Asymptotic transfer maps in parametrized K-theory
aa r X i v : . [ m a t h . K T ] F e b ASYMPTOTIC TRANSFER MAPS IN PARAMETRIZED K -THEORY GUNNAR CARLSSON AND BORIS GOLDFARB
Abstract.
We define asymptotic transfers in bounded K -theory togetherwith a context where this can be done in great generality. Controlled alge-bra plays a central role in many advances in geometric topology, includingrecent work on Novikov, Borel, and Farrell-Jones conjectures. One of the fea-tures that appears in various manifestations throughout the subject, startingwith the original work of Farrell and Jones, is an asymptotic transfer whosemeaning and construction depend on the geometric circumstances. We firstdevelop a general framework that allows us to construct a version of asymp-totic transfer maps for any finite aspherical complex. This framework is theequivariant parametrized K -theory with fibred control. We also include severalfibrewise excision theorems for its computation and a discussion of where thestandard tools break down and which tools replace them. Contents
1. Introduction 12. Pedersen-Weibel categories, Karoubi filtrations 43. Parametrized K -theory with fibred control 94. Equivariant parametrized K -theory 145. Example: absence of Karoubi filtrations 196. Bounded actions and a cone construction 217. Asymptotic transfer in parametrized K -theory 24References 291. Introduction
Controlled algebra of geometric modules uses geometric control conditions well-suited for generating K -theory spectra out of the associated additive controlledcategories of free modules. Specifically, the bounded controlled algebra was devel-oped by E.K. Pedersen and C. Weibel in [22, 23]. It has been used extensively fordelooping K -theory of rings and other algebraic objects and in the study of theNovikov conjecture [8, 9, 14] about the assembly map for algebraic K -theory and L -theory of certain group rings.There are two innovations in this paper: the setting of parametrized K -theorywith fibred control which generalizes bounded K -theory and a general asymptotictransfer that can be defined in this new theory. Date : February 5, 2020.
We start by explaining the work of Pedersen/Weibel as an algebraic theory offree modules, possibly infinitely generated, parametrized by a metric space. Theinput is a proper metric space X (i.e. a metric space in which every closed boundedsubset is compact) and a ring R . Given this data, consider triples ( M, B, ϕ ), where M is a free left R -module with basis B , and where ϕ : B → X is a referencefunction with the property that the inverse image of any bounded subset is finite.A morphism from ( M, B, ϕ ) to ( M ′ , B ′ , ϕ ′ ) is an R -module homomorphism f from M to M ′ which has the property that there exists a bound b ≥ β ∈ B , f ( β ) is in the span of basis elements β ′ ∈ B ′ for which d ( ϕ ( β ) , ϕ ′ ( β ′ )) ≤ b . This is the simplest version of a control condition that canbe imposed on homomorphisms to construct various categories of modules, and itleads to the category of geometric modules C ( X, R ). This is an additive category towhich one can apply the usual algebraic K -theory construction.A formulation of this theory that was observed already in [22] allows one touse objects from an arbitrary additive category A as “coefficients” in place of thefinitely generated free R -modules. We will spell out the details of this further inthe paper. So one obtains a new additive category C ( X, A ). This observation leadsto the possibility of iterating the geometric control construction: if there are twoproper metric spaces X and Y , then we have an additive category C ( X, C ( Y, R )).This is precisely a case of fibred control we want to study in this paper.To give an idea for what kind of control is implied by this construction, let’s parsethe implications for objects viewed as R -modules. In this setting an object is a freemodule M viewed as parametrized over the product X × Y , so we have a pair ( M, B )with a reference function ϕ : B → X × Y . The control condition on homomorphisms f from ( M , B , ϕ ) to ( M , B , ϕ ) amounts to the existence of a number b X ≥ c : X → [0 , ∞ ) such that for β ∈ B , we have that f ( β ) is a linearcombination of basis elements β ′ ∈ B so that d ( π X ( ϕ ( β )) , π X ( ϕ ( β ′ ))) ≤ b X and d ( π Y ( ϕ ( β )) , π Y ( ϕ ( β ′ ))) ≤ c ( π X ( ϕ ( β ))). Contrast this with the control conditionin C ( X × Y, R ) which is exactly as above except c can be chosen to be a constantfunction. This should suggest that the condition of fibred control, with X regardedas a base and Y regarded as a fibre, relaxes the usual control condition over X × Y by letting c be a quantity varying with x ∈ X .It is shown in [8] that C ( X, R ) enjoys a certain excision property, which per-mits among other things a comparison with Borel-Moore homology spectra withcoefficients in the K -theory spectrum K ( R ). This property becomes crucial in theequivariant homotopy theoretic approach to the Novikov conjecture. In this paperwe prove a number of excision results in categories with fibred control and somerelated categories of fixed-point objects with respect to naturally occurring actionson the metric spaces. These theorems are required for further work on assemblymaps in K -theory.We briefly address the methods used and the interesting phenomena that comeup. The go-to technique for proving localization and excision theorems in controlledalgebra is the use of Karoubi filtrations in additive categories. This technique isindeed used also in our proofs of general non-equivariant fibrewise excision results(section 3) and in some specific equivariant situations (section 4). However, wewant point out one curious situation that will interest the experts. The most usefulnovelty of the fibred control is that it allows introduction of constraints on features,including actions, that vary across fibres. When this is done to actions, it becomes SYMPTOTIC TRANSFER MAPS IN PARAMETRIZED K -THEORY 3 unnatural to insist that the action is by maps that are necessarily isometries onthe nose or, if the action is by more general coarse equivalences, that the samenumerical constraint is satisfied across all fibres. In this situation we can statedesired excision statements about fixed point object categories for multiple actionsbut discover that they are inaccessible to the Karoubi filtration technique. We givean example in section 5 where we are able to pinpoint the deficiency that makesthe Karoubi filtrations unavailable.Before we move on to the asymptotic transfer, we want to point out that thegeneralization of controlled algebra in this paper is useful for a variety of otherapplications. It is most immediately the natural controlled theory to consider forbundles on non-compact manifolds where the control in the fibre direction canvary with the fibre. This holds more generally for stacks or the coarse analogues ofbundles that have already appeared in the literature [26]. Another manifestation offibred control is found in the active current research area seeking sufficient geometricconditions for verifying the coarse Baum-Connes conjecture. The original conditionin [27, 28] was existence of a coarse embedding of the group into a Hilbert space. Theembedding can be viewed as a way to gain geometric control for performing stableexcision over the group or the space with a proper cocompact action by the group.That condition has been relaxed by several authors to fibred coarse embeddings [1, 15, 16, 18, 20]. The results of this paper allow to address the Borel conjecturein K -theory for groups admitting a fibred coarse embedding into Hilbert space.A transfer map is a very useful tool in algebraic topology. For a continuousmap between spaces f : X → Y and a covariant functor F on spaces to some alge-braic category, a transfer T ( f ) is a morphism F ( Y ) → F ( X ), usually with severalproperties that depend on the context. This is a “wrong way” map as opposed tothe usual induced maps F ( f ) : F ( X ) → F ( Y ). A survey of the general homotopytheoretic construction for projections in Hurewicz fibrations with homotopy finitefibers and numerous points of views and applications is given by its creators inBecker/Gottlieb [5]. Infinite transfers with no finiteness assumption on the fibersappear naturally in the area of topology related to modern proofs of the Novikovand Farrell-Jones conjectures. Here the functor F is usually either locally finite ho-mology or controlled K -theory. We refer to [7] for a general introduction to the ideasand to [8] for the details required for the proofs of the Novikov conjecture aboutinjectivity of assembly maps in K -theory. In this paper, we define a parametrizedversion of the infinite transfer and establish some of its properties needed for thestudy of surjectivity of those maps.Asymptotic transfers are a crucial feature in modern proofs of cases of the Borelrigidity conjecture. This is most notable in the pioneering work of Farrell and Jones[17]. The basic idea in controlled topology is that improvement in control leads totrivialization of an appropriate h-cobordism or a surgery problem. In the case ofclosed hyperbolic manifolds, Farrell and Jones were able to define a special as-ymptotic transfer to the unit sphere bundle of the manifold and use the geodesicflow on the bundle to control the lifted homotopies and eventually trivialize theobstructions to geometric problems. Further work on the Farrell-Jones conjecturerelied in similar ways on an asymptotic transfer to a setting where a given algebraicrepresentation of the geometric problem can be manipulated to have progressivelybetter control. This can be done in cases where subtle consequences of immanentmanifestations of non-positive curvature can be exploited for controlling geometric GUNNAR CARLSSON AND BORIS GOLDFARB problems. We refer to instances of this in section 6 of [3] or section 7 of [4]. Incontrast to these cases, we construct in this paper a K -theory transfer that usesno geometric constraints. The target of this transfer is the parametrized K -theorywhere, in analogous fashion, a particular trivialization exists as indicated at variousplaces, for example in Theorem 4.8. For the application to aspherical manifolds, thefibre for the parametrized K -theory is the universal cover of the normal disk bundleto the embedding of the manifold in a Euclidean space. It is a metric space withthe metric constructed in section 6. Trivialization in parametrized K -theory, aftercertain algebraic constraints are imposed on the coefficient ring, is an outcome of ageneral equivariant excision theorem proven elsewhere [13]. We address the relationbetween that G -theoretic excision theorem and the excision theorems from section4 in Remark 5.4.The paper starts with a paced introduction to bounded K -theory and Karoubifiltration methods used to compute it. After that, we introduce fibred control andthe equivariant fibred K -theory in sections 3 and 4. The material from section 2 isused to motivate and construct the proofs of similar controlled excision theorems inthe parametrized setting. We also use it to carefully describe in section 5 how thestandard techniques fail in proving a natural desired statement in the parametrizedtheory. This motivates the development of fibred G -theory in [12, 13] where theequivariant fibred excision theorems do hold. Finally, the last two sections 6 and 7are about the asymptotic transfer.2. Pedersen-Weibel categories, Karoubi filtrations
Bounded K -theory introduced in Pedersen [21] and Pedersen–Weibel [22] asso-ciates a nonconnective spectrum K −∞ ( M, R ) to a proper metric space M (a metricspace where closed bounded subsets are compact) and an associative ring R withunity. We are going to start with a careful, at times revisionist, review of the well-known features of this theory that will need generalization.The metric spaces in this subject are often understood in the generalized senseas follows. Definition 2.1. A generalized metric space is a set X and a function d : X × X → [0 , ∞ ) ∪ {∞} which is reflexive, symmetric, and satisfies the triangle inequalityin the usual way. Classical metric spaces are the generalized metric spaces withdistance function d assuming only finite values. We will use the term metric space to mean a generalized metric space.We call a subset M ′ of a metric space M a metric subset if all values of d | M ′ × M ′ are finite. The maximal metric subsets are called metric components . A metric spaceis proper if it is a countable disjoint union of metric components M i , and all closedmetric balls in M are compact. The metric topology on a metric space is definedas usual.In this section, all metric spaces and facts about them may be viewed in this gen-eralized context. We will restrict to spaces with a single metric component startingin the next section. Definition 2.2. C ( M, R ) is the additive category of geometric R -modules whoseobjects are functions F : M → Free fg ( R ) which are locally finite assignments of freefinitely generated R -modules F m to points m of M . The local finiteness conditionrequires precisely that for any bounded subset S ⊂ M the restriction of F to S has SYMPTOTIC TRANSFER MAPS IN PARAMETRIZED K -THEORY 5 finitely many nonzero values. Let d be the distance function in M . The morphismsin C ( M, R ) are the R -linear homomorphisms φ : M m ∈ M F m −→ M n ∈ M G n with the property that the components F m → G n are zero for d ( m, n ) > b for somefixed real number b = b ( φ ) ≥
0. The associated K -theory spectrum is denoted by K ( M, R ), or K ( M ) when the choice of ring R is implicit, and is called the boundedK-theory of M . Notation . For a subset S ⊂ M and a real number r ≥ S [ r ] will stand for themetric r -enlargement { m ∈ M | d ( m, S ) ≤ r } . In this notation, the metric ball ofradius r centered at x is { m } [ r ] or simply m [ r ].Now for every object F and subset S there is a free R -module F ( S ) = L m ∈ S F m .The condition that φ is controlled as above is equivalent to existence of a number b ≥ φF ( S ) ⊂ F ( S [ b ]) for all choices of S . Proposition 2.4.
The description of C ( M, R ) in the introduction defines a categoryadditively equivalent to the one in Definition , establishing a dictionary betweenterminology in various papers in the literature.Proof. Given a triple (
M, B, ϕ ) described in the introduction, define M x as the R -submodule freely generated by ϕ − ( x ). It is clear that this gives an additive functorin one direction. The inverse functor is constructed by selecting a finite basis in each F x and defining B to be the union of these bases. The map ϕ sends a basis element b to x if b ∈ F x . (cid:3) Inclusions of metric spaces induce additive functors between the correspondingbounded K -theory spectra. The main result of Pedersen–Weibel [22] is a deloopingtheorem which can be stated as follows. Theorem 2.5 (Nonconnective delooping of bounded K -theory) . Given a propermetric space M and the standard Euclidean metric on the real line R , the naturalinclusion M → M × R induces isomorphisms K n ( M ) ≃ K n − ( M × R ) for allintegers n > . If one defines the spectrumK −∞ ( M, R ) = hocolim −−−−→ k Ω k K ( M × R k ) , then the stable homotopy groups of K −∞ ( R ) = K −∞ (pt , R ) coincide with the alge-braic K -groups of R in positive dimensions and with the Bass negative K -theory of R in negative dimensions. When we develop the equivariant theory, we will want to consider group actionsby maps that are more general than isometries. Let X and Y be proper metricspaces with metric functions d X and d Y . This means, in particular, that closedbounded subsets of X and Y are compact. Definition 2.6.
A function f : X → Y between proper metric spaces is uniformlyexpansive if there is a real positive function l such that for pairs of points x and x the inequality d X ( x , x ) ≤ r implies the inequality d Y ( f ( x ) , f ( x )) ≤ l ( r ).This is the same concept as bornologous maps in Roe [24, Definition 1.8]. It shouldbe emphasized that the function is not assumed to be necessarily continuous. The GUNNAR CARLSSON AND BORIS GOLDFARB function f is proper if f − ( S ) is a bounded subset of X for each bounded subset S of Y . We say f is a coarse map if it is proper and uniformly expansive. Theorem 2.7.
Coarse maps between proper metric spaces induce additive functorsbetween bounded categories.
We include a proof of this basic fact that will be generalized in parametrizedsetting.
Proof.
Let f : X → Y be coarse. The additive functor f ∗ : C ( X, R ) → C ( Y, R ). isinduced on objects by the assignment( f ∗ F ) y = M x ∈ f − ( y ) F x . Since f is proper, f − ( y ) is a bounded set for all y in Y . So the direct sum inthe formula is finite, and ( f ∗ F ) y is a finitely generated free R -module. If S ⊂ Y is a bounded subset then f − ( S ) is bounded. There are finitely many F z = 0 for z ∈ f − ( S ) and therefore finitely many ( f ∗ F ) y = 0 for y ∈ S . This shows f ∗ F islocally finite.Notice that f ∗ F = M y ∈ Y ( f ∗ F ) y = M y ∈ Y M z ∈ f − ( y ) F z = F. Suppose we are given a morphism φ : F → G in C ( X, R ). Interpreting f ∗ F and f ∗ G as the same R -modules, as in the formula above, we define f ∗ φ : f ∗ F → f ∗ G equal to φ . We must check that f ∗ φ is bounded. Suppose φ is bounded by D , and f is l -coarse. We claim that f ∗ φ is bounded by l ( D ). Indeed, if d Y ( y, y ′ ) > l ( D )then d X ( x, x ′ ) > D for all x , x ′ ∈ X such that f ( x ) = y and f ( x ′ ) = y ′ . So allcomponents φ x,x ′ = 0, therefore all components ( f ∗ φ ) y,y ′ = 0. (cid:3) Corollary 2.8. K −∞ is a covariant functor from the category of proper metricspaces and coarse maps to the category of spectra. The 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. Examples 2.9.
The isometric embedding of a metric subspace is a coarse map. Anisometry, which is a bijective isometric map, is a coarse equivalence. An isometricembedding onto a subspace that has the property that its bounded enlargement isthe whole target metric space is also a coarse equivalence.Any bounded function f : X → X , that is a function with d X ( x, f ( x )) ≤ D for all x ∈ X and a fixed D ≥
0, is a coarse equivalence. More generally, a quasi-isometry f : X → Y onto a subset U ⊂ Y such that for some number s ≥ U [ s ] = Y is a coarse equivalence.The following definition makes precise a useful class of metrics one has on afinitely generated group. Definition 2.10.
The word-length metric d = d Ω on a group Γ with a fixed finitegenerating set Ω closed under taking inverses is the length metric induced from thecondition that d ( γ, γω ) = 1, whenever γ ∈ Γ and ω ∈ Ω. In other words, d ( α, β ) isthe minimal length t of sequences α = γ , γ , . . . γ t = β in Γ where each consecutivepair of elements differs by right multiplication by an ω from Ω. This metric makesΓ a proper metric space with a free action by Γ via left multiplication. SYMPTOTIC TRANSFER MAPS IN PARAMETRIZED K -THEORY 7 If one considers a different choice of a finite generating set Ω ′ , it is well-knownthat the identity map on the group with the two metrics d Ω and d Ω ′ is a quasi-isometry, cf. [24, Proposition 1.15]. We see from the combination of Corollary 2.8and the examples above that the bounded K -theory of Γ is independent from thechoice of a finite generating set, up to an equivalence.Our special interest in this paper is in actions by bounded coarse equivalences. Definition 2.11.
A left action of Γ on a metric space X is bounded if for eachelement γ ∈ Γ there is a number B γ ≥ d ( x, γx ) ≤ B γ for all x ∈ X .Such actions do appear naturally but only in special circumstances. If Γ is agroup acting on itself via left multiplication, then this action is a bounded actionby isometries when the group is a finitely generated abelian group. However, theleft multiplication action of the integral lattice in the three-dimensional Heisenberggroup is already not bounded because of the well-known warping of the metric.In section 6 we will define a useful conversion of any action by isometries to abounded action which is sufficient for our and many other purposes.We next review an excision theorem that makes bounded K -theory computablein special but crucial geometric situations. This review includes some details thatwe will use to prove some excision results for parametrized K -theory in the nextsection.Suppose U is a subset of M . Let C ( M, R ) D forsome fixed number D > F . This is an additive subcategory of C ( M, R )with the associated K -theory spectrum K −∞ ( M, R ) D and d ( m, V ) > D for some numbers D , D ≥
0. It is easy to see that C ( U, R ) is asubcategory which is in fact equivalent to C ( M, R )
A pair of subsets S , T of a proper metric space M is called coarsely antithetic if S and T are proper metric subspaces with the subspace metricand for each pair of numbers d S , d T ≥ d ′ ≥ S [ d S ] ∩ T [ d T ] ⊂ ( S ∩ T )[ d ′ ] . Examples of coarsely antithetic pairs include complementary closed half-spacesin a Euclidean space, as well as any two non-vacuously intersecting closed subsets of
GUNNAR CARLSSON AND BORIS GOLDFARB a simplicial tree. In the latter example, a tree is viewed as a geodesic metric spacewhere the metric is induced from the local condition that each edge is isometric toa closed real interval.
Corollary 2.14. If U and V is a coarsely antithetic pair of subsets of M whichform a cover of M , then the commutative squareK −∞ ( U ∩ V ) / / (cid:15) (cid:15) K −∞ ( U ) (cid:15) (cid:15) K −∞ ( V ) / / K −∞ ( M ) is a homotopy pushout. We want to outline the proof of Theorem 2.12 in specific terms that will be usedlater. Another reason for recording rather standard details is to refer to them laterwhen we study a failure of these methods in section 5.The notion of Karoubi filtrations in additive categories is central to the proof ofthis theorem as developed by Cardenas/Pedersen [6].
Definition 2.15.
An additive category C is Karoubi filtered by a full subcategory A if every object C of C has a family of decompositions { C = E α ⊕ D α } with E α ∈ A and D α ∈ C , called a Karoubi filtration of C , satisfying the following properties. • For each object C of C , there is a partial order on Karoubi decompositionssuch that E α ⊕ D α ≤ E β ⊕ D β whenever D β ⊂ D α and E α ⊂ E β . • Every morphism A → C with A ∈ A and C ∈ C factors as A → E α → E α ⊕ D α = C for some value of α . • Every morphism C → A with C ∈ C and A ∈ A factors as C = E α ⊕ D α → E α → A for some value of α . • For each pair of objects C and C ′ with the corresponding filtrations { E α ⊕ D α } and { E ′ α ⊕ D ′ α } , the filtration of C ⊕ C ′ is the family { C ⊕ C ′ =( E α ⊕ E ′ α ) ⊕ ( D α ⊕ D ′ α ) } .A morphism f : C → D in C is A - zero if f factors through an object of A . Onedefines the Karoubi quotient C / A as the additive category with the same objects as C and morphism sets Hom C / A ( C, D ) = Hom(
C, D ) / {A− zero morphisms } .The following is the main theorem of Cardenas–Pedersen [6, Theorem 7.1]. Theorem 2.16 (Fibration Theorem) . Suppose C is an A -filtered category, thenthere is a homotopy fibration K ( A ∧ K ) −→ K ( C ) −→ K ( C / A ) . Here A ∧ K is a certain subcategory of the idempotent completion of A with the samepositive K -theory as A . The next statement is a consequence of the Fibration Theorem obtained in thelast paragraph of [6].
Corollary 2.17.
Suppose C is an A -filtered category, then there is a homotopyfibration K −∞ ( A ) −→ K −∞ ( C ) −→ K −∞ ( C / A ) . Theorem 2.12 follows from this Corollary by the following device.
SYMPTOTIC TRANSFER MAPS IN PARAMETRIZED K -THEORY 9 Sketch of the proof of Theorem . The first crucial observation is that C = C ( M )is A = C ( M )
0, are free direct summands of F ( M ) which can be given thestructure of a geometric module over M in the obvious way. Now the decompositions F = F ( U [ k ]) ⊕ F ( M \ U [ k ])is the family we need for a Karoubi filtration. Suppose, for illustration, we have f : A → F bounded by b ≥
0. Then A ( M ) = A ( U [ r ]) for some r ≥ f ( A ) ⊂ F ( U [ r + b ]). So, indeed, f factors through this direct summand.Since C ( M ) is C ( M ) The new category has the same objects as C ( X × Y, R ) but aweaker control condition on the morphisms. For a choice of a base point x in X ,any function f : [0 , + ∞ ) → [0 , + ∞ ), and a real number D ≥ 0, define N ( D, f )( x, y ) = x [ D ] × y [ f ( d ( x, x ))] , the ( D, f )-neighborhood of ( x, y ) in X × Y . A homomorphism φ : F → G is called fibrewise ( D, f )- bounded for some choiceof D ≥ f : [0 , + ∞ ) → [0 , + ∞ ), or simply fibrewisebounded , if the components F ( x,y ) → G ( x ′ ,y ′ ) are zero maps for ( x ′ , y ′ ) outside ofthe ( D, f )-neighborhood of ( x, y ). Lemma 3.2. The notion of a fibrewise bounded homomorphism is independent ofthe choice of x in X .Proof. It suffices to show that if φ is fibrewise ( D, f )-bounded with respect toa choice of a base point x , and if given a different choice x ′ of a base point,then φ is fibrewise ( D ′ , f ′ )-bounded for some choice of the parameters ( D ′ , f ′ ). Wechoose D ′ = D and define f ′ ( t ) = f ( t + d ( x , x ′ )). From the triangle inequality, d ( x, x ) ≤ d ( x, x ′ ) + d ( x , x ′ ), so f ′ ( d ( x, x ′ )) = f ( d ( x, x ′ ) + d ( x , x ′ )) ≥ f ( d ( x, x )) . This shows that x [ D ′ ] × y [ f ′ ( d ( x, x ′ ))] always contains x [ D ] × y [ f ( d ( x, x ))]. (cid:3) We define the category of geometric modules over the product X × Y with fibrewisecontrol over X as the category of usual geometric R -modules over the product metricspace and fibrewise bounded homomorphisms. Notation . The notation for this category which emphasizes the special roleof the factor X is C X ( Y, R ). This is very much in line with the original notation C X ( R ) used by Pedersen and Weibel for the bounded category of R -modules over X and specifically C i ( R ) for the theory over R i , see for example [22, Remark 1.2.3].To streamline the notation even further, we often omit the the ring R from thenotation C X ( Y ) as it usually plays the role of a dummy variable. Definition 3.4. The connective bounded K -theory of geometric R -modules overthe product X × Y with fibrewise control over X is the spectrum K X ( Y ) associatedto the additive category C X ( Y ).It follows from Example 1.2.2 of Pedersen-Weibel [22] that in general the fibredbounded category C X ( Y ) is not isomorphic to C ( X × Y, R ). The proper generalityof that work, as explained in [22, 23], starts with a general additive category A embedded in a cocomplete additive category, generalizing the setting of free finitelygenerated R -modules as a subcategory of all free R -modules. All of the results ofPedersen-Weibel hold for C ( X, A ).In these terms, the category C X ( Y ) can be seen isomorphic to the category C ( X, A ), where A = C ( Y, R ). Unfortunately this concise description is not sufficientfor defining features such as the necessary choice of Karoubi filtrations and otherdetails in forthcoming arguments.The difference between C X ( Y ) and C ( X × Y, R ) is made to disappear in [22] bymaking C ( Y, R ) “remember the filtration” of morphisms when viewed as a filteredadditive category with Hom D ( F, G ) consisting of all morphisms φ ∈ Hom( F, G )which are bounded by D . Identifying a small category with its set of morphisms,one can think of the bounded category as C ( Y, R ) = colim −−−−→ D ∈ R C D ( Y, R ) , SYMPTOTIC TRANSFER MAPS IN PARAMETRIZED K -THEORY 11 where C D ( Y, R ) = Hom D ( C ( Y, R )) is the collection of all Hom D ( F, G ). This inter-pretation gives an exact embedding C ( X × Y, R ) = colim −−−−→ D ∈ R C ( X, C D ( Y, R )) ι −−→ C ( X, colim −−−−→ D ∈ R C D ( Y, R )) = C ( X, C ( Y, R )) , which induces a map of K -theory spectra K ( ι ) : K ( X × Y, R ) → K X ( Y ).We want to develop some results for a variety of categories with fibred boundedcontrol where the Karoubi filtration techniques suffice. Notation . Let C k = C X ( Y × R k ) , C + k = C X ( Y × R k − × [0 , + ∞ )) , C − k = C X ( Y × R k − × ( −∞ , . We will also use the notation C < + k = colim −−−−→ D ≥ C X ( Y × R k − × [ − D, + ∞ )) , C < − k = colim −−−−→ D ≥ C X ( Y × R k − × ( −∞ , D ]) , C < k = colim −−−−→ D ≥ C X ( Y × R k − × [ − D, D ]) . Clearly C k is C < − k -filtered and that C < + k is C < k -filtered. There are equivalencesof categories C < k ≃ C k − , C < − k ≃ C − k , and C k / C < − k ≃ C < + k / C < k , as explained at theend of section 2. By Theorem 2.16, the commutative diagram K (( C < k ) ∧ K ) −−−−→ K ( C < + k ) −−−−→ K ( C < + k / C < k ) y y y ∼ = K (( C < − k ) ∧ K ) −−−−→ K ( C k ) −−−−→ K ( C k / C < − k )where all maps are induced by inclusions on objects, is in fact a map of homotopyfibrations. The categories C < + k and C < − k are flasque, that is, possess an endofunctorSh such that Sh( F ) ∼ = F ⊕ Sh( F ), which can be seen by the usual Eilenberg swin-dle argument. Therefore K ( C < + k ) and K ( C < − k ) are contractible by the AdditivityTheorem, cf. Pedersen–Weibel [22]. This gives a map K ( C k − ) → Ω K ( C k ) whichinduces isomorphisms of K -groups in positive dimensions. Definition 3.6. The nonconnective fibred bounded K -theory is the spectrum K −∞ X ( Y ) def = hocolim −−−−→ k> Ω k K ( C k ) . If Y is the single point space then the delooping K −∞ X (pt) is clearly equiva-lent to the nonconnective delooping K −∞ ( X, R ) of Pedersen–Weibel via the map K ( ι ) : K −∞ ( X × pt , R ) → K −∞ X (pt). Remark 3.7.