Excision in equivariant fibred G-theory
aa r X i v : . [ m a t h . K T ] N ov EXCISION IN EQUIVARIANT FIBRED G -THEORY GUNNAR CARLSSON AND BORIS GOLDFARB
Abstract.
This paper provides a generalization of excision theorems in con-trolled algebra in the context of equivariant G -theory with fibred control andfamilies of bounded actions. It also states and proves several characteristic fea-tures of this theory such as existence of the fibred assembly and the fibrewisetrivialization. Contents
1. Introduction 12. Homotopy fixed points in categories with action 33. Bounded K -theory and the K -theory of group rings 64. Fibred homotopy fixed points in K -theory 75. Summary of bounded G -theory with fibred control 86. Fibrewise excision in equivariant fibred G -theory 127. Other properties of equivariant fibred G -theory 18References 231. Introduction
The bounded K -theory construction due to Pedersen and Weibel [19] has beenshown to be extremely useful in the analysis of versions of the Novikov conjecture([2, 5, 12, 13, 20]). This conjecture asserts the split injectivity of a natural trans-formation called the assembly . The present paper is the culmination of a series ofpapers of the authors [6, 10, 11] that extend the techniques sufficient to addressthe much more difficult Borel conjecture, which is very closely related to the ques-tion of whether the assembly map is an isomorphism. What we have found is thatsubstantial extensions are necessary. In order to explain these extensions, we firstrecall some of the properties of the original construction of Pedersen and Weibel. • The construction begins with a metric space X and a commutative ring R of coefficients and constructs a bounded K -theory spectrum K ( X, R ).The spectrum is constructed by considering potentially infinitely generatedbased R -modules equipped with a reference map ϕ from the chosen basis B to the metric space X , and considering only morphisms satisfying acondition of control, defined by a single parameter P . The requirement fora homomorphism f from ( M, B M , ϕ M ) to ( N, B N , ϕ N ) to be bounded withparameter P is that for any b ∈ B M , f ( b ) lies in the span of the basis Date : November 22, 2019. elements in B N which lie within a distance P of ϕ M ( b ). We will call suchmodules geometric R -modules over X . • The construction is functorial for maps f : X → Y of metric spaces that are unifomly expansive , which are maps such that there is a function c : R → R so that if x, x ′ ∈ X are any two points with d ( x, x ′ ) ≤ t , then d ( f ( x ) , f ( x ′ )) ≤ c ( t ). • The construction is coarse invariant . For example, if we have an isometricinclusion
X ֒ → Y of metric spaces, and every element of Y is within afixed distance of an element of X , then the map K ( X, R ) → K ( Y, R ) is anequivalence of spectra. In particular, it is possible to show that for compact K (Γ , X , the inclusion of a single orbit Γ · x in the universalcover of X is an equivalence. • For X a Riemannian manifold, the construction is closely related to the locally finite or Borel homology of X with coefficients in the algebraic K -theory spectrum K ( R ). Specifically, there is a bounded assembly map h lf ( X, K ( R )) −→ K ( X, R )which is an equivalence for a large class of manifolds. • For free and proper actions by isometries of a discrete group on a Riemann-ian manifold X , there is an equivariant construction K Γ , ( X, R ) explainedin section 3 which is equivalent to the original construction K ( X, R ) non-equivariantly, and whose fixed point spectrum is the ordinary homology of X/ Γ + with coefficients in K ( R ).There are various directions in which one can generalize the Pedersen-Weibelframework. The following are the extensions and issues that need to be addressed,with some comments and motivation. • In algebraic K -theory, one usually considers categories of free or projec-tive finitely generated modules, however in many situations one can useother categories of modules such as finitely generated or finitely presentedmodules. When studying Noetherian commutative rings, one can considerthe category of finitely generated modules and perform an algebraic K -theoretic construction on it. The result which is often easier to computeis referred to as G -theory . The authors have performed analogues of thiskind of construction in the context of bounded K -theory and group ringsin [6, 9]. • The issue that naturally comes up is construction of equivariant theoriesbeyond the case of isometric actions. We are led to study actions of a groupΓ on a metric space through uniformly expansive self-maps of metric spaces.These actions can no longer be assumed to be free, and each self-map is nolonger necessarily an injection. • Next, there is the related new requirement of more general notions of control.Specifically, given two metric spaces X and Y , we will need to consider thenotion of fibrewise control for geometric R -modules over X × Y . Given twogeometric modules M = ( M, B M , ϕ M ) and N = ( N, B N , ϕ N ) over X × Y , XCISION IN EQUIVARIANT FIBRED G -THEORY 3 equipped with the sum metric D = d X + d Y , we say a homomorphism f : M → N is fibrewise controlled if it satisfies two conditions.(1) There exists a number P > b ∈ B M , f ( b ) lies in the span of the those basis elements b ′ in B N for which π X ( ϕ ( b ′ )) lies in a ball of radius P of π X ( ϕ M ( b )) in X .(2) For any bounded subset U in X , there is a number P ( U ) > b ∈ ( π X ◦ ϕ M ) − ( U ), f ( b ) lies in the span of the set of elements b ′ of B N for which d ( π Y ◦ ϕ M ( b ) , π Y ◦ ϕ N ( b ′ )) ≤ P ( U ).Intuitively, this notion of control provides maps which are controlled in the“base direction" in the bundle X × Y → X , and which are controlled ineach of the "fibers", but not necessarily requiring that there exist a uni-form bound over all fibers. We will also require G -theoretic and equivariantversions of such notions of control. • Fibrewise control is the analogue for K - and G -theory of the notion of“parametrized homotopy theory" or “homotopy theory over a base", where X is the base and Y is the fiber. In the equivariant case, where X is equalto a group Γ regarded as a metric space using the word length metric.The fixed points of the Γ action on K Γ ( Y ) is analogous to the study ofthe bundles over a classifying space B Γ obtained from Γ-spaces Y by theconstruction Y → E Γ × Γ Y . To realize our results, we will need excisionproperties holding for coverings of Y . • Finally, we will need results that imply that K Γ ( Y ) is canonically equivalentto K Γ ( Y ), where Y is defined to be Y equipped with the trivial action.We say that an action of a group Γ on a metric space is bounded if for every γ ∈ Γ, there exists an R γ > d ( x, γ ( x )) ≤ R γ for all x ∈ X . Wewill be able to prove that for any bounded action of Γ on a metric space Y , K Γ ( Y ) is canonically equivalent to K Γ ( Y ).That such a result should be possible is suggested by the analoguethat occurs in ordinary parametrized homotopy theory, where E Γ × Γ BH is canonically equivalent to B Γ × BH whenever the Γ-action is specifiedby a homomorphism to H followed by the homomorphism H → Inn ( H ),where Inn ( H ) is the inner automorphism group of H . The reason this is anapt analogy is that in the case where we are considering a bounded actionof Γ on a metric space Y , the action can be considered as conjugation bybounded automorphisms of Y , all of which are contained in the categorydefining K ( Y ).In this paper we prove excision results that incorporate all generalizations si-multaneously. Because we will only need the excision results where the action onthe space Y is bounded in the sense defined above, we will only prove them inthat situation. That is, we prove excision theorems (Theorems 6.11 and 6.12) forequivariant G -theory with fibred control of bounded Γ-spaces Y . Additionally weobtain the results suggested above as part of equivariant fibred G -theory.2. Homotopy fixed points in categories with action
Given an action of a group Γ on a space X , one has the subspace of fixed points X Γ . This subspace often has geometric significance for the study of X and Γ. Adifferent powerful idea in topology is to model an interesting space or spectrum as GUNNAR CARLSSON AND BORIS GOLDFARB the fixed point space or spectrum X Γ for a specifically designed X with an actionby a related group Γ. In either case, there is always the homotopy fixed pointspectrum X h Γ which is easier to understand than X Γ and the canonical referencemap ρ : X Γ → X h Γ .Now suppose we have a group action on a category. This automatically producesan action on the nerve and therefore a space. Suppose the category is then fed intoa machine such as the algebraic K -theory, and we are interested in the fixed pointsof the K -theory. Therefore we want to look at the homotopy fixed points. In manyimportant cases it is possible to construct a spatial or categorical description ofwhat we get. Thomason defined the lax limit category whose K -theory turns outto be exactly the homotopy fixed points of the old action. Definition 2.1.
Let E Γ be the category with the object set Γ and the uniquemorphism µ : γ → γ for any pair γ , γ ∈ Γ. There is a left Γ-action on E Γ induced by the left multiplication in Γ. If C is a category with a left Γ-action, thenthe category of functors Fun( E Γ , C ) is another category with the Γ-action givenon objects by the formulas γ ( F )( γ ′ ) = γF ( γ − γ ′ ) and γ ( F )( µ ) = γF ( γ − µ ). It isnonequivariantly equivalent to C .The category Fun( E Γ , C ) is an interesting and useful object in its own right.There are several manifestations of this, for example in the work of Mona Merlingand coauthors [14, 16, 17] or the work of these authors [2, 4, 5, 7]. While in both ap-plications it is crucial to work with the category itself, in this paper we concentrateon approximating the fixed points in Fun( E Γ , C ).The following construction has been used by Thomason [22]. We will refer to itas the homotopy fixed points of a category , following Merling [17]. Definition 2.2 (Homotopy fixed points) . The fixed point subcategory Fun( E Γ , C ) Γ of the category of functors Fun( E Γ , C ) consists of equivariant functors and equi-variant natural transformations. We will denote it by C h Γ .Explicitly, the objects of C h Γ are the pairs ( C, ψ ) where C is an object of C and ψ is a function from Γ to the morphisms of C with ψ ( γ ) ∈ Hom(
C, γC ) thatsatisfies ψ ( e ) = id for the identity group element e , and satisfies the cocycle identity ψ ( γ γ ) = γ ψ ( γ ) ψ ( γ ) for all pairs γ and γ in Γ. These conditions imply that ψ ( γ ) is always an isomorphism. The set of morphisms ( C, ψ ) → ( C ′ , ψ ′ ) consists ofthe morphisms φ : C → C ′ in C such that the squares C ψ ( γ ) / / φ (cid:15) (cid:15) γC γφ (cid:15) (cid:15) C ′ ψ ′ ( γ ) / / γC ′ commute for all γ ∈ Γ. Remark 2.3.
As pointed out in [17], the homotopy fixed points of a category arenot necessarily identical with space level constructions. It is for example not truein general that the nerve of the homotopy fixed point category of a category is thesame as the geometric homotopy fixed points of the nerve of the category. It ishowever true in the case where the category is a discrete Γ-groupoid.
XCISION IN EQUIVARIANT FIBRED G -THEORY 5 Example 2.4.
Let C denote a category, and equip it with the trivial action by Γ.Then the category C h Γ is the category of representations of Γ in C . In particular, if C is the category of R -modules for a commutative ring R , then C h Γ may be identifiedwith the category of (left) R [Γ]-modules. Example 2.5.
Let F ⊆ E denote a Galois field extension, with Galois group G .We consider the skew group ring Λ = E t [ G ], and consider the category C E whoseobjects are F t [ G ]-modules and whose morphisms are the E -linear maps. There is a G action on C E , which is the identity on objects and which is defined by the groupaction on the morphisms. In this case, C h Γ E is equivalent to the category of F -vectorspacesIn Example 2.4, we saw that the group ring of a group Γ with coefficients in acommutative ring R may be realized as the fixed point subcategory of the actionof Γ on Fun( E Γ , C ), where C denotes the category of all R -modules. In many cases,however, it is important to understand the category of free and finitely generatedleft R [Γ]-modules as a fixed point category. This is the case in the papers [3] and [5],for instance, where the injectivity of the assembly map is proved in a large familyof cases. In the case of these two papers, this is achieved by defining a subcategoryof Fun( E Γ , C ) by restricting the morphisms ψ ( γ ). The restriction in this case arisesby the selection of a subcategory of the category of all R -modules based on thePedersen-Weibel construction, which is endowed with a filtration and an action ofthe group Γ. The restricted version of C h Γ E requires that all of the morphisms ψ ( γ )have filtration zero. In order to attack the surjectivity problem for the assembly, weare led to the construction of more general forms of restriction of the maps ψ ( γ ).This leads us to the concept of the relative homotopy fixed points of a category ,which we now define. Definition 2.6 (Relative homotopy fixed points) . The category C h Γ ( M ) is definedusing input data consisting of a category C equipped with an action by a group Γand a subcategory M ⊂ C closed under the action of Γ. It is the full subcategoryof C h Γ on objects ( C, ψ ) with the additional condition that ψ ( γ ) is in M for allelements γ ∈ Γ. Example 2.7.
Clearly, if M is the entire category C , the relative homotopy fixedpoints are the genuine homotopy fixed points. Example 2.8.
In the case where C is a filtered category, we can consider thesituation where M is the subcategory of the filtration zero morphisms. This is thesituation used in [3] and [5].We will exploit the relative homotopy fixed points in two applications. The firstconstruction required in [2] by the first author allows us to model the K -theory of agroup ring whenever the group has a finite classifying space. It is based on bounded K -theory of the group given a word metric with the isometric action on itself givenby the left multiplication. It turns out that the categorical homotopy fixed pointconstruction equires a constraint. We review that construction in section 3.A generalization of bounded K -theory has appeared in recent work on the Borelisomorphism conjecture [7, 9, 10, 11]. The fibred homotopy fixed points definedin [10] are particularly intriguing because the standard tools in controlled algebra,based on Karoubi filtrations that generate long exact sequences, fail to computethem on a very basic level. The details can be found in [10, Example 5.2]. Section GUNNAR CARLSSON AND BORIS GOLDFARB G -theory which we areable to compute.3. Bounded K -theory and the K -theory of group rings Bounded control is the simplest version of a “control condition” that can beimposed in various categories of modules, to which one can apply the algebraic K -theory construction. It was introduced in Pedersen [18] and Pedersen/Weibel [19]and has become crucial for K -theory computations in geometric topology.Let X be a metric space and let R be an arbitrary associative ring with unity. Wewill always assume that metric spaces are proper in the sense that closed boundedsubsets are compact. Definition 3.1.
The objects of the category of geometric R -modules over X arelocally finite functions F from points of X to the category of finitely generated free R -modules Free fg ( R ). Following Pedersen and Weibel, we will denote by F x themodule assigned to the point x of X and denote the object itself by writing downthe collection { F x } . The local finiteness condition requires precisely that for everybounded subset S ⊂ X the restriction of F to S has finitely many nonzero modulesas values.Let d be the distance function in X . The morphisms φ : { F x } → { G x } are collec-tions of R -linear homomorphisms φ x,x ′ : F x → G x ′ , for all x and x ′ in X , with theproperty that φ x,x ′ is the zero homomorphism whenever d ( x, x ′ ) > D for some fixedreal number D = D ( φ ) ≥
0. One says that φ is bounded by D . The composition oftwo morphisms φ : { F x } → { G x } and ψ : { G x } → { H x } is given by the formula( ∗ ) ( ψ ◦ φ ) x,x ′ = X z ∈ X ψ z,x ′ ◦ φ x,z . This sum is finite because of the local finiteness property of G .We will want to enlarge this category, and so we use instead an equivalent cate-gory B ( X, R ) that is better for this purpose.The objects are functors F : P ( X ) → Free ( R ) from the power set P ( X ) tothe category of free modules, both viewed as posets ordered by split inclusions.There are two additional requirements. For every bounded subset C of X the value F ( C ) has to belong to the subcategory of finitely generated modules Free fg ( R ). Inthe codomain, the values are required to satisfy the equality F ( S ) = L x ∈ S F ( x )for all S ⊂ X . The morphisms in this reformulation are R -linear homomorphisms φ : F ( X ) → G ( X ) such that the components φ x,x ′ : F ( x ) → G ( x ′ ) are zero whenever d ( x, x ′ ) > D for some D . The composition of two morphisms φ : F → G and ψ : G → H is the usual composition of R -linear homomorphisms; its componentsare the maps ( ψ ◦ φ ) x,x ′ in the formula ( ∗ ) above. Definition 3.2.
A map f : X → Y between metric spaces is called uniformlyexpansive if there is a function λ : [0 , ∞ ) → [0 , ∞ ) such that d X ( x , x ) ≤ r implies d Y ( f ( x ) , f ( x )) ≤ λ ( r ). A map f is proper if f − ( S ) is a bounded subset of X foreach bounded subset S of Y . We say f is a coarse map if it is eventually continuousand proper. XCISION IN EQUIVARIANT FIBRED G -THEORY 7 Extensively used instances of coarse maps in geometry are quasi-isometries.It is elementary to check that the geometric R -modules over X is an additivecategory and that coarse maps between metric spaces induce additive functors. Acoarse map f is a coarse equivalence if there is a coarse map g : Y → X such that f ◦ g and g ◦ f are bounded maps. It follows that an action of a group on a metricspace by coarse equivalences induces an additive action on B ( X, R ).We will treat the group Γ equipped with a finite generating set Ω closed undertaking inverses as a metric space. The word-length metric d = d Ω is induced fromthe condition that d ( γ, γω ) = 1 whenever γ ∈ Γ and ω ∈ Ω. It is well-knownthat varying Ω only changes Γ to a quasi-isometric metric space. The word-lengthmetric makes Γ a proper metric space with a free Γ-action by isometries via leftmultiplication.
An important observation.
A free action of Γ on X by isometries always gives afree action on C = B ( X, R ). In contrast, Fun( E Γ , C ) with the induced group actiondoes have the subcategory C h Γ of equivariant functors. These homotopy fixed points,however, is not the correct notion for modeling the finitely generated free modulesover R [ G ] and the K -theory of R [ G ]. Definition 3.3.
The category B Γ , ( X, R ) Γ are relative homotopy fixed points C h Γ ( M ) with the following data: C is the category of geometric modules B ( X, R )and M consists of those morphisms in C that are bounded by 0.The additive category B Γ , ( X, R ) Γ has the associated nonconnective K -theoryspectrum K −∞ ( X, R ) Γ constructed as in [19]. There is now the following desiredidentification. Theorem 3.4.
Suppose Γ acts on X freely, properly discontinuously by isometriesso that the orbit space X/ Γ with the orbit metric is bounded.It follows that K −∞ ( X, R ) Γ is weakly homotopy equivalent to the nonconnectivespectrum K −∞ ( R [Γ]) . The stable homotopy groups of the nonconnective spectrumare the Quillen K -groups of R [ G ] in nonnegative dimensions and the negative K -groups of Bass in negative dimensions.Proof. The result follows from Corollary VI.8 in [2]. (cid:3)
This geometric situation occurs, for example, when Γ acts cocompactly, freelyproperly discontinuously on a contractible connected Riemannian manifold X orwhen it acts on itself with a word metric via left multiplication.4. Fibred homotopy fixed points in K -theory Let A be an additive category. Generalizing Definition 3.1, one has the boundedcategory with coefficients in A . Notation . Given a subset S of a metric space and a number k ≥ S [ k ] is usedfor the k - enlargement of S defined as the set of all points x with d ( x, S ) ≤ k .Recall that A is a subcategory of its cocompletion A ∗ which is closed undercolimits. For example, a construction based on the presheaf category was given byKelly in 6.23 of [15]. Definition 4.2. B ( X, A ) has objects which are covariant functors F : P ( X ) → A ∗ from the power set P ( X ) to A ∗ , both ordered by inclusion. Just as in Definition3.1, there are several requirements: GUNNAR CARLSSON AND BORIS GOLDFARB • F ( x ) is an object of A for every point x in X , • the resulting function F : X → A is locally finite, so only finitely manyvalues are non-zero when restricted to any compact subset of X , • for all subsets S ⊂ X , F ( S ) = M x ∈ S F ( x ) , • the inclusion F ( S ⊂ X ) is onto a direct summand for each subset S .A morphism in B ( X, A ) is a morphism φ : F ( X ) → G ( X ) in A ∗ with a number D ≥ φ restricted to F ( S ) factors through G ( S [ D ]) for all S ⊂ X . Wesay a morphism which admits such a number D is D -controlled This context, which produces a category isomorphic to B ( X, R ) when A is thecategory of free finitely generated R -modules, allows us to iterate the boundedcontrol construction as follows. Definition 4.3 (Fibred control for geometric modules) . Given two metric spaces X and Y and any ring R , the category B X ( Y, R ), or simply B X ( Y ) when the choiceof ring R is clear, is the bounded category B ( X, A ) with A = B ( Y, R ).Among many options for relativizing homotopy fixed points in this setting, thereis one of specific interest.Let A = B ( Y, R ) as before and A ′ = Mod ( R ) be the category of arbitrary R -modules. There is a forget control functor t : B ( Y, R ) → Mod ( R ) which onlyremembers that the objects are R -modules and the morphisms are R -linear homo-morphisms. From t we may induce the functor T : B ( X, A ) → B ( X, A ′ ).For this construction we assume that Γ acts on X by isometries and so, therefore,on B ( X, A ′ ). On the other hand, we allow the action of Γ on Y to be by coarseequivalences. This can also be used to induce an action on B ( X, A ). Definition 4.4 (Fibred homotopy fixed points in bounded K -theory) . These arerelative homotopy fixed points with the following choice of ingredients:– the category C is B X ( Y, R ),– the subcategory M consists of all morphisms φ such that T ( φ ) is a con-trolled morphism bounded by 0. Notation . When X is the group Γ itself with the left multiplication action andthe word metric with respect to some choice of a finite set of generators, we willuse the special notation B w Γ ( Y ) for the fibred homotopy fixed points C h Γ ( M ).5. Summary of bounded G -theory with fibred control A comprehensive exposition of bounded G -theory with fibred control is availablein [11]. This is a summary of that theory and a number of facts in the form we canrefer to in the next section.Throughout the rest of the paper, R will be a Noetherian ring.At the basic level, bounded G -theory with fibred control is an analogue of thealgebraic K -theory of B X ( Y, R ) locally modeled on finitely generated R -modules.The result is an exact category B X ( Y ) where the exact sequences are not necessarilysplit but which contains B X ( Y ) as an exact subcategory. XCISION IN EQUIVARIANT FIBRED G -THEORY 9 Definition 5.1.
Given an R -module F , an ( X, Y ) -filtration of F is a functor φ F : P ( X × Y ) → I ( F ) from the power set of the product metric space to thepartially ordered family of R -submodules of F ( X × Y ), both ordered by inclusion.When there is no ambiguity, we find it convenient to denote the values φ F ( U ) by F ( U ). We assume that F is reduced in the sense that F ( ∅ ) = 0.The associated X -filtered R -module F X is given by F X ( S ) = F ( S × Y ). Similarly,for each subset S ⊂ X , one has the Y -filtered R -module F S given by F S ( T ) = F ( S × T ). In particular, F X ( T ) = F ( X × T ).We will use the following notation generalizing enlargements in a metric space.Given a subset U of X × Y and a function k : X → [0 , + ∞ ), let U [ k ] = { ( x, y ) ∈ X × Y | there is ( x, y ′ ) ∈ U with d ( y, y ′ ) ≤ k ( x ) } . If in addition we are given a number K ≥ U [ K, k ] = { ( x, y ) ∈ X × Y | there is ( x ′ , y ) ∈ U [ k ] with d ( x, x ′ ) ≤ K } . For a product set U = S × T , it is more convenient to use the notation ( S, T )[ K, k ]in place of ( S × T )[ K, k ]. We will refer to the pair (
K, k ) in the notation U [ K, k ] asthe enlargement data .Let x be a chosen fixed point in X . Given a monotone function h : [0 , + ∞ ) → [0 , + ∞ ), there is a function h x : X → [0 , + ∞ ) defined by h x ( x ) = h ( d X ( x , x )) . Given two (
X, Y )-filtered modules F and G , an R -homomorphism f : F ( X × Y ) → G ( X × Y ) is boundedly controlled if there are a number b ≥ θ : [0 , + ∞ ) → [0 , + ∞ ) such that( † ) f F ( U ) ⊂ G ( U [ b, θ x ])for all subsets U ⊂ X × Y and some choice of x ∈ X . It is easy to see that thiscondition is independent of the choice of x . If a homomorphism f is boundedlycontrolled with respect to some choice of parameters b and θ , we will say that f is( b, θ )- controlled .The unrestricted fibred bounded category U X ( Y ) has ( X, Y )-filtered modules asobjects and the boundedly controlled homomorphisms as morphisms. Theorem 3.1.6of [11] shows that U X ( Y ) is a cocomplete semi-abelian category.When Y is the one point space, this construction recovers the controlled cat-egory U ( X, R ) of X -filtered R -modules used to construct bounded G -theory in[6] and chapter 2 of [11]. In this case, boundedly controlled homomorphisms arecharacterized by a single parameter b , so one can specify that by abbreviating theterm to simply b - controlled . The construction of an X -filtration F X from a given( X, Y )-filtration in Definition 5.1 allows us to view a ( b, θ )-controlled homomor-phism in U X ( Y ) as a b -controlled homomorphism in U ( X, R ) via the forgetfulfunctor T : U X ( Y ) → U ( X, R ).We now want to restrict to a subcategory of U X ( Y ) that is full on objects withparticular properties. This process consists of two steps that result in a theory withbetter localization properties. Definition 5.2.
An (
X, Y )-filtered module F is called • split or ( D, ∆)- split if there is a number D ≥ , + ∞ ) → [0 , + ∞ ) so that F ( U ∪ U ) ⊂ F ( U [ D, ∆ ′ x ]) + F ( U [ D, ∆ x ])for each pair of subsets U and U of X × Y , • lean/split or ( D, ∆ ′ )- lean/split if there is a number D ≥ ′ : [0 , + ∞ ) → [0 , + ∞ ) so that – the X -filtered module F X is D - lean , in the sense that F X ( S ) ⊂ X x ∈ S F X ( x [ D ])for every subset S of X , while – the ( X, Y )-filtered module F is ( D, ∆ ′ )-split, • insular or ( d, δ )- insular if there is a number d ≥ δ : [0 , + ∞ ) → [0 , + ∞ ) so that F ( U ) ∩ F ( U ) ⊂ F (cid:0) U [ d, δ x ] ∩ U [ d, δ x ] (cid:1) for each pair of subsets U and U of X × Y .There are two subcategories nested in U X ( Y ). The category LS X ( Y ) is the fullsubcategory of U X ( Y ) on objects F that are lean/split and insular. The category B X ( Y ) is the full subcategory of LS X ( Y ) on objects F such that F ( U ) is a finitelygenerated submodule whenever U ⊂ X × Y is bounded.We proceed to define appropriate exact structures in these categories. The ad-missible monomorphisms are precisely the morphisms isomorphic in U X ( Y ) to thefiltration-wise monomorphisms and the admissible epimorphisms are those mor-phisms isomorphic to the filtration-wise epimorphisms. In other words, the exactstructure E in U X ( Y ) consists of sequences isomorphic to those E · : E ′ i −−→ E j −−→ E ′′ which possess filtration-wise restrictions E · ( U ) : E ′ ( U ) i −−→ E ( U ) j −−→ E ′′ ( U )for all subsets U ⊂ ( X, Y ), and each E · ( U ) is an exact sequence of R -modules.Both LS X ( Y ) and B X ( Y ) are closed under extensions in U X ( Y ). Therefore,they are themselves exact categories, and the inclusion B X ( Y ) → B X ( Y ) is anexact embedding, as we projected.There is a useful invariant of a finitely generated group Γ that is defined interms of the exact category B Γ (point) in [9]. Here Γ can be given the word metricassociated to any of the finite generating sets. The left multiplication action givesan action of Γ on B Γ (point).Recall that Theorem 3.4 provides an interpretation to the K -theory of a groupring R [Γ] in terms of relative homotopy fixed points of the additive category B ( X, R ),which can be viewed as B Γ (point). Example 5.3 (Bounded G -theory of a finitely generated group) . In the case where C is the exact category B Γ (point) and M is the subcategory of the filtration zeromorphisms, the bounded G -theory of Γ is defined to be the nonconnective K -theoryof the relative homotopy fixed points B Γ (point) h Γ , denoted G −∞ ( R [Γ]). XCISION IN EQUIVARIANT FIBRED G -THEORY 11 Notice that this definition makes sense even when the group ring is not Noether-ian unlike the much more restrictive situation with the usual G -theory defined onlyfor Noetherian rings. Theorem 5.4.
There is an exact subcategory of finitely generated Γ -modules for anarbitrary finitely generated group Γ such that its relative homotopy fixed points haveQuillen K -theory with features similar to G -theory of group rings. In particular, ithas a Cartan map from the K -theory of R [Γ] .Proof. The category is equivalent to B Γ (point). We refer to sections 2 and 3 of [9] fordetails. The clear resemblance to Definition 3.3 and the identification of B Γ , ( X, R ) Γ with B Γ (point) h Γ allow us to induce the Cartan map K −∞ ( R [Γ]) → G −∞ ( R [Γ])from the exact inclusion B Γ (point) → B Γ (point) above. (cid:3) Suppose C is a subset of Y . Let B X ( Y ) X, C )[ D, δ x ] for some choices of a subset C ⊂ Y , a number D ≥ 0, and a function δ ∈ M ≥ . Definition 5.5. Given an object F of B X ( Y ), a Y - grading of F is a functor F : P X ( Y ) → I ( F ) with the following properties: • the submodule F (( X, C )[ D, δ x ]), with the standard ( X, Y )-filtration in-duced from F , is an object of B X ( Y ), • there is an enlargement data ( K, k ) such that F (( X, C )[ D, δ x ]) ⊂ F (( X, C )[ D, δ x ]) ⊂ F (( X, C )[ D + K, δ x + k x ]) , for all subsets in P X ( Y ).We say that an object F of B X ( Y ) is Y - graded if there exists a Y -grading of F ,but the grading itself is not specified, and define G X ( Y ) as the full subcategory of B X ( Y ) on Y -graded filtered modules.We will summarize some additional required results from section 3.4 of [11]. Theorem 5.6. The subcategory G X ( Y ) is closed under both isomorphisms andexact extensions in B X ( Y ) . Therefore, G X ( Y ) is an exact subcategory of B X ( Y ) .The restriction to Y -gradings in B X ( Y ) Fibrewise excision in equivariant fibred G -theory It is well-known that Quillen K -theory of an exact category can be obtainedequivalently as Waldhausen’s K -theory of bounded chain complexes in the category.The cofibrations are then the chain maps which are the degree-wise admissiblemonomorphisms. The weak equivalences are the chain maps whose mapping conesare homotopy equivalent to acyclic complexes. An exposition with a number ofdetails verified specifically for bounded G -theory can be found in [6, section 4].The Waldhausen theory setting is crucial in proving the excision theorem in thatthe Approximation Theorem [6, Theorem 4.5] becomes essential. We will indicate XCISION IN EQUIVARIANT FIBRED G -THEORY 13 passage from an exact category to the derived category of bounded chain complexesby prefixing “ch” in front of the name of the exact category.We proceed to define the equivariant fibred G -theory.The basic setting consists of • two proper metric spaces X and Y , • an arbitrary subset Y ′ of Y , • a Γ-action on X by isometries, and • a bounded action of Γ on Y . This is an action such that for each γ in Γ theset of real numbers W γ = { d ( x, γ ( x )) } is bounded from above. Remark 6.1. In a number of situations, we will be specifying subcategories closedunder the Γ-action by subsets that are arbitrary, therefore certainly not closedunder the action. This works due to the boundedness of the action. For example, ifwe have a subset C ⊆ X and define a subcategory as the set of modules supportedon some neighborhood of C , then this subcategory is closed under the Γ-actionprovided the action is bounded. This would definitely not hold were the action notbounded.Consider the exact category G Γ ( Y ) with the induced action by Γ, in the case X is the group Γ with a word metric, acting on itself by isometries via the left mul-tiplication. Since the action on Y is bounded, we have the quotient exact category G Γ ( Y, Y ′ ). Notation . If Z is another arbitrary subset of Y , it is also useful to consider thefull exact subcategory G Γ ( Y, Y ′ ) The equivariant fibred G -theory is G Γ ( Y, Y ′ , Z ) = Ω K ( | wS. G Γ , ( Y, Y ′ , Z ) | ) . This is a functor from the category of triples ( Y, Y ′ , Z ), where both Y ′ and Z aresubspaces of Y but not necessarily subspaces of each other, and uniformly expansivemaps of triples to the category of spectra.Now we turn to the construction of fibred homotopy fixed points. There is aforget control functor T : G X ( Y, Y ′ , Z ) → U X ( Y, Y ′ , Z ) sending F to F X . SinceΓ acts on X by isometries, it also acts on U X ( Y, Y ′ , Z ). The combination of thisaction and a bounded action on Y induces an action on G X ( Y, Y ′ , Z ). With thesechoices, T is an equivariant functor. Definition 6.4 (Fibred homotopy fixed points in bounded G -theory) . This is aspecial case of a relative homotopy fixed points, as defined in 4.4, with the choicesof C and M as follows.– the category C is G X ( Y, Y ′ , Z ),– the subcategory M consists of all controlled morphisms φ in C with theproperty that T ( φ ) is bounded by 0 as homomorphisms controlled over X .Let us recapitulate what this definition entails in the case X is the group Γ witha word metric.The fibred homotopy fixed points of a triple ( Y, Y ′ , Z ) is the category G h Γ ( Y, Y ′ , Z )with objects which are sets of data ( { F γ } , { ψ γ } ) where • F γ is an object of G Γ ( Y, Y ′ , Z ) for each γ in Γ, • ψ γ is an isomorphism F e → F γ in G Γ ( Y, Y ′ , Z ), • ψ γ is 0-controlled when viewed as a morphism in U Γ ( Y, Y ′ , Z ), • ψ e = id, • ψ γ γ = γ ψ γ ◦ ψ γ for all γ , γ in Γ.The morphisms ( { F γ } , { ψ γ } ) → ( { F ′ γ } , { ψ ′ γ } ) are collections of morphisms φ γ : F γ → F ′ γ in G X ( Y, Y ′ , Z ) such that the squares F e ψ γ / / φ e (cid:15) (cid:15) F γφ γ (cid:15) (cid:15) F ′ e ψ ′ γ / / F ′ γ commute for all γ .The exact structure on G h Γ ( Y, Y ′ , Z ) is induced from that on G Γ ( Y, Y ′ , Z ) asfollows. A morphism φ in G h Γ ( Y, Y ′ , Z ) is an admissible monomorphism if φ e : F → F ′ is an admissible monomorphism in G Γ ( Y, Y ′ , Z ). This of course implies that allstructure maps φ γ are admissible monomorphisms. Similarly, a morphism φ is anadmissible epimorphism if φ e : F → F ′ is an admissible epimorphism. This gives G h Γ ( Y, Y ′ , Z ) an exact structure.Since the induced Γ-action on S. G Γ , ( Y, Y ′ , Z ) commutes with taking fixed points,we have the following fact. Proposition 6.5. The fixed point spectrum G Γ ( Y, Y ′ , Z ) Γ is equivalent to the K -theory of the relative homotopy fixed point category G h Γ ( Y, Y ′ , Z ) . We proceed to consider multiple bounded actions of Γ on Y . Let β ( Y ) be theset of all such actions. Let F be the functor that assigns to a set Z the partiallyordered set of finite subsets of Z . Definition 6.6. For any S in F ( β ( Y )) we define Y S as the metric space which isthe disjoint union F s ∈ S Y s , where Y s are copies of Y with the specified action. Themetric on Y S is induced by the requirement that it restricts to the metric from Y in each Y s and for the same point y in different components the distance d ( y s , y s ′ )equals 1.Clearly, the action of Γ on Y S is bounded.As a consequence of Proposition 6.5, for each choice of finite subset S of β ( Y ),the spectrum G Γ ( Y S , Y ′ S , Z ) Γ is the Quillen K -theory spectrum of G h Γ ( Y S , Y ′ S , Z ),where Z is a subset of Y S . Theorem 6.7. Let C be an arbitrary subset of Y . There is a homotopy fibration G Γ ( Y S , Y ′ S , Z ) Γ Taking into account Remark 6.1, the fact that G Γ ( Y S , Y ′ S , Z ) The nonconnective delooping of algebraic K -theory of the fibredhomotopy fixed points is the spectrum e G Γ ( Y S ) Γ = hocolim −−−−→ k> Ω k G Γ ( Y S × R k ) Γ . In the case Y is the one point space, e G Γ ( Y ) Γ coincides with the nonconnective G -theory of the group ring R [Γ] defined by the authors in [9].The discussion leading up to Definition 6.9 can be repeated verbatim for otherSerre subcategory pairs. For example, the subcategory G Γ ( Y S × R k ) Γ Let Y ′ , Y and Y be arbitrary subsets of Y so that Y and Y form a covering of Y . There are corresponding subsets Y ′ S , Y ,S and Y ,S of Y S obtained as Y ′ S = F Y ′ s , Y ,S = F Y ′ ,s and Y ,S = F Y ′ ,s . It is now straightforwardto define nonconnective spectra e G Γ ( Y S , Y ′ S ) Γ = hocolim −−−−→ k> Ω k G Γ ( Y S × R k , Y ′ S × R k ) Γ , e G Γ ( Y S , Y ′ S ) Γ Suppose Y and Y are subsets of a metric space Y , and Y = Y ∪ Y . There is a homotopy pushout diagram of spectra e G Γ ( Y S , Y ′ S ) Γ There is a homotopy pushout e G Γ ( Y S , Y ′ S ) Γ This observation is warranted as we contrast Theorem 6.11and its proof with inability to use other, more standard methods in bounded alge-bra based on Karoubi filtrations in order to prove similar facts in K -theory. Thekey idea in the proof is still the commutative diagram from Cardenas/Pedersen [1,section 8] transported from bounded K -theory to fibred G -theory. Cardenas andPedersen use Karoubi quotients and the Karoubi fibrations in order to establishtheir diagram. One of the crucial points in [1] is that the functor I between theKaroubi quotients is an isomorphism of categories. In fibred G -theory the situationis more complicated: I is not necessarily full and, therefore, not an isomorphismof categories. However we can see here just as in the analogous Theorem 4.4.2 in[11], the Approximation Theorem suffices to prove that K ( I ) is nevertheless a weakequivalence.We make an explicit statement that does not hold in K -theory. Let V Γ ( Y ) bethe K -theory of the fibred homotopy fixed points B w Γ ( Y ) defined in 4.5. In theseterms, we don’t know whether, or under what conditions on Γ and Y ,hocolim −−−−→ U ∈U V Γ ( U ) −→ V Γ ( Y )is an equivalence in the context of Theorem 6.11. Now Example 5.2 in [10] demon-strates in the most basic geometric situation that B w Γ ( Y ) fails to be Karoubi filteredby the natural choice of subcategory. Through indirect ways related to the work onthe Borel conjecture sketched in section 4, we know that Karoubi filtrations shouldbe impossible to use to compute fibred homotopy fixed points in full generalitybecause of the well-known counterexamples to the isomorphism conjecture for theassembly map in cases of non-regular rings R .Suppose U is a finite covering of Y that is closed under intersections and suchthat the family of all subsets U in U together with Y ′ are pairwise coarsely anti-thetic. The extra conditions in the second statement ensure that the covering is infact by complete representatives of a covering by “coarse families” in the languageintroduced in [11, section 4.3].We define the homotopy colimit E Γ ( Y, Y ′ ) < U = hocolim −−−−→ U ∈U E Γ ( Y, Y ′ )
The usual notion of metric assumes only finite values.We will require a generalized metric on a set X . It is a function d : X × X → [0 , ∞ ) ∪ {∞} which is reflexive, symmetric, and satisfies the triangle inequality inthe obvious way. The generalized metric space is proper if it is a countable disjointunion of metric spaces X i on each of which the generalized metric d is finite, and all XCISION IN EQUIVARIANT FIBRED G -THEORY 19 closed metric balls in X are compact. The metric topology on a generalized metricspace is defined as usual.The basic fibred assembly map A ( X, Y ) : h lf ( X ; G −∞ ( Y )) −→ G −∞ X ( Y ) , for a proper generalized metric spaces X and Y , sends the locally finite homology of X with coefficients in the spectrum G −∞ ( Y ) to the nonconnective fibred G -theory G −∞ X ( Y ) defined in section 5.The locally finite homology h lf ( X ; S ) we use was introduced in [2, DefinitionII.5] for any coefficient spectrum S . Let b S k X be the collection of all locally finitefamilies F of singular k -simplices in X which are uniformly bounded, in the sensethat each family possesses a number N such that the diameter of the image im( σ ) isbounded from above by N for all simplices σ ∈ F . For any spectrum S , the theory b h lf ( X ; S ) is the realization of the simplicial spectrum k hocolim −−−−→ C ∈ b S k X h lf ( C, S ) . There is an equivalence of spectra b h lf ( X ; S ) → h lf ( X ; S ), for any proper generalizedmetric space X , from [2, Corollary II.21].A similar theory J h ( X, A ) is obtained as the realization of the simplicial spec-trum k hocolim −−−−→ C ∈ b S k X K −∞ ( C, A )by viewing C as a discrete metric space and using the notation K −∞ ( C, A ) for thenonconnective delooping of the K -theory of B ( C, A ) from Definition 4.2. Using thecoefficients A = B C ( Y ), we obtain J h ( X, A ) which we denote J h ( X, Y ). The proofof [2, Corollary III.14] gives a weak homotopy equivalence η : h lf ( X ; G −∞ ( Y )) −→ J h ( X, Y )of functors from proper locally compact metric spaces and coarse maps to spectra.We next define a natural transformation ℓ : J h ( X, Y ) −→ G −∞ X ( Y ) . In the case Y is a point and the coefficients are finitely generated free R -modules,this kind of transformation is defined as part of the proof of Proposition III.20 of [2].The definition is entirely in terms of maps between singular simplices in X , so theconstruction can be generalized to give ℓ as above. For convenience of the reader,we present the necessary details.Let us first note that controlled algebra can be used to build equivalent bounded K -theory spectra using the symmetric monoidal category approach which we willfind useful in the rest of the paper. For the details we refer to section 6 of [3].Let D be any collection of singular n -simplices of X and ζ be any point of thestandard n -simplex. Define a function ϑ ζ : D → X by ϑ ζ ( σ ) = σ ( ζ ). Since D isviewed as a discrete metric space, if D is locally finite then ϑ ζ is coarse, so we havethe induced functor B ( D , A ) → B ( X, A ) given by M d ∈D F d −→ M x ∈ X M ϑ ζ ( d )= x F d which is the identity for each d ∈ D . Therefore, there is the induced map of spectra K ( ϑ ζ , A ) : K ( D , A ) −→ K ( X, A ) . Suppose further that D ∈ b S k X and that N is a bound required to exist for D in b S k X . If ζ and θ are both points in the standard n -simplex, we have a symmet-ric monoidal natural transformation N θζ : K ( ϑ ζ , A ) → K ( ϑ θ , A ) induced from thefunctors which are identities on objects in the cocompletion of A . Both of thoseidentity morphisms are isomorphisms in B ( X, A ) because they and their inversesare bounded by N .Recall that the standard n -simplex can be viewed as the nerve of the ordered set n = { , , . . . , n } , with the natural order, viewed as a category. Let D ∈ b S n X .We define a functor l ( D , n ) : i B ( D , A ) × n → i B ( X, A ) as follows. On objects,( l ( D , n ) F ) x = L ϑ ( i )= x F d , where i denotes the vertex of ∆ n = N. n correspondingto i . On morphisms, l ( D , n ) is defined by the requirement that the restriction to thesubcategory i B ( D , A ) × j is the functor induced by θ j , and that (id × ( i ≤ j ))( F )is sent to N ji ( F ). This is compatible with the inclusion of elements in b S n X , so weobtain a functor colim −−−−→D∈ b S n X i B ( D , A ) × n −→ i B ( X, A ) , and therefore a maphocolim −−−−→D∈ b S n X N. i B ( D , A ) × ∆ n −→ N. i B ( X, A ) . If M is a symmetric monoidal category, let the t -th space in Spt( M ) be denotedby Spt t ( M ), and let σ t : S ∧ Spt t ( M ) → Spt t +1 ( M ) be the structure map forSpt( M ). The fact that the natural transformations N ji are symmetric monoidalshows in particular that we obtain mapsΛ t : hocolim −−−−→D∈ b S n X Spt t ( i B ( D , A )) × ∆ n −→ Spt t ( i B ( X, A ))so that the diagramshocolim −−−−→D∈ b S n X ( S ∧ Spt t ( i B ( D , A ))) × ∆ n / / σ t × id (cid:15) (cid:15) S ∧ Spt t ( i B ( X, A )) σ t (cid:15) (cid:15) hocolim −−−−→D∈ b S n X Spt t +1 ( i B ( D , A )) × ∆ n Λ t +1 / / Spt t +1 ( i B ( X, A ))commute. Further, for each t we obtain a map (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k hocolim −−−−→D∈ b S n X Spt t ( i B ( D , A )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −→ Spt t ( i B ( X, A ))respecting the structure maps in Spt t . This gives a map ℓ : c J h ( X ; A ) → K ( X, A )where c J h ( X ; A ) stands for the realization of the simplicial spectrum k hocolim −−−−→ C ∈ b S k X K ( C, A ) . XCISION IN EQUIVARIANT FIBRED G -THEORY 21 Since ℓ is natural in X and is compatible with delooping, it generalizes to thehomotopy natural transformation ℓ K : J h ( X ; A ) → K −∞ ( X, A ). Composing thiswith the Cartan natural transformation K −∞ ( X, A ) → G −∞ X ( Y ) gives ℓ : J h ( X, Y ) −→ G −∞ X ( Y ) . Definition 7.1 (Fibred assembly map in G -theory) . The homotopy natural trans-formation A ( X, Y ) : h lf ( X ; G −∞ ( Y )) −→ G −∞ X ( Y )is the composition of η and ℓ .7.2. Fibrewise trivialization. In this section we want to justify the claim fromthe introduction that in the new equivariant theory we have built there are fibrewisetrivializations. First, we will state the desired fact precisely.Recall the proper metric space Y S described in section 6. We assume that thetrivial action s is in S and use the notation Y for the space Y with the trivialaction. Theorem 7.2. The equivariant inclusion of metric spaces Y → Y S induces anequivalence e G Γ ( Y ) Γ → e G Γ ( Y S ) Γ . Therefore, there is an equivalence e G Γ ( Y ) → E Γ ( Y ) . We start with several facts about filtrations on modules. Let Φ d : P ( X ) → P ( X )denote the functor that assigns to a subset of X its d -neighborhood in X . We canthink of an object of G ( X ) as a pair ( F, θ ), where θ has a grading with the propertiesspelled out in Definition 5.5. Given two X -filtrations θ and η , we say θ is containedin η if θ ( S ) ⊂ η ( S ) for all S ⊂ X and write θ ≤ η . We say two X -filtrations θ and η are similar if there is a number d so that θ ≤ η ◦ Φ d and η ≤ θ ◦ Φ d . Lemma 7.3. If θ and η are similar then the objects ( F, θ ) and ( F, η ) are isomorphicin G ( X ) .Proof. The conditions ensure that the identity homomorphism is boundedly con-trolled in both directions. (cid:3) Let f : X → Y be a coarse map of proper metric spaces, so that for every d ≥ L ( d ) ≥ d ( x, y ) ≤ d implies d ( f ( x ) , f ( y )) ≤ L ( d ). Suppose furtherthat f is proper. Given an X -filtration θ on an R -module F , we define f ∗ ( θ ) to bethe Y -filtration on F given by f ∗ ( θ )( U ) = θ ( f − ( U )). Similarly, given a Y -filtration θ on F , we define f ∗ ( θ ) to be the X -filtration on F given by f ∗ ( θ )( U ) = θ ( f ( U )).Recall the definition of Y S in 6.6. Let i : Y → Y S be the inclusion y → ( y, s ),an isometric embedding, and let π : Y S → Y denote the projection, a distancenon-increasing map. Lemma 7.4. Let F be any R -module. Then any Y S -filtration on F is similar toone of the form i ∗ θ , where θ is a Y -filtration on F .Proof. Let ex : P ( Y S ) → P ( Y S ) be defined by ex ( U ) = π ( U ) × S . It is clear fromthe definition that U ⊂ ex ( U ). It is also readily checked that ex ( U ) ⊂ Φ ( U ), whichshows that any Y S -filtration θ on an R -module F is similar to the Y S -filtration θ ◦ ex .Let θ denote the Y -filtration on F given by θ ( U ) = θ ( U × S ). Then it is clear that θ ◦ ex = π ∗ ( θ ). It therefore suffices to show that for any Y -filtration η on F , wehave that i ∗ η and π ∗ η are similar. But it is clear that π ∗ η ≤ i ∗ η ◦ Φ , which givesthe result. (cid:3) We also have the following useful fact. Lemma 7.5. Suppose that we are given two X -filtrations θ and η on an R -module F , and that f : F → F is bounded as a morphism from ( F, θ ) to ( F, η ) . Supposefurther that θ ′ and η ′ are also X -filtrations, and that θ ′ and η ′ are similar to θ and η , respectively. Then f : F → F is bounded as a morphism from from ( F, θ ′ ) to ( F, η ′ ) . Suppose a metric space Y has an action by a discrete group Γ through coarseequivalences. Recall that we say the action is bounded if for each γ ∈ Γ, there is b ( γ ) ≥ d ( y, γy ) ≤ b ( γ ) for all y ∈ Y . The following is an elementaryobservation. Lemma 7.6. Suppose Y is a proper metric space equipped with a bounded Γ -actionby coarse maps. Then, given any Y -filtration θ on an R -module F , and any γ ∈ Γ we have that θ and γ ∗ θ are similar. We have the following equivalent interpretation of the category G h Γ ( Y ) intro-duced in Definition 6.6. An object of G h Γ ( Y ) is given by data ( F, θ, { f γ,γ ′ } γ,γ ′ ∈ Γ )where(1) F is an R -module,(2) θ is a Y -filtration on F ,(3) f γ,γ ′ is an automorphism of F ,(4) f γ,γ = id F and f γ,γ ′ ◦ f γ ′ ,γ ′′ = f γ,γ ′′ ,(5) f γ,γ ′ is bounded when regarded as a homomorphism ( F, γ ′∗ θ ) → ( F, γ ∗ θ ).Lemmas 7.5 and 7.6 give that condition 5 on f γ,γ ′ is equivalent to f γ,γ ′ beingbounded as a homomorphism from ( F, θ ) to ( F, θ ).Now we are ready to prove Theorem 7.2. Proof. First, observe it suffices to verify that the inclusion induces an equivalenceof categories i G h Γ ( Y ) → i G h Γ ( Y S ). The equivalence then clearly extends to cate-gories of diagrams of objects in G h Γ ( Y S ), and Waldhausen’s S. -construction usedto produce the spectra gives simplicial spaces which in every level are the nerves ofcategories of isomorphisms of diagrams of cofibrations of objects in G h Γ ( Y S ).The inclusion exhibits i G h Γ ( Y ) as a full subcategory of i G h Γ ( Y S ), and it followsthat it’s enough to prove that every object of i G h Γ ( Y S ) is isomorphic to an objectof i G h Γ ( Y ). An object of i G h Γ ( Y ) is given by data ( F, θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ), where θ is an Y -filtration on F , and where f is an automorphism of F which is boundedas a homomorphism from ( F, θ ) to ( F, θ ). Note that the transformations by γ ’s donot occur in this situation because the action of Γ on Y is trivial. The inclusionfunctor i G h Γ ( Y ) ֒ → i G h Γ ( Y S ) is given by( F, θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ) → ( F, i ∗ θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ) , so an object ( F, θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ) is in the subcategory i G h Γ ( Y ) if and only if θ isof the form i ∗ η for some Y -filtration η .Next, we observe that if ( F, θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ) is an object of i G h Γ ( Y S ), and if θ ′ is an Y S -filtration on F which is similar to θ , then (a) ( F, θ ′ , { f γ,γ ′ } γ,γ ′ ∈ Γ ) isalso an object of ( F, i ∗ θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ), and (b) ( F, θ, { f γ,γ ′ } γ,γ ′ ∈ Γ ) is isomorphicto ( F, θ ′ , { f γ,γ ′ } γ,γ ′ ∈ Γ ). But we have already observed in Lemma 7.4 that every Y S -filtration on F is equivalent to one of the form i ∗ η , for some Y -filtration η on F , proving the result. (cid:3) XCISION IN EQUIVARIANT FIBRED G -THEORY 23 References [1] M. Cardenas and E.K. Pedersen, On the Karoubi filtration of a category , K -theory, (1997),165–191.[2] G. 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