ r ( π ( M ); R ) ([ M, u M ]) = 0 ∈ K top n ( C ∗ r ( π ( M ); R )) . If M carries a Riemannian metric with positive scalar curvature, then the indexof the Dirac operator must vanish by the Bochner-Lichnerowicz formula [99]. Theconverse statement that the vanishing of the index implies the existence of a Rie-mannian metric with positive scalar curvature is the hard part of the conjecture.The unstable version of Conjecture 8.5, where one does not stabilize with B k , isnot true in general, see [103].A sketch of the proof of the following result can be found in Stolz [107, Section 3]. Theorem 8.6 (The Baum-Connes Conjecture implies the Stable Gromov-Law-son-Rosenberg Conjecture) . If the assembly map for the real version of the Baum-Connes Conjecture 8.2 is injective for the group G , then the Stable Gromov-Lawson-Rosenberg Conjecture 8.5 is true for all closed Spin -manifolds of dimension ≥ with π ( M ) ∼ = G . Knot theory.
Cochran-Orr-Teichner give in [23] new obstructions for a knotto be slice which are sharper than the Casson-Gordon invariants. They use L -signatures and the Baum-Connes Conjecture 8.2. We also refer to the survey arti-cle [22] about non-commutative geometry and knot theory. SSEMBLY MAPS 21
The status of the Baum-Connes Conjecture.
Let BC be the class ofgroups for which the Baum-Connes Conjecture with coefficients, which implies theBaum-Connes Conjecture 8.2, is true. Theorem 8.7 (Status of the Baum-Connes Conjecture 8.2) . (1) The following classes of groups belong to BC .(a) A-T-menable groups;(b) Hyperbolic groups;(c) One-relator groups;(d) Fundamental groups of compact -manifolds (possibly with boundary);(2) The class BC has the following inheritance properties:(a) Passing to subgroups
Let H ⊆ G be an inclusion of groups. If G belongs to BC , then H belongs to BC ;(b) Passing to finite direct products
If the groups G and G belong to BC , the also G × G belongs to BC ;(c) Group extensions
Let → K → G → Q → be an extension of groups. Suppose thatfor any finite subgroup F ⊆ Q the group p − ( F ) belongs to BC andthat the group Q belongs to BC .Then G belongs to BC ;(d) Directed unions
Let { G i | i ∈ I } be a direct system of subgroups of G indexed by thedirected set I such that G = S i ∈ I G i . Suppose that G i belongs to BC for every i ∈ I .Then G belongs to BC ;(e) Actions on trees
Let G be a countable discrete group acting without inversion on a tree T . Then G belongs to BC if and only if the stabilizers of each of thevertices of T belong to BC .In particular BC is closed under amalgamated products and HNN-extensions.Proof. See [4, 21, 46, 60, 85, 90, 91]. (cid:3)
It is not known whether finite-dimensional CAT(0)-groups and SL n ( Z ) for n ≥ BC .For more information about the Baum-Connes Conjecture and its applicationswe refer for instance to [14, 45, 48, 49, 50, 75, 78, 87, 93, 101, 104, 109].8.6. Relating the assembly maps of Farrell-Jones to the one of Baum-Connes.
One can construct the following commutative diagram (8.8) H Gn ( EG ; L h−∞i Z )[1 / / / l ∼ = (cid:15) (cid:15) L h−∞i n ( Z G )[1 / id ∼ = (cid:15) (cid:15) H Gn ( EG ; L h−∞i Z [1 / / / L h−∞i n ( Z G )[1 / H Gn ( EG ; L p Z [1 / i ∼ = (cid:15) (cid:15) i ∼ = O O / / L pn ( Z G )[1 / j ∼ = (cid:15) (cid:15) j ∼ = O O H Gn ( EG ; L p Q [1 / i ∼ = (cid:15) (cid:15) / / L pn ( Q G )[1 / j (cid:15) (cid:15) H Gn ( EG ; L p R [1 / i ∼ = (cid:15) (cid:15) / / L pn ( R G )[1 / j (cid:15) (cid:15) H Gn ( EG ; L pC ∗ r (?; R ) [1 / / / L pn ( C ∗ r ( G ; R ))[1 / H Gn ( EG ; K top R [1 / / / i ∼ = O O K top n ( C ∗ r ( G ; R ))[1 / j ∼ = O O H Gn ( EG ; K top R )[1 / (cid:15) (cid:15) i (cid:15) (cid:15) l ∼ = O O K top n ( C ∗ r ( G ; R ))[1 / (cid:15) (cid:15) j (cid:15) (cid:15) ∼ =id O O H Gn ( EG ; K top C )[1 / / / K n ( C ∗ r ( G ))[1 / Groupoids → Spectra . These transformations areinduced by change of rings maps except the one from K top R [1 /
2] to L pC ∗ r (?; R ) [1 / i and j are bijections.For any finite group H each of the following maps is known to be a bijectionbecause of [97, Proposition 22.34 on page 252] and R H = C ∗ r ( H ; R ) L pn ( Z H )[1 / ∼ = −→ L pn ( Q H )[1 / ∼ = −→ L pn ( R H )[1 / ∼ = −→ L pn ( C ∗ r ( H ; R )) . The natural map L pn ( RG )[1 / → L h−∞i n ( RG )[1 /
2] is an isomorphism for any n ∈ Z ,group G and ring with involution R by the Rothenberg sequence, see [98, Theo-rem 17.2 on page 146]. Hence we conclude from the equivariant Atiyah Hirzebruchspectral sequence that the vertical arrows i , i , and i are isomorphisms. Thearrow j is bijective by [96, page 376]. The maps l are isomorphisms for generalresults about localizations.The lowermost vertical arrows i and j are known to be split injective be-cause the inclusion C ∗ r ( G ; R ) → C ∗ r ( G ; C ) induces an isomorphism C ∗ r ( G ; R ) → C ∗ r ( G ; C ) Z / for the Z / C → C . SSEMBLY MAPS 23
The following conjecture is already raised as a question in [58, Remark 23.14 onpage 197], see also [62, Completion Conjecture in Subsection 5.2].
Conjecture 8.9 (Passage for L -theory from Q G to R G to C ∗ r ( G ; R )) . The maps j and j appearing in diagram (8.8) are bijective. One easily checks
Lemma 8.10.
Let G be a group.(1) Suppose that G satisfies the L -theoretic Farrell-Jones Conjecture 7.3 withcoefficients in the ring R for R = Q and R = R and the Baum-ConnesConjecture 8.2. Then G satisfies Conjecture 8.9;(2) Suppose that G satisfies Conjecture 8.9. Then G satisfies the L -theoreticFarrell-Jones Conjecture 7.3 for the ring Z after inverting , if and onlyif G satisfies the real version of the Baum-Connes Conjecture 8.2 afterinverting ;(3) Suppose that the assembly map appearing in the Baum-Connes Conjec-ture 8.2 is (split) injective after inverting . Then the assembly map ap-pearing in L -theoretic Farrell-Jones Conjecture 7.3 with coefficients in thering for R = Z is (split) injective after inverting . Topological cyclic homology
Let R be a (well-pointed connective) symmetric ring spectrum and p be a prime.There are covariant functors respecting equivalences THH R : Groupoids → Spectra ; TC R ,p : Groupoids → Spectra , such that for every group G and all n ∈ Z we have π n ( THH R ( G )) ∼ = π n ( THH ( R [ G ])); π n ( TC R ; p ( G )) ∼ = π n ( TC ( R [ G ]; p )) , where THH ( R [ G ]) is the topological Hochschild homology and THH ( R [ G ]; p ) isthe topological cyclic homology of the group ring spectrum R [ G ].9.1. Topological Hochschild homology.
If we now take the functor
THH R and the family CY of cyclic subgroups, we obtain from [80, Theorem 1.19] that theFarrell-Jones Conjecture for topological Hochschild homology is true for all groups. Theorem 9.1 (Topological Hochschild homology) . The assembly maps induced bythe projection pr : E CY ( G ) → G/GH Gn (pr; THH R ) : H Gn ( E CY ( G ); THH R ) → H Gn ( G/G ; THH R ) = π n ( THH ( R [ G ])) is bijective for all n ∈ Z . Topological cyclic homology.
If we take the functor TC R ; p and the family FIN of cyclic subgroups, we obtain from [79, Theorem 1.5] that the injectivitypart of the Farrell-Jones Conjecture for topological cyclic homology is true undercertain finiteness assumptions
Theorem 9.2 (Split injectivity for topological cyclic homology) . Assume that onefor the following conditions hold for the family F :(1) We have F = FIN and there is a model for EG of finite type;(2) We have F = VCY and G is hyperbolic or virtually abelian. Then the assembly maps induced by the projection pr : EG → G/GH Gn (pr; TC R ; p ) : H Gn ( EG ; TC R ; p ) → H Gn ( G/G ; THH R ; p ) = π n ( THH ( R [ G ]; p )) is split injective for all n ∈ Z . Moreover, we also have, see [79, Theorem 1.2]
Theorem 9.3 (Topological cyclic homology and finite groups) . If G is a finitegroup, then the assembly map for the family CY of cyclic subgroups H Gn (pr; TC R ; p ) : H Gn ( E CY ( G ); TC R ; p ) → H Gn ( G/G ; TC R ; p ) = π n ( TC ( R [ G ]; p )) is bijective for all n ∈ Z . Remark 9.4 (The Farrell Jones Conjecture for topological cyclic homology is nottrue in general) . There are examples, where the assembly map H Gn (pr; TC R ; p ) : H Gn ( EG ; TC R ; p ) → π n ( THH ( R [ G ]; p ))not surjective, see [79, Theorem 1.6]. At least there is a pro-isomorphism for TC R ; p with respect to the family CY , see [79, Theorem 1.4]. The complications occurringwith topological cyclic homology are due to the fact that smash products andhomotopy inverse limit do not commute in general, see [81].9.3. Relating the assembly maps of Farrell-Jones to the one for topolog-ical cyclic homology via the cyclotomic trace.
There is an important trans-formation from algebraic K -theory to topological cyclic homology, the so called cyclotomic trace . It relates the assembly maps for the algebraic K -theory of Z G to the cyclic topological homology of the spherical group ring of G and is a keyingredient in proving the rational injectivity of K -theoretic assembly maps. Theconstruction of the cyclotomic trace and the proof of the K -theoretic Novikov con-jecture is carried out in the celebrated paper by Boekstedt-Hsiang-Madsen [17].The passage from T R to FIN , thus detecting a much larger portion of the alge-braic K -theory of Z G and proving new results about the Whitehead group Wh( G ),is presented in [80].For more information about topological cyclic homology we refer for instanceto [17, 30, 89]. 10. The global point of view
At various occasions it has turned out that one should take a global point of view,i.e., one should not consider each group separately, but take into account that ingeneral there is a theory which can be applied to every group and the values forthe various groups are linked. This appears for instance in the following definitiontaken from [67, Section 1].Let α : H → G be a group homomorphism. Given an H -space X , define the induction of X with α to be the G -space ind α X := G × α X , which is the quotient of G × X by the right H -action ( g, x ) · h := ( gα ( h ) , h − x ) for h ∈ H and ( g, x ) ∈ G × X . Definition 10.1 (Equivariant homology theory) . An equivariant homology theorywith values in Λ -modules H ? n assigns to each group G a G -homology theory H G ∗ with values in Λ-modules (in the sense of Definition 2.1) together with the followingso called induction structure :Given a group homomorphism α : H → G and a H - CW -pair ( X, A ), there arefor every n ∈ Z natural homomorphismsind α : H Hn ( X, A ) → H Gn (ind α ( X, A ))(10.2)satisfying:
SSEMBLY MAPS 25 • Compatibility with the boundary homomorphisms ∂ Gn ◦ ind α = ind α ◦ ∂ Hn ; • Functoriality
Let β : G → K be another group homomorphism. Then we have for n ∈ Z ind β ◦ α = H Kn ( f ) ◦ ind β ◦ ind α : H Hn ( X, A ) → H Kn (ind β ◦ α ( X, A )) , where f : ind β ind α ( X, A ) ∼ = −→ ind β ◦ α ( X, A ) , ( k, g, x ) ( kβ ( g ) , x ) is thenatural K -homeomorphism; • Compatibility with conjugation
For n ∈ Z , g ∈ G and a (proper) G - CW -pair ( X, A ) the homomorphismind c ( g ): G → G : H Gn ( X, A ) → H Gn (ind c ( g ): G → G ( X, A )) agrees with H Gn ( f )for the G -homeomorphism f : ( X, A ) → ind c ( g ): G → G ( X, A ) which sends x to (1 , g − x ) in G × c ( g ) ( X, A ); • Bijectivity
If ker( α ) acts freely on X \ A , then ind α : H Hn ( X, A ) → H Gn (ind α ( X, A )) isbijective for all n ∈ Z .Because of the following theorem it will pay off that in Subsection 6.4 we con-sidered functors defined on Groupoids and not only on Or ( G ). Theorem 10.3 (Constructing equivariant homology theories using spectra) . Con-sider a covariant functor E : Groupoids → Spectra respecting equivalences.Then there is an equivariant homology theory H ? ∗ ( − ; E ) satisfying H Gn ( G/H ; E ) ∼ = H Hn ( {•} ; E ) ∼ = π n ( E ( H )) for every subgroup H ⊆ G of every group G and every n ∈ Z .Proof. See [78, Proposition 5.6 on page 793]. (cid:3)
The global point of view has been taken up and pursued by Stefan Schwede onthe level of spectra in his book [106], where global equivariant homotopy theory forcompact Lie groups is developed. To deal with spectra is much more advanced andsophisticated than with equivariant homology.11.
Relative assembly maps
In the formulations of the Isomorphism Conjectures above such as the one dueto Farrell-Jones and Baum-Connes it is important to make the family F as small aspossible. The largest family we encounter is VCY , but there are special cases, whereone can get smaller families. In particular it is desirable to get away with
FIN ,since there are often finite models for EG = E FIN ( G ), whereas conjecturally thereis a finite model for EG = E VCY ( G ) only if G itself is virtually cyclic, see [52,Conjecture 1].The general problem is to study and hopefully to prove bijectivity of relativeassembly map associated to two families F ⊆ F ′ , i.e., of the map induced by theup to G -homotopy unique G -map E F ( G ) → E F ′ ( G )asmb F⊆F ′ : H Gn ( E F ( G )) → H Gn ( E G ( G ))for a G -homology theory H G ∗ with values in Λ-modules. In studying this the globalpoint of view becomes useful.The main technical result is the so called Transitivity Principle, which we explainnext. For a family F of subgroups of G and a subgroup H ⊂ G we define a familyof subgroups of H F| H = { K ∩ H | K ∈ F} . Theorem 11.1 (Transitivity Principle) . Let H ? ∗ ( − ) be an equivariant homologytheory with values in Λ -modules. Suppose F ⊂ F ′ are two families of subgroups of G . If for every H ∈ F ′ and every n ∈ Z the assembly map asmb F| H ⊆ALL : H Hn ( E F| H ( H )) → H Hn ( {•} ) is an isomorphism, then for every n ∈ Z the relative assembly map asmb F⊆F ′ : H Gn ( E F ( G )) → H Gn ( E F ′ ( G )) is an isomorphism.Proof. See [78, Theorem 65 on page 742]. (cid:3)
One has the following results about diminishing the family of subgroups. De-note by
VCY I the family of subgroups of G which are either finite or admit anepimorphism onto Z with finite kernel. Obviously FIN ⊆ VCY I ⊆ VCY . Theorem 11.2 (Relative assembly maps) . (1) The relative assembly map for K -theory asmb T R⊆VCY : H Gn ( E T R ( G ); K R ) → H Gn ( E VCY ( G ); K R ) is bijective for all n ∈ Z , provided that G is torsionfree and R is regular;(2) The relative assembly map for K -theory asmb FIN ⊆VCY : H Gn ( E FIN ( G ); K R ) → H Gn ( E VCY ( G ); K R ) is bijective for all n ∈ Z , provided that R is a regular ring containing Q ;(3) The relative assembly map for K -theory asmb VCY I ⊆VCY : H Gn ( E VCY I ( G ); K R ) ∼ = −→ H n ( E VCY ( G ); K R ) is bijective for all n ∈ Z ;(4) The relative assembly map for K -theory asmb FIN ⊆VCY ⊗ Z id Q : H Gn ( E FIN ( G ); K R ) ⊗ Z Q → H Gn ( E VCY ( G ); K R ) ⊗ Z Q is bijective for all n ∈ Z , provided that R is regular;(5) The relative assembly map for L -theory asmb T R⊆VCY : H Gn ( E T R ( G ); L h−∞i R ) → H Gn ( E VCY ( G ); L h−∞i R ) is an isomorphism for all n ∈ Z , provided that G is torsionfree;(6) There relative assembly map for L -theory asmb FIN ⊆VCY I : H Gn (cid:0) E FIN ( G ); L h−∞i R (cid:1) → H Gn (cid:0) E VCY I ( G ); L h−∞i R (cid:1) is bijective for all n ∈ Z ;(7) The relative assembly map for L -theory asmb FIN ⊆VCY [1 /
2] : H Gn ( E FIN ( G ); L h−∞i R )[1 / → H Gn ( E VCY ( G ); L h−∞i R )[1 / is bijective for all n ∈ Z ;(8) The relative assembly map for topological K -theory asmb FCY⊆FIN : H Gn ( E FCY ( G ); K top ) → H Gn ( E FIN ( G ); K top ) is bijective for all n ∈ Z . This is also true for the real version;(9) The relative assembly maps for K -theory and L -theory asmb FIN ⊆VCY : H Gn ( E FIN ( G ); K R ) → H Gn ( E VCY ( G ); K R );asmb FIN ⊆VCY : H Gn ( E FIN ( G ); L h−∞i R ) → H Gn ( E VCY ( G ); L h−∞i R ) , are split injective for all n ∈ Z ; SSEMBLY MAPS 27
Proof.
See [12, Theorem 1.3], [6, Theorem 0.5], [28], [72, Lemma 4.2], [78, Sec-tion 2.5], [83, Theorem 0.1 and Theorem 0.3], and [84]. (cid:3)
Remark 11.3 (Torsionfree groups) . A typical application is that for a torsionfreegroup G and a regular ring the K -theoretic Farrell-Jones Conjecture 7.3 impliestogether with Theorem 11.2 (1) that the assembly map H n ( BG ; K ( R )) = π n ( BG + ∧ K ( R ) → K n ( RG )is bijective for all n ∈ Z . Analogously, for a torsionfree group G the L -theoreticFarrell-Jones Conjecture 7.3 implies together with Theorem 11.2 (5) that the as-sembly map H n ( BG ; L h−∞i ( R )) = π n ( BG + ∧ L h−∞i ) → L h−∞i n ( RG )is bijective for all n ∈ Z . 12. Computationally tools
Most computations of K - and L -groups of group rings are done using the Farrell-Jones Conjecture 7.3 and the Baum-Connes Conjecture 8.2. The situation inthe Farrell-Jones Conjecture 7.3 is more complicated than in the Baum-Connessetting, since the family VCY is much harder to handle than the family
FIN .One can consider H Gn ( EG ; K R ) and H Gn ( EG, EG ; K R ) separately because of The-orem 11.2 (9), where one considers EG as a G - CW -subcomplex of EG . The term H Gn ( EG, EG ; K R ) involves Nil-terms and UNil-terms, which are hard to deter-mine. For H Gn ( EG ; K R ), H Gn ( EG ; L h−∞i R ) and H Gn ( EG ; K top ) one can use theequivariant Atyiah-Hirzebruch spectral sequence or the p -chain spectral sequence,see Davis-Lueck [26]. Rationally these groups can often be computed explicitlyusing equivariant Chern characters, see [67, Section 1]. Notice that these can onlybe constructed since we take on the global point of view as explained in Section 10.Often an important input is that one obtains from the geometry of the underlyinggroup nice models for EG and can construct EG from EG by attaching a tractablefamily of equivariant cells.Here are two example, where these ideas lead to a explicite computation, whoseoutcome is as simple as one can hope. Theorem 12.1 (Farrell-Jones Conjecture for torsionfree hyperbolic groups for K -theory) . Let G be a torsionfree hyperbolic group.(1) We obtain for all n ∈ Z an isomorphism H n ( BG ; K ( R )) ⊕ M C (cid:0) NK n ( R ) ⊕ NK n ( R ) (cid:1) ∼ = −→ K n ( RG ) , where C runs through a complete system of representatives of the conju-gacy classes of maximal infinite cyclic subgroups. If R is regular, we have NK n ( R ) = 0 for all n ∈ Z ;(2) The abelian groups K n ( Z G ) for n ≤ − , e K ( Z G ) , and Wh( G ) vanish;(3) We get for every ring R with involution and n ∈ Z an isomorphism H n ( BG ; L h−∞i ( R )) ∼ = −→ L h−∞i n ( RG ) . For every j ∈ Z , j ≤ , and n ∈ Z , the natural map L h j i n ( Z G ) ∼ = −→ L h−∞i n ( Z G ) is bijective;(4) We get for every n ∈ Z an isomorphism K top n ( BG ) ∼ = −→ K top n ( C ∗ r ( G )) . Proof.
See [82, Theorem 1.2]. (cid:3)
Theorem 12.2.
Suppose that G satisfies the Baum-Connes Conjecture 8.2 and K -theoretic Farrell-Jones Conjecture 7.3 with coefficients in the ring C . Let con( G ) f be the set of conjugacy classes ( g ) of elements g ∈ G of finite order. We denote for g ∈ G by C G h g i the centralizer in G of the cyclic subgroup generated by g .Then we get the following commutative square, whose horizontal maps are iso-morphisms and complexifications of assembly maps, and whose vertical maps areinduced by the obvious change of theory homomorphisms L p + q = n L ( g ) ∈ con( G ) f H p ( C G h g i ; C ) ⊗ Z K q ( C ) ∼ = / / (cid:15) (cid:15) K n ( C G ) ⊗ Z C (cid:15) (cid:15) L p + q = n L ( g ) ∈ con( G ) f H p ( C G h g i ; C ) ⊗ Z K top q ( C ) ∼ = / / K top n ( C ∗ r ( G )) ⊗ Z C Proof.
See [67, Theorem 0.5]). (cid:3)
The challenge of extending equivariant homotopy theory toinfinite groups
We have seen that it is important to study equivariant homology and homotopyalso for groups which are not necessarily finite. In particular equivariant KK-theoryhas developed into a whole industry, which, however, does not really take the pointof view of spectra instead of K -groups and cycles into account. So we encounter Problem 13.1.
Extend equivariant homotopy theory for finite groups to infinitegroups, at least in the case of proper G -actions. A few first steps are already in the literature. We have already explained thenotion of an equivariant homology and the existence of equivariant Chern charac-ters, see [67], where the global point of view enters. There is also a cohomologicalversion, see [71]. Topological K -theory has systematically been studied in [76, 77],and an attempt of defining Burnside rings and equivariant cohomotopy for proper G -spaces is presented in [70]. These do include multiplicative structures.An important ingredient in equivariant homotopy theory for finite groups is tostabilize with unit spheres in finite-dimensional orthogonal representations. How-ever, there are infinite groups such that any finite-dimensional representation istrivial and therefore one has to stabilize with equivariant vector bundles, see [70,Remark 6.17]. Or one may have to pass even to Hilbert bundles and equivariantFredholm operator between these, see [92] and also [77].There are various interesting pairings on the group level in the literature, suchas Kasparov products, the action of Swan groups on algebraic K-theory and so on.They all should be implemented on the spectrum level. So a systematical study ofhigher structures for equivariant spectra over infinite groups has to be carried outand one has to find the right equivariant homotopy category. This applies also tomultiplicative structures and smash products. First steps will be presented in [29]using orthogonal spectra. This seems to work well for topological K -theory, but isprobably not adequate for algebraic K -theory. This remark also holds for globalequivariant homotopy theory.A general description of Mackey structures and induction theorems in the senseof Dress is described in [6]. There are more sophisticated Mackey structure andtransfers in the equivariant homotopy of finite groups, but it is not at all clearwhether and how they extend to infinite groups.Topological K -theory and the Baum-Connes Conjecture make sense and arestudied also for topological groups, e.g., reductive p -adic groups and Lie groups. SSEMBLY MAPS 29
It is conceivable that also the Farrell-Jones Conjecture has an analogue for Heckealgebras of totally disconnected groups, see [78, Conjecture 119 on page 773]. Soone can ask Problem 13.1 also for (not necessarily compact) topological groupsinstead of infinite (discrete) groups.
References [1] J. F. Adams.
Stable homotopy and generalised homology . University of Chicago Press,Chicago, Ill., 1974. Chicago Lectures in Mathematics.[2] A. Bartels. On proofs of the Farrell-Jones conjecture. In
Topology and geometric grouptheory , volume 184 of
Springer Proc. Math. Stat. , pages 1–31. Springer, [Cham], 2016.[3] A. Bartels and M. Bestvina. The Farrell-Jones Conjecture for mapping class groups.Preprint, arXiv:1606.02844 [math.GT], 2016.[4] A. Bartels, S. Echterhoff, and W. L¨uck. Inheritance of isomorphism conjectures under colim-its. In Cortinaz, Cuntz, Karoubi, Nest, and Weibel, editors,
K-Theory and noncommutativegeometry , EMS-Series of Congress Reports, pages 41–70. European Mathematical Society,2008.[5] A. Bartels, F. T. Farrell, and W. L¨uck. The Farrell-Jones Conjecture for cocompact latticesin virtually connected Lie groups.
J. Amer. Math. Soc. , 27(2):339–388, 2014.[6] A. Bartels and W. L¨uck. Induction theorems and isomorphism conjectures for K - and L -theory. Forum Math. , 19:379–406, 2007.[7] A. Bartels and W. L¨uck. The Borel conjecture for hyperbolic and CAT(0)-groups.
Ann. ofMath. (2) , 175:631–689, 2012.[8] A. Bartels, W. L¨uck, and H. Reich. The K -theoretic Farrell-Jones conjecture for hyperbolicgroups. Invent. Math. , 172(1):29–70, 2008.[9] A. Bartels, W. L¨uck, and H. Reich. On the Farrell-Jones Conjecture and its applications.
Journal of Topology , 1:57–86, 2008.[10] A. Bartels, W. L¨uck, H. Reich, and H. R¨uping. K- and L-theory of group rings over GL n ( Z ). Publ. Math., Inst. Hautes ´Etud. Sci. , 119:97–125, 2014.[11] A. Bartels, W. L¨uck, and S. Weinberger. On hyperbolic groups with spheres as boundary.
Journal of Differential Geometry , 86(1):1–16, 2010.[12] A. C. Bartels. On the domain of the assembly map in algebraic K -theory. Algebr. Geom.Topol. , 3:1037–1050 (electronic), 2003.[13] H. Bass. Euler characteristics and characters of discrete groups.
Invent. Math. , 35:155–196,1976.[14] P. Baum, A. Connes, and N. Higson. Classifying space for proper actions and K -theory ofgroup C ∗ -algebras. In C ∗ -algebras: 1943–1993 (San Antonio, TX, 1993) , pages 240–291.Amer. Math. Soc., Providence, RI, 1994.[15] P. Baum, E. Guentner, and R. Willett. Expanders, exact crossed products, and the Baum-Connes conjecture. Ann. K-Theory , 1(2):155–208, 2016.[16] P. Baum and M. Karoubi. On the Baum-Connes conjecture in the real case.
Q. J. Math. ,55(3):231–235, 2004.[17] M. B¨okstedt, W. C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K -theoryof spaces. Invent. Math. , 111(3):465–539, 1993.[18] S. Cappell, A. Ranicki, and J. Rosenberg, editors.
Surveys on surgery theory. Vol. 1 . Prince-ton University Press, Princeton, NJ, 2000. Papers dedicated to C. T. C. Wall.[19] S. Cappell, A. Ranicki, and J. Rosenberg, editors.
Surveys on surgery theory. Vol. 2 . Prince-ton University Press, Princeton, NJ, 2001. Papers dedicated to C. T. C. Wall.[20] G. Carlsson and E. Pedersen. Controlled algebra and the baum-connes conjecture. in prepa-ration, 1996.[21] J. Chabert and S. Echterhoff. Permanence properties of the Baum-Connes conjecture.
Doc.Math. , 6:127–183 (electronic), 2001.[22] T. D. Cochran. Noncommutative knot theory.
Algebr. Geom. Topol. , 4:347–398, 2004.[23] T. D. Cochran, K. E. Orr, and P. Teichner. Knot concordance, Whitney towers and L -signatures. Ann. of Math. (2) , 157(2):433–519, 2003.[24] D. Crowley, W. L¨uck, and T. Macko. Surgery Theory: Foundations. book, in preparation,2019.[25] J. F. Davis and W. L¨uck. Spaces over a category and assembly maps in isomorphism con-jectures in K - and L -theory. K -Theory , 15(3):201–252, 1998.[26] J. F. Davis and W. L¨uck. The p -chain spectral sequence. K -Theory , 30(1):71–104, 2003.Special issue in honor of Hyman Bass on his seventieth birthday. Part I. [27] J. F. Davis and W. L¨uck. The topological K -theory of certain crystallographic groups. Journal of Non-Commutative Geometry , 7:373–431, 2013.[28] J. F. Davis, F. Quinn, and H. Reich. Algebraic K -theory over the infinite dihedral group: acontrolled topology approach. J. Topol. , 4(3):505–528, 2011.[29] D. Degrijse, M. Hausmann, W. L¨uck, I. Patchkoria, and S. Schwede. Proper equivariantstable homotopy theory. in preparation, 2019.[30] B. I. Dundas, T. G. Goodwillie, and R. McCarthy.
The local structure of algebraic K-theory ,volume 18 of
Algebra and Applications . Springer-Verlag London Ltd., London, 2013.[31] W. Dwyer, M. Weiss, and B. Williams. A parametrized index theorem for the algebraic K -theory Euler class. Acta Math. , 190(1):1–104, 2003.[32] S. Echterhoff, W. L¨uck, N. C. Phillips, and S. Walters. The structure of crossed products ofirrational rotation algebras by finite subgroups of SL ( Z ). J. Reine Angew. Math. , 639:173–221, 2010.[33] N.-E. Enkelmann, W. L¨uck, M. Pieper, M. Ullmann, and C. Winges. On the Farrell–Jonesconjecture for Waldhausen’s A–theory.
Geom. Topol. , 22(6):3321–3394, 2018.[34] F. T. Farrell. The Borel conjecture. In F. T. Farrell, L. G¨ottsche, and W. L¨uck, editors,
High dimensional manifold theory
Algebraic and geometric topology (Proc.Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1 , Proc. Sympos. PureMath., XXXII, pages 325–337. Amer. Math. Soc., Providence, R.I., 1978.[36] F. T. Farrell and L. E. Jones. Rigidity in geometry and topology. In
Proceedings of theInternational Congress of Mathematicians, Vol. I, II (Kyoto, 1990) , pages 653–663, Tokyo,1991. Math. Soc. Japan.[37] F. T. Farrell and L. E. Jones. Isomorphism conjectures in algebraic K -theory. J. Amer.Math. Soc. , 6(2):249–297, 1993.[38] F. T. Farrell, L. E. Jones, and W. L¨uck. A caveat on the isomorphism conjecture in L -theory. Forum Math. , 14(3):413–418, 2002.[39] T. Farrell, W. L¨uck, and W. Steimle. Approximately fibering a manifold over an asphericalone.
Math. Ann. , 370(1-2):669–726, 2018.[40] S. Ferry, W. L¨uck, and S. Weinberger. On the stable Cannon Conjecture. Preprint,arXiv:1804.00738 [math.GT], 2018.[41] S. C. Ferry, A. A. Ranicki, and J. Rosenberg, editors.
Novikov conjectures, index theoremsand rigidity. Vol. 1 . Cambridge University Press, Cambridge, 1995. Including papers fromthe conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach,September 6–10, 1993.[42] S. C. Ferry, A. A. Ranicki, and J. Rosenberg, editors.
Novikov conjectures, index theoremsand rigidity. Vol. 2 . Cambridge University Press, Cambridge, 1995. Including papers fromthe conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach,September 6–10, 1993.[43] I. Hambleton and E. K. Pedersen. Identifying assembly maps in K - and L -theory. Math.Ann. , 328(1-2):27–57, 2004.[44] F. Hebestreit, W. L¨uck, M. Land, and O. Randal-Williams. A vanishing theorem for tauto-logical classes of aspherical manifolds. Preprint, arXiv:1705.06232 [math.AT], 2017.[45] N. Higson. The Baum-Connes conjecture. In
Proceedings of the International Congress ofMathematicians, Vol. II (Berlin, 1998) , pages 637–646 (electronic), 1998.[46] N. Higson and G. Kasparov. E -theory and KK -theory for groups which act properly andisometrically on Hilbert space. Invent. Math. , 144(1):23–74, 2001.[47] N. Higson, V. Lafforgue, and G. Skandalis. Counterexamples to the Baum-Connes conjec-ture.
Geom. Funct. Anal. , 12(2):330–354, 2002.[48] N. Higson and J. Roe. Mapping surgery to analysis. I. Analytic signatures. K -Theory ,33(4):277–299, 2005.[49] N. Higson and J. Roe. Mapping surgery to analysis. II. Geometric signatures. K -Theory ,33(4):301–324, 2005.[50] N. Higson and J. Roe. Mapping surgery to analysis. III. Exact sequences. K -Theory ,33(4):325–346, 2005.[51] M. Joachim. K -homology of C ∗ -categories and symmetric spectra representing K -homology. Math. Ann. , 327(4):641–670, 2003.
SSEMBLY MAPS 31 [52] D. Juan-Pineda and I. J. Leary. On classifying spaces for the family of virtually cyclicsubgroups. In
Recent developments in algebraic topology , volume 407 of
Contemp. Math. ,pages 135–145. Amer. Math. Soc., Providence, RI, 2006.[53] H. Kammeyer, W. L¨uck, and H. R¨uping. The Farrell–Jones conjecture for arbitrary latticesin virtually connected Lie groups.
Geom. Topol. , 20(3):1275–1287, 2016.[54] G. G. Kasparov. Operator K -theory and its applications: elliptic operators, group repre-sentations, higher signatures, C ∗ -extensions. In Proceedings of the International Congressof Mathematicians, Vol. 1, 2 (Warsaw, 1983) , pages 987–1000, Warsaw, 1984. PWN.[55] G. G. Kasparov. Equivariant KK -theory and the Novikov conjecture. Invent. Math. ,91(1):147–201, 1988.[56] G. G. Kasparov. K -theory, group C ∗ -algebras, and higher signatures (conspectus). In Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) , pages 101–146. Cambridge Univ. Press, Cambridge, 1995.[57] D. Kasprowski, M. Ullmann, C. Wegner, and C. Winges. The A -theoretic Farrell-Jonesconjecture for virtually solvable groups. Bull. Lond. Math. Soc. , 50(2):219–228, 2018.[58] M. Kreck and W. L¨uck.
The Novikov conjecture: Geometry and algebra , volume 33 of
Oberwolfach Seminars . Birkh¨auser Verlag, Basel, 2005.[59] P. K¨uhl, T. Macko, and A. Mole. The total surgery obstruction revisited.
M¨unster J. Math. ,6:181–269, 2013.[60] V. Lafforgue. The Baum-Connes conjecture with coefficients for hyperbolic groups. (Laconjecture de baum-connes `a coefficients pour les groupes hyperboliques.).
J. Noncommut.Geom. , 6(1):1–197, 2012.[61] M. Land. The analytical assembly map and index theory.
J. Noncommut. Geom. , 9(2):603–619, 2015.[62] M. Land and T. Nikolaus. On the relation between K - and L -theory of C ∗ -algebras. Math.Ann. , 371(1-2):517–563, 2018.[63] M. Langer and W. L¨uck. Topological K -theory of the group C ∗ -algebra of a semi-directproduct Z n ⋊ Z /m for a free conjugation action. J. Topol. Anal. , 4(2):121–172, 2012.[64] X. Li and W. L¨uck. K -theory for ring C ∗ -algebras – the case of number fields with higherroots of unity. Journal of Topology and Analysis 4 (4) , pages 449–479, 2012.[65] W. L¨uck.
Transformation groups and algebraic K -theory , volume 1408 of Lecture Notes inMathematics . Springer-Verlag, Berlin, 1989.[66] W. L¨uck. A basic introduction to surgery theory. In F. T. Farrell, L. G¨ottsche, and W. L¨uck,editors,
High dimensional manifold theory K -and L -theory. J. Reine Angew. Math. , 543:193–234, 2002.[68] W. L¨uck. L -Invariants: Theory and Applications to Geometry and K -Theory , volume 44of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveysin Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of ModernSurveys in Mathematics] . Springer-Verlag, Berlin, 2002.[69] W. L¨uck. The relation between the Baum-Connes conjecture and the trace conjecture.
In-vent. Math. , 149(1):123–152, 2002.[70] W. L¨uck. The Burnside ring and equivariant stable cohomotopy for infinite groups.
PureAppl. Math. Q. , 1(3):479–541, 2005.[71] W. L¨uck. Equivariant cohomological Chern characters.
Internat. J. Algebra Comput. , 15(5-6):1025–1052, 2005.[72] W. L¨uck. K - and L -theory of the semi-direct product of the discrete 3-dimensional Heisen-berg group by Z / Geom. Topol. , 9:1639–1676 (electronic), 2005.[73] W. L¨uck. Survey on classifying spaces for families of subgroups. In
Infinite groups: geo-metric, combinatorial and dynamical aspects , volume 248 of
Progr. Math. , pages 269–322.Birkh¨auser, Basel, 2005.[74] W. L¨uck. Survey on aspherical manifolds. In A. Ran, H. te Riele, and J. Wiegerinck, editors,
Proceedings of the 5-th European Congress of Mathematics Amsterdam 14 -18 July 2008 ,pages 53–82. EMS, 2010.[75] W. L¨uck. Isomorphism Conjectures in K - and L -theory. in preparation, 2020.[76] W. L¨uck and B. Oliver. Chern characters for the equivariant K -theory of proper G -CW-complexes. In Cohomological methods in homotopy theory (Bellaterra, 1998) , pages 217–247.Birkh¨auser, Basel, 2001.[77] W. L¨uck and B. Oliver. The completion theorem in K -theory for proper actions of a discretegroup. Topology , 40(3):585–616, 2001. [78] W. L¨uck and H. Reich. The Baum-Connes and the Farrell-Jones conjectures in K - and L -theory. In Handbook of K -theory. Vol. 1, 2 , pages 703–842. Springer, Berlin, 2005.[79] W. L¨uck, H. Reich, J. Rognes, and M. Varisco. Assembly maps for topological cyclic ho-mology of group algebras. Preprint, arXiv:1607.03557 [math.KT], to appear in Crelle, 2016.[80] W. L¨uck, H. Reich, J. Rognes, and M. Varisco. Algebraic K-theory of group rings and thecyclotomic trace map. Adv. Math. , 304:930–1020, 2017.[81] W. L¨uck, H. Reich, and M. Varisco. Commuting homotopy limits and smash products. K -Theory , 30(2):137–165, 2003. Special issue in honor of Hyman Bass on his seventiethbirthday. Part II.[82] W. L¨uck and D. Rosenthal. On the K - and L -theory of hyperbolic and virtually finitelygenerated abelian groups. Forum Math. , 26(5):1565–1609, 2014.[83] W. L¨uck and W. Steimle. Splitting the relative assembly map, Nil-terms and involutions.
Ann. K-Theory , 1(4):339–377, 2016.[84] M. Matthey and G. Mislin. Equivariant K -homology and restriction to finite cyclic sub-groups. Preprint, 2003.[85] I. Mineyev and G. Yu. The Baum-Connes conjecture for hyperbolic groups. Invent. Math. ,149(1):97–122, 2002.[86] A. S. Miˇsˇcenko and A. T. Fomenko. The index of elliptic operators over C ∗ -algebras. Math-ematics of the USSR-Izvestiya , 15(1):87–112, 1980.[87] G. Mislin and A. Valette.
Proper group actions and the Baum-Connes conjecture . AdvancedCourses in Mathematics. CRM Barcelona. Birkh¨auser Verlag, Basel, 2003.[88] P. D. Mitchener. C ∗ -categories, groupoid actions, equivariant KK -theory, and the Baum-Connes conjecture. J. Funct. Anal. , 214(1):1–39, 2004.[89] T. Nikolaus and P. Scholze. On topological cyclic homology. Preprint, arXiv:1707.01799[math.AT], 2017.[90] H. Oyono-Oyono. Baum-Connes Conjecture and extensions.
J. Reine Angew. Math. ,532:133–149, 2001.[91] H. Oyono-Oyono. Baum-Connes conjecture and group actions on trees. K -Theory ,24(2):115–134, 2001.[92] N. C. Phillips. Equivariant K -theory for proper actions . Longman Scientific & Technical,Harlow, 1989.[93] P. Piazza and T. Schick. Bordism, rho-invariants and the Baum-Connes conjecture. J. Non-commut. Geom. , 1(1):27–111, 2007.[94] F. Quinn. B (TOP n ) ∗∗∗ bt ∗∗ and the surgery obstruction. Bull. Amer. Math. Soc. , 77:596–600, 1971.[95] F. Quinn. Assembly maps in bordism-type theories. In
Novikov conjectures, index theoremsand rigidity, Vol. 1 (Oberwolfach, 1993) , pages 201–271. Cambridge Univ. Press, Cambridge,1995.[96] A. A. Ranicki.
Exact sequences in the algebraic theory of surgery . Princeton UniversityPress, Princeton, N.J., 1981.[97] A. A. Ranicki.
Algebraic L -theory and topological manifolds . Cambridge University Press,Cambridge, 1992.[98] A. A. Ranicki. Lower K - and L -theory . Cambridge University Press, Cambridge, 1992.[99] J. Rosenberg. C ∗ -algebras, positive scalar curvature and the Novikov conjecture. III. Topol-ogy , 25:319–336, 1986.[100] J. Rosenberg. Analytic Novikov for topologists. In
Novikov conjectures, index theorems andrigidity, Vol. 1 (Oberwolfach, 1993) , pages 338–372. Cambridge Univ. Press, Cambridge,1995.[101] J. Rosenberg. Structure and applications of real C ∗ -algebras. In Operator algebras and theirapplications , volume 671 of
Contemp. Math. , pages 235–258. Amer. Math. Soc., Providence,RI, 2016.[102] H. R¨uping. The Farrell–Jones conjecture for S -arithmetic groups. J. Topol. , 9(1):51–90,2016.[103] T. Schick. A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture.
Topology , 37(6):1165–1168, 1998.[104] T. Schick. Index theory and the Baum-Connes conjecture. In
Geometry Seminars. 2001-2004(Italian) , pages 231–280. Univ. Stud. Bologna, Bologna, 2004.[105] T. Schick. Real versus complex K -theory using Kasparov’s bivariant KK -theory. Algebr.Geom. Topol. , 4:333–346, 2004.[106] S. Schwede.
Global homotopy theory , volume 34 of
New Mathematical Monographs . Cam-bridge University Press, Cambridge, 2018.[107] S. Stolz. Manifolds of positive scalar curvature. In T. Farrell, L. G¨ottsche, and W. L¨uck,editors,
High dimensional manifold theory , number 9 in ICTP Lecture Notes, pages 661–708.
SSEMBLY MAPS 33
Arch. Math. (Basel) , 23:307–317,1972.[109] A. Valette.
Introduction to the Baum-Connes conjecture . Birkh¨auser Verlag, Basel, 2002.From notes taken by Indira Chatterji, With an appendix by Guido Mislin.[110] W. Vogell. Algebraic K -theory of spaces, with bounded control. Acta Math. , 165(3-4):161–187, 1990.[111] W. Vogell. Boundedly controlled algebraic K -theory of spaces and its linear counterparts. J. Pure Appl. Algebra , 76(2):193–224, 1991.[112] F. Waldhausen. Algebraic K -theory of topological spaces. I. In Algebraic and geometrictopology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1 , Proc.Sympos. Pure Math., XXXII, pages 35–60. Amer. Math. Soc., Providence, R.I., 1978.[113] F. Waldhausen. Algebraic K -theory of spaces. In Algebraic and geometric topology (NewBrunswick, N.J., 1983) , pages 318–419. Springer-Verlag, Berlin, 1985.[114] F. Waldhausen. Algebraic K -theory of spaces, concordance, and stable homotopy theory. In Algebraic topology and algebraic K -theory (Princeton, N.J., 1983) , pages 392–417. Prince-ton Univ. Press, Princeton, NJ, 1987.[115] F. Waldhausen. An outline of how manifolds relate to algebraic K -theory. In Homotopytheory (Durham, 1985) , volume 117 of
London Math. Soc. Lecture Note Ser. , pages 239–247. Cambridge Univ. Press, Cambridge, 1987.[116] F. Waldhausen, B. Jahren, and J. Rognes.
Spaces of PL manifolds and categories of simplemaps , volume 186 of
Annals of Mathematics Studies . Princeton University Press, Princeton,NJ, 2013.[117] C. T. C. Wall.
Surgery on compact manifolds , volume 69 of
Mathematical Surveys andMonographs . American Mathematical Society, Providence, RI, second edition, 1999. Editedand with a foreword by A. A. Ranicki.[118] C. Wegner. The K -theoretic Farrell-Jones conjecture for CAT(0)-groups. Proc. Amer. Math.Soc. , 140(3):779–793, 2012.[119] C. Wegner. The Farrell-Jones conjecture for virtually solvable groups.
J. Topol. , 8(4):975–1016, 2015.[120] M. Weiss and B. Williams. Assembly. In
Novikov conjectures, index theorems and rigidity,Vol. 2 (Oberwolfach, 1993) , pages 332–352. Cambridge Univ. Press, Cambridge, 1995.[121] M. Weiss and B. Williams. Automorphisms of manifolds. In
Surveys on surgery theory, Vol.2 , volume 149 of
Ann. of Math. Stud. , pages 165–220. Princeton Univ. Press, Princeton, NJ,2001.
E-mail address : [email protected] URL :