A USER'S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA 1
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1. AJH The functor K A OK D x 0 1 A from the category of G CW com plexes to the category of spectra is G homotopy invariant and G excisive 4 Section 5 As in the unequivariant case the proof of this fact makes use of Karoubi filtrations and Kilenberg swindles In this sense K A defines a G equivariant homology theory The corresponding equivariant homology groups are denoted by H X K 4 7 K X A The collapse map X pt induces the continuously controlled assembly map AF K X A K pt A which yields a map on homology AS H amp X Ky gt H amp pt Ky K A G In light of Example 2 2 this construction interprets assembly as a forget control map It is also worth noting that the continuously controlled assembly map is a map of fixed point spectra since K commutes with taking fixed sets That is if G acts on an additive category B then there is an induced action of on K B and the fixed point set of this spectrum K B is equivalent to the spectrum K B This implies that AF is the map induced by the equivariant map of spectra K X A K pt A The continuously controlled version of the assembly map is homotopy equivalent to the one defined by Bartels and Reich who used the Davis Ltick machinery for constructing assembly maps 7 16 In the case A Fr Hambleton and Pedersen used Lemma 3 3 to identify the Davis Liick2. A Yt B X AX It is an exercise to show that when C is a closed subset of X the inclusion func tor B C A B X AN produces an equivalence of categories Therefore BX AV AETS BX A ay and BNA AT ESB UFA a One also checks that B X A Y B X U Y A A Y t4 has an inverse induced by projection Thus diagram 4 becomes 5 B X AY A Kx B X A X B X A X Vx B Y A B X UY A ZVO gt B X UY A X Vx11 Applying K to 5 yields a commutative diagram of spectra in which each row is a fibration Since fibration sequences are also cofibration sequences in the category of spectra the two rows of the induced diagram are cofibration sequences with equivalent cofibers Therefore diagram 3 is a homotopy co Cartesian square which means it is a homotopy Cartesian square since we are working in the category of spectra The above argument can be found in 17 Corollary 9 3 15 p 48 and 4 Proposition 4 3 For a proof of homotopy invariance the reader is referred to 4 Section 5 The proof employs a trick dating back to Pedersen and Weibel 22 that makes clever use of Eilenberg swindles 3 EQUIVARIANT HOMOLOGY AND THE ASSEMBLY MAP Given a discrete group G and a ring R the classical assembly map in algebraic K theory introduced by Loday 21 is a map that relates the K theory of the group ring R G to the homology of BG with coefficients in the K theory spectrum of R In the re
3. has trivial K theory The idea for constructing the necessary endofunctor is to shift the objects and morphisms towards 1 Let A and B be objects of B 0 1 1 A and let 6 A B be a morphism Let S B 0 1 1 A B 0 1 1 A be the endofunctor defined by S A z A 1 SCP 03624 A2s 1 gt Bort Now define the endofunctor X B 0 1 1 A B 0 1 1 A by A Bs A n gt 1 Ele Bsr 4 n gt 1 For each object A there is a continuously controlled isomorphism U4 A S A defined by fida if s 2t 1 Ua l 0 otherwise i The desired natural equivalence 7 id PE amp is given by n A n gt o Usna There fore K B 0 1 1 A is weakly contractible i e it has trivial homotopy groups At first glance it appears that B X x 0 1 X x 1 A admits an Eilenberg swindle for any X by sliding the objects and morphisms towards 1 as was done in Example 2 3 But this does not work in general Remember that the continuous control condition on morphisms says that the components of a morphism must become shorter and shorter as they approach 1 This must happen in the X direction as well as in the 0 1 direction For example consider the two point space S 1 1 By the continuous control condition the components of a morphism in B S x 0 1 S x 1 A are zero between 1 s and 1 t if s or t is sufficiently close to 1 Therefore a morphism with a n
4. assembly map with the continuously controlled assembly map 15 They also showed that both versions were equivalent to the classical definition of the assembly map The Farrell Jones conjecture with coefficients predicts that the assembly map is an iso morphism for all coefficients A when X Eycy G where VCyc is the family of virtually 12 D ROSENTHAL cyclic subgroups of G t This is a very strong statement For example if the Farrell Jones conjecture with coefficients is true for a group G then the Fibered Farrell Jones conjec ture is true for G 7 Furthermore it would imply that the Farrell Jones conjecture with coefficients is true for every subgroup of G It also behaves well with respect to extensions see 16 By the universal property there is a map ErinG Eycy G where Fin is the family of finite subgroups of G Bartels 2 proved that for every discrete group G the induced map on homology HS E inG K4 gt Ho Eyey G Ky is a split injection He showed this for A Fr but the proof works in the general case as well As a result the Farrell Jones conjecture implies that the assembly map for the family of finite subgroups H amp E inG Ky K A G should be a split injection for every discrete group G 4 PROVING ISOMORPHISM AND INJECTIVITY RESULTS In this section we present an alternate description of the continuously controlled assembly map that is useful for proofs Consider the following commutati
5. implies that there are only finitely many t 0 1 with Aes 0 where e is the identity of G By equivariance Aig 4 A e for every t 0 1 Therefore A is a finitely generated free R G module Since the continuous control condition is vacuous equivariance tells us that the morphisms in D pt x 0 1 Fr o are just R G homomorphisms Thus D pt x 0 1 Fr g is equivalent to Fag In fact D X x 0 1 Fr g is equivalent to Frig for every X because of the relatively G compact condition on objects As in Example 2 9 since D pt x 0 1 Fr is flasque QK D pt x 0 1 Fr is weakly homotopy equivalent to K Fjq A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA 11 More generally we have the following lemma about orbits G H Lemma 3 3 QK D G H x 0 1 A gt K A H Proof Equivariance tells us that the objects of DO G H x 0 1 A are determined by their value over the points in eH x 0 1 Since the isotropy at every point in eH x 0 1 is H the objects over eH x 0 1 are objects in A H Since G H is a discrete space and we are taking germs at 1 the morphisms in D G H x 0 1 A g cannot jump between different points of G H compare with Lemma 2 7 Combining this with the equivariance of morphisms we see that D G H x 0 1 A is equivalent to Dt pt x 0 1 A H Since Dt pt x 0 1 A H is flasque QK D G H x 0 1 A K7
6. invariant subcomplex of X The category D X Y A has objects A A consisting of collections of objects of A indexed by G x X Y such that the support of A supp A z G x X Y A 0 is 1 locally finite in G x X Y and 2 relatively G compact in G x X A morphism A gt B is a collection 7 A gt By of morphisms in A such that 1 for every z Gx X Y the set z Gx X Y 0 or G2 0 is finite and 2 A B is continuously controlled at Y which means that for every y Y and every G invariant neighborhood U C X of y there is a G invariant neighborhood V CX of y such that 2 0 and 2 0 whenever z G x V and 2 G x U Adding a factor of G to the control space an idea due to Pedersen creates an interesting action of G on D X Y A The right G action on D X Y A is induced by the diagonal action of G on G x X and the right G action on A It is given by 10 D ROSENTHAL In the corresponding fixed point category D X Y A every object A and every mor phism satisfy A g A g Agz and g g 2 for every g in G and every z z in G x X Y If A Fr with the trivial G action then the objects of D X x 0 1 Fr are free R G modules and the morphisms are R G homomorphisms Furthermore the direct sum of the pieces of an object A over an entire orbit B yec Agz Pec Agr for some x in X Y is a finitely gen
7. A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA DAVID ROSENTHAL ABSTRACT Continuously controlled algebra is an important tool for proving the Farrell Jones conjecture and the Novikov conjecture The purpose of this expository article is to present an accessible introduction to continuously controlled algebra and the continuously controlled version of assembly maps 1 INTRODUCTION Controlled algebra was first considered by Connell and Hollingsworth in 12 and its first major applications were developed by Quinn in 24 Continuously controlled algebra was introduced by Anderson Connolly Ferry and Pedersen 1 and was used by Carlsson and Pedersen 10 11 to analyze assembly maps in algebraic K and L theory Carlsson and Pedersen used this theory to prove the Novikov conjecture for a large class of groups by showing that the assembly map was a split injection Motivated by the groundbreaking work of Farrell and Jones 14 Bartels L ck and Reich 5 have recently used the continu ously controlled version of the assembly map in their proof of the K theoretic Farrell Jones conjecture for word hyperbolic groups and Bartels and L ck have announced the analogous result in L theory These two isomorphism results combine to prove the Borel conjecture for these groups The aim of this article is to provide a friendly introduction to continuously controlled algebra hopefully making the general framework of this proof technique a little more
8. RA 13 category vanishes when X Eycy G Because of the nice inheritance properties of the assembly map with coefficients their result proves the K theoretic Farrell Jones conjecture for all subgroups of finite products of word hyperbolic groups Recently Bartels and Liick have announced the corresponding result in algebraic L theory Combining these two theorems proves the Borel conjecture for all subgroups of finite products of word hyperbolic groups The ultimate result about the assembly map is that it is an isomorphism Proving that the obstruction category vanishes achieves this goal but it is a difficult task However with a bit less one can still prove that the assembly map is a split injection using a trick known as the descent principle Using the fact that the continuously controlled assembly map is a map of fixed spectra one can employ homotopy fixed point sets to prove the following theorem a proof can be found in 8 Theorem 4 1 The Descent Principle Let G be a discrete group A be an additive G category and E inG be a finite dimensional G CW complex Assume that the K theory of the category DY E inG x 0 1 A vanishes for every finite subgroup H of G Then AG HC ErinG KI gt K A G is a split injection Consider the case when G is torsion free Then the descent principle says that one can prove the assembly map is a split injection by proving that it is an equivalence unequiv ariantly This is the origi
9. X Y A B X Y A is a Karoubi filtration sequence where W is an open subset of Y Example 2 9 An instance of 1 that is useful for studying assembly maps is the sequence B X x 0 1 X x 1 AJo gt B X x 0 1 X x 1 A gt B X x 0 1 X x 1 A When X is a point the associated long exact sequence implies that Kns B 0 1 1 A Kn B 0 1 1 Alo Ka A since K7 B 0 1 1 A is weakly contractible Therefore QK7 B 0 1 1 A t is weakly homotopy equivalent to K7 A The functor QK B x 0 1 x 1 A t4 from the category of CW complexes to the category of spectra is homotopy invariant and excisive thereby yielding the gen eralized homology theory H X K y 7 QIKK B x 0 1 x 1 A t4 By Example 2 9 its value at a point is the K theory of A Karoubi filtrations are needed to establish the fact that H X K is in fact an homology theory Excision is proved by showing that the pushout diagram of CW complexes XAY X Y XUY induces a homotopy Cartesian square 3 K B X A Y AYA K B X At KK BY AR 03 gt K 9 BUX UY Ae 8 D ROSENTHAL where B x 0 1 x 1 A is denoted by B A To prove this consider the following commutative diagram of Karoubi filtrations induced by inclusion 4 BA eds B X A X BRUKA 0 gt B X UY A O xt gt B X UY
10. a closed subcomplex and A be a small additive category A subset K C X is called relatively compact if the closure of K in X is compact A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA 3 It is called locally finite if for every point x in K there is a neighborhood U of x such that UNK r Definition 2 1 The continuously controlled category B X Y A is an additive category whose objects A A are collections of objects of A indexed by the points in the space X Y such that the support of A supp A x X Y A 0 is 1 locally finite in X Y and 2 relatively compact in X A morphism A gt B is a collection Az gt Bw of morphisms in A such that 1 for every x X Y the set x X Y 6 0 or 6 0 is finite and 2 6 A gt B is continuously controlled at Y That is for every y Y and every neighborhood U C X of y there is a neighborhood V C X of y such that 6 0 and 0 whenever x V and a U The continuously controlled category depends functorially on the CW pair X Y for a proof see for example 4 Section 3 3 A map f X Y X Y induces a functor fe B X Y A gt BX Y A defined by f A y Presen A Given a morphism A Bin B X Y A f Q f A gt f B is defined in the obvious way Example 2 2 Forget control Consider the categories B X x 0 1 X x 1 A and B CX A where CX X x 0 1 X x 1 i
11. accessible To achieve this modest goal we focus on the continuously controlled algebra approach to equivariant homology theories and assembly maps The objects of study in this theory are additive categories of so called geometric mod ules The notation used in the literature varies from paper to paper and is often quite involved We hope the presentation here will serve to demystify these categories and to Date January 20 2008 2000 Mathematics Subject Classification 18F25 Key words and phrases K theory assembly map controlled algebra 1 2 D ROSENTHAL provide the reader with an understanding of the underlying concepts Continuously con trolled categories are introduced without group actions and are illustrated with several examples Group actions are then incorporated into the definition so that an equivari ant homology theory and corresponding assembly map can be defined The presentation focuses on algebraic K theory To study algebraic L theory one has to be careful with in volutions but the treatment is virtually the same We conclude with a general discussion of how this theory has been used to prove isomorphism and split injectivity results The author would like to thank Marco Varisco for useful comments and discussions The author would also like to thank Erik K Pedersen for his expertise and for his many invaluable insights 2 CONTINUOUSLY CONTROLLED ALGEBRA Let us begin by recalling some categorical ter
12. ch LEquivariant covers for hyperbolic groups arXiv math GT 0609685 A Bartels and H Reich Coefficients for the Farrell Jones Conjecture Adv Math 209 1 2007 337 362 A Bartels and D Rosenthal On the algebraic K and L theory of groups with finite asymptotic dimension To appear in J Reine Agnew Math arXiv math KT 0605088 M Cardenas and E K Pedersen On the Karoubi filtration of a category K theory 12 1997 165 191 G Carlsson and E K Pedersen Controlled algebra and the Novikov conjectures for K and L theory Topology 34 1995 731 758 G Carlsson and E K Pedersen Cech homology and the Novikov conjectures for K and L theory Math Scand 82 1998 5 47 E H Connell and J Hollingsworth Geometric groups and Whitehead torsion Trans Amer Math Soc 140 1969 161 181 J Davis and W L ck Spaces over a category and assembly maps in isomorphism conjectures in K and L theory K theory 15 1998 201 252 F T Farrell and L E Jones Isomorphism conjectures in algebraic K theory J Amer Math Soc 6 1993 no 3 249 297 I Hambleton and E K Pedersen Identifying assembly maps in K and L theory Math Ann 328 2004 27 57 I Hambleton E K Pedersen and D Rosenthal Assembly maps for group extensions in K and L theory arXiv math KT 07090437 N Higson E K Pedersen and J Roe C algebras and controlled topology K theory 11 1997 209 239 W C Hsiang Geometric applicat
13. cts of A by the letters A through F and the objects of U by the letters U through W Then U is said to be A filtered if every object U has a family of decompositions U Ea 6 Ua where i the decomposition forms a filtered poset under the partial order in which EF U lt Eg Ug whenever Ug C Ua and Ea C Eg ii every map A U factors through EF for some a iii every map U A factors through Ea for some a iv for each U and V the filtration on U V is equivalent to the sum of the filtrations U Ea U and V Fs Va that is U 6 V Ea Fs Ua Vp Karoubi also defined the quotient category U A to have the same objects as U but morphisms Y U V are identified if their difference y factors through A The key fact is that the induced sequence K A gt KS U gt K U A is a homotopy fibration of spectra which yields a long exact sequence of homotopy groups For this reason Karoubi filtrations play a major role in the subject It is straightforward to check that is a Karoubi filtration where C is a closed subset of Y Furthermore the corresponding quotient category is precisely the germ category B X Y A Therefore the sequence 1 B X Y A o gt B X Y A gt B X Y A S A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA 7 yields a homotopy fibration of spectra after applying K Similarly it is an exercise to show that 2 B X Y AW gt B
14. erated free R G module Thus we can think of objects in D X Y Fp as being built out of finitely generated free R G modules where x is a point in X In particular notice that DS pt 0 FR is equivalent to Fric the category of finitely generated free R G modules For this reason the category DC pt 0 A is denoted by AJG As in the unequivariant case D X Y A depends functorially on the G CW pair X Y 4 Section 3 3 In order to construct an equivariant homology theory and an assembly map we will work with the category D X x 0 1 X x 1 A where X isa G CW complex For brevity denote this category by D X x 0 1 A Let D X x 0 1 A p be the full subcategory of D X x 0 1 A on objects A such that the intersection of X x 1 with the closure of supp A in X x 0 1 is the empty set Notice that as in Example 2 9 the quotient category which we denote by D X x 0 1 A is a germ category The objects are the same as in D X x 0 1 A but morphisms are identified if they agree close to G x X x 1 i e on the complement of a neighborhood of Gx X x 0 It is not difficult to check that taking germs and taking fixed categories commute Furthermore there is a corresponding Karoubi filtration sequence D X x 0 1 A g DE X x 0 1 A DE X x 0 1 A Example 3 2 Let A be an object in D pt x 0 1 Fpr o Since objects have empty support at infinity the local finiteness condition
15. gories B S x 0 1 S x 1 A and B 0 1 1 A Ue B 0 1 1 A 4 are equivalent The reason is that we are taking germs at 1 which implies that every morphism has a representative that does not jump between levels What is meant by this is the following Let be a morphism in B S x 0 1 S x 1 A The continuous control condition implies that there is a neighborhood V C 9 x 0 1 of the point 1 1 such that ban 0 por whenever 1 t is in V Now define a morphism w such that ve ae ae for all s and t and Y otherwise Therefore and w represent the same morphism in the germ category B S x 0 1 S x 1 A S x and the components of y are zero between points in 1 x 0 1 and 1 x 0 1 More generally we have Lemma 2 7 IfT is a discrete space then B L x 0 1 T x 1 A 7 4 amp CB B 0 1 1 A 6 D ROSENTHAL Sketch of proof As with the two point space since we are taking germs at 1 every mor phism has a representative that does not jump between levels The reason for the direct sum is the relative compactness condition which implies that each object in B T x 0 1 T x 1 A T can only be non zero for finitely many points in T The categories from Definitions 2 4 and 2 6 have a special relationship In 20 Karoubi introduced the following notion Definition 2 8 Let A bea full subcategory of an additive category U Denote the obje
16. ions of algebraic K theory Proceedings of the ICM Vols 1 2 Warsaw 1983 P W N Warsaw 1984 99 118 L Ji The integral Novikov conjectures for linear groups containing torsion elements Preprint M Karoubi Foncteur derivees et K theorie Lecture Notes in Mathematics vol 136 Springer Verlag Berlin New York 1970 J L Loday K th orie alg brique et repr sentations de groupes Ann Sci Ecole Norm Sup 4 9 1976 309 377 E K Pedersen and C Weibel A non connective delooping of algebraic K theory Algebraic and Geometric Topology Rutgers 1983 Lecture Notes in Mathematics vol 1126 Springer Berlin 1985 166 181 E K Pedersen and C Weibel K theory homology of spaces Algebraic Topology Arcata 1986 Lectures Notes in Mathematics vol 1370 Springer Verlag Berlin New York 1989 346 361 F Quinn Ends of maps I Invent Math 68 1982 353 424 D Rosenthal Splitting with continuous control in algebraic K theory K theory 32 2004 139 166 DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE ST JOHN S UNIVERSITY 8000 UTOPIA Pkwy JAMAICA NY 11439 USA E mail address rosenthd st johns edu
17. minology Two functors F and G between categories A and B are naturally equivalent if there is a natural transformation from F to G that is an isomorphism for every object in A Two categories A and B are equivalent if there are functors F A B and G B gt A such that FG is naturally equivalent to idg and GF is naturally equivalent to idy An additive category A is a small category in which every hom set i e the set of mor phisms between two objects is an abelian group morphism composition is bilinear there is a zero object and for every finite collection of objects A A in A the biproduct A19 An is an object of A a biproduct is both a product and a coproduct An additive functor between additive categories A and B is a functor that is a group homomorphism for every hom set in A Two additive categories are equivalent when they are equivalent by additive functors Let A be a small additive category and let K7 A denote the non connective K theory spectrum associated with the symmetric monoidal category obtained from A by restricting to isomorphisms That is 7 K A K A for every integer n In particular if Fr is the category of finitely generated free R modules then 7 K Fr Kn R for every integer n K7 is a functor from the category of small additive categories to the category of spectra For a more detailed description of this functor see 22 9 Let X be a CW complex Y C X be
18. nal version of the descent principle It was used by Carlsson and Pedersen in 10 to prove the Novikov conjecture for torsion free groups whose universal space EG satisfied certain geometric conditions Since Carlsson and Pedersen s seminal work the descent principle and continuously controlled algebra have been used to prove split injectivity results see for example 11 2 3 25 8 This method was recently used to prove that the assembly map is a split injection for all discrete subgroups of virtually connected Lie groups 8 and for all S arithmetic subgroups of algebraic groups defined over global fields regardless of rank by Ji 19 REFERENCES 1 D R Anderson F Connolly S C Ferry and E K Pedersen Algebraic K theory with continuous control at infinity J Pure Appl Algebra 94 1994 25 47 2 A Bartels On the domain of the assembly map in algebraic K theory Algebr Geom Topol 3 2003 1037 1050 3 A Bartels Squeezing and higher algebraic K theory K theory 28 2003 19 37 14 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 D ROSENTHAL A Bartels F T Farrell L Jones and H Reich On the isomorphism conjecture in algebraic K theory Topology 43 1 2004 157 213 A Bartels W Liick and H Reich The K theoretic Farrell Jones Conjecture for hyperbolic groups To appear in Invent Math arXiv math KT 0701434 A Bartels W L ck and H Rei
19. on zero component between 1 s and 1 t for some s and t away from 1 cannot be shifted arbitrarily close to S x 1 A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA 5 The support at infinity of an object A in B X Y A is the set of limit points of supp A in Y Definition 2 4 Let C be a closed subset of Y The category B X Y A c is the full subcategory of B X Y A on objects whose support at infinity is contained in C Example 2 5 Consider B X x 0 1 X x 1 A p the full subcategory of B X x 0 1 X x 1 A whose objects have empty support at infinity Because of the relatively compact and locally finite conditions the support of every object is finite Furthermore the con tinuous control condition is vacuous Therefore B X x 0 1 X x 1 A and A are equivalent additive categories Definition 2 6 Let W be an open subset of Y The germ category B X Y A has the same objects as B X Y A but morphisms are identified if they agree in a neighborhood of W Specifically Y A B are identified if there is a neighborhood U C X of W such that x U or a U implies that y Germ categories are very interesting As noted above B S x 0 1 S x 1 A is not equivalent to B 0 1 1 A4 6 B 0 1 1 A since B 0 1 1 A 6 B 0 1 1 A is flasque because each summand is flasque while B S x 0 1 S x 1 A is not However it is true that the germ cate
20. s the cone on X and x is the cone point The control spaces for these two categories are the same namely X x 0 1 but the control conditions are different In B X x 0 1 X x 1 A the continuous control condition on morphisms is defined with respect to the boundary X x 1 whereas in B CX A the continuous control condition is defined with respect to the one point boundary Therefore the continuous control condition on morphisms in B X x 0 1 X x 1 A is more re strictive than the one in B CX x A The quotient map X x 0 1 X x 0 1 X x 1 induces a functor B X x 0 1 X x 1 A B CX x A that is essentially the identity on objects and morphisms For this reason it is known as a forget control functor In addition because of the relatively compact condition on objects the map CX pt x 0 1 which collapses X to a point induces an equivalence of categories B CX x A B 0 1 1 A This helps to illustrate the difference between B X x 0 1 X x 1 A and B CX x A Compare with Example 2 3 below 4 D ROSENTHAL An additive category B is called flasque if it admits an endofunctor B B and a natural equivalence between idg E and X By the Additivity Theorem such a category has trivial K theory Using an argument of this type to prove that a category has trivial K theory is referred to as an Etlenberg swindle Example 2 3 We will use an Eilenberg swindle to show that B 0 1 1 A
21. ve diagram D X x 0 1 A g gt D X x 0 1 A DF X x 0 1 A 5 DC pt x 0 1 AJga gt D pt x 0 1 A D pt x 0 1 A Since DF pt x 0 1 A is flasque K DF pt x 0 1 A is weakly contractible There fore it is a simple diagram chase to verify that the assembly map A is homotopy equiv alent to the connecting map QK D X x 0 1 A K DF X x 0 1 Ap This variant of the continuously controlled assembly map shows us that in order to IIe prove isomorphism results one needs to prove that the category D X x 0 1 A has trivial K theory For this reason Bartels Ltick Reich call this category the obstruction category In 5 they were able to prove the K theoretic Farrell Jones conjecture for all word hyperbolic groups and all coefficient categories by showing that the K theory of this Recall that if F is a family of subgroups of G that is closed under conjugation and taking subgroups then the universal space for G with isotropy in F EFG is a G CW complex with the property that ExG is contractible if H is in F and is empty otherwise Such spaces are universal for G actions with isotropy in F meaning that given any G CW complex Y whose isotropy groups belong to F there is a map Y EFG that is unique up to G equivariant homotopy equivalence Thus E G is unique up to G equivariant homotopy equivalence A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEB
22. volutionary work of Farrell and Jones 14 a more general assembly map relating the K theory of RIG to a generalized G equivariant homology theory was formulated using Quinn s homology of simplicially stratified fibrations 24 Davis and Liick 13 used the orbit category A USER S GUIDE TO CONTINUOUSLY CONTROLLED ALGEBRA 9 to create an abstract approach to Farrell Jones assembly and Hambleton and Pedersen formalized the continuously controlled version of the Farrell Jones assembly map in 15 Recently Bartels and Reich 7 used the Davis Liick machinery to construct an assembly map with coefficients in an additive category on which G acts This more general formu lation allows one to study for example twisted group rings It also possesses interesting inheritance properties see 7 16 In this section the assembly map with coefficients is defined using continuously controlled algebra In order to do this the corresponding equivariant homology theory must be developed An additive category with right G action A is an additive category together with a collection of covariant functors g A A g G such that g o h h o g and e idy A subset S C X is called G compact if S G K for some compact subset K CX It is called relatively G compact if its closure in X is G compact Definition 3 1 Let G be a discrete group A be an additive category with right G action X be a G CW complex and Y C X be a closed G
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