ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC
DIMENSION
ARTHUR BARTELS AND DAVID ROSENTHAL
Abstract.It is proved that the assembly maps in algebraic K- and L-theory
with respect to the family of finite subgroups is injective for groups *
* with
finite asymptotic dimension that admit a finite model for E_ . The resul*
*t also
applies to certain groups that admit only a finite dimensional model for*
* E_ . In
particular, it applies to discrete subgroups of virtually connected Lie *
*groups.
Introduction
Assembly maps in algebraic K- and L-theory are designed to study the K- and
L-theory of group rings R[ ], which contain important geometric information abo*
*ut
manifolds with fundamental group . Similarly, the Baum-Connes map is used to
analyze the topological K-theory of the reduced C*-algebra of . An important
class of groups that has been studied in recent years, by topologists and analy*
*sts
alike, is the class of discrete groups with finite asymptotic dimension. Yu pro*
*ved
the Novikov conjecture for groups with finite asymptotic dimension that admit a
finite classifying space [Yu98 ]. He achieved this by using controlled techniqu*
*es to
prove the injectivity of the Baum-Connes map for such groups. Later on, Higson
was able to remove the finite classifying space assumption [Hig00]. In [Bar03b]*
*, the
first author proved the algebraic K- and L-theory versions of Yu's work by deve*
*l-
oping a squeezing theorem for higher algebraic K-theory to use with the approach
established in [Yu98 ]. For algebraic K-theory, this was also achieved in [CG04*
*a ].
The purpose of this paper is to extend the results from [Bar03b] by relaxing*
* the
finite B assumption to allow for groups with torsion. Specifically, we prove*
* the
following theorem.
Theorem A. Let be discrete group and R a ring. Assume that there is a finite
dimensional -CW -model for the universal space for proper -actions, E_ , and
assume that there is a -invariant metric on E_ such that E_ is uniformly Fin-
contractible and has finite asymptotic dimension. Then the assembly map,
(0.1) H*(E_ ; KR ) ! K*(R[ ]),
in algebraic K-theory, is a split injection.
The notion of uniformly Fin-contractibility is a strengthening of uniform co*
*n-
tractibility and is introduced in Definition 1.1. The asymptotic dimension of a
metric space was introduced by Gromov [Gro93] and is reviewed in Section 2.
____________
Date: May 3, 2006.
This work was supported by the SFB 478 "Geometrische Strukturen in der Mathe*
*matik".
1
2 ARTHUR BARTELS AND DAVID ROSENTHAL
Note that Theorem A has interesting consequences for Whitehead groups. The
classical assembly map
(0.2) Hn(B ; KR ) ! Kn(R[ ])
considered in [Bar03b, CG04a ], factors through the assembly map (0.1)via
(0.3) Hn(B ; KR ) ~=Hn(E ; KR ) ! Hn(E_ ; KR ).
In particular, the cokernel of (0.2)contains the cokernel of (0.3)in the situat*
*ion
of Theorem A. This cokernel can be evaluated using an Atiyah-Hirzebruch spectral
sequence (see [DL98 , Theorem 4.7]), or computed rationally using the equivaria*
*nt
Chern character from [L"uc02, Theorem 0.3], [LR05 , Theorem 173]. If R = Z and
n = 1, then the cokernel of (0.2)is the Whitehead group. In this way, Theorem A
implies non-vanishing results for the Whitehead group.
In Section 3 it is shown that the assumptions of Theorem A are satisfied for *
*all
discrete subgroups of Lie groups with a finite number of components. Also, if t*
*here
is a cocompact -CW -model for E_ , then, with any -invariant metric, it will *
*be
quasi-isometric to when is equipped with a word length metric. By Lemma 1.5,
such a model is uniformly Fin-contractible. Thus, Theorem A implies the followi*
*ng
corollary.
Corollary. Let be a group and assume that one of the following two conditions
is satisfied:
(i) is a discrete subgroup of a virtually connected Lie group.
(ii) has finite asymptotic dimension and admits a cocompact -CW -model
for E_ .
Then the assembly map (0.1)in algebraic K-theory is split injective for every r*
*ing
R.
The techniques used to prove Theorem A allow for a very similar proof of the
corresponding result in L-theory if ultimate lower quadratic L-theory, L<-1>*, *
*is
used. The only difference is that the compatibility of L-theory with infinite p*
*roducts
is only known under an additional K-theory assumption. This forces the extra
hypothesis in the following theorem.
Theorem B. Let be discrete group and R a ring with involution. Assume that
there is a finite dimensional -CW -model for the universal space for proper -
actions, E_ , and assume that there is a -invariant metric on E_ such that E_*
* is
uniformly Fin-contractible and has finite asymptotic dimension. Further assume
that for every finite subgroup G of , the group K-i(R[G]) vanishes for suffici*
*ently
large i. Then the assembly map,
(0.4) H*(E_ ; LR ) ! L<-1>*(R[ ]),
in L-theory, is a split injection.
As in the K-theory case, Theorem B can be used to obtain non-vanishing resul*
*ts
for the cokernel of the assembly map
H*(B ; LR ) ! L<-1>*(R[ ]).
It is well-known that the Novikov conjecture on the homotopy invariance of high*
*er
signatures is implied by the rational injectivity of this map. Clearly, Theorem*
* B
also implies the following result.
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 3
Corollary. Let be a group and assume that one of the following two conditions
is satisfied:
(i) is a discrete subgroup of a virtually connected Lie group.
(ii) has finite asymptotic dimension and admits a cocompact -CW -model
for E_ .
Then the assembly map (0.4)in L-theory is split injective for every ring R with
involution such that for every finite subgroup G of , the group K-i(R[G]) vani*
*shes
for sufficiently large i.
It is interesting to compare the finiteness conditions in Theorems A and B
with the finiteness assumption in the rational injectivity result of B"okstedt-*
*Hsiang-
Madsen [BHM93 ] for the assembly map (0.2)(with R = Z), where it is assumed
that the integral homology, Hn(B ; Z), is finitely generated for every n. The
only other injectivity results that we are aware of that apply to all subgroups
of virtually connected Lie groups are those for the Baum-Connes assembly map
by Kasparov [Kas88] and Higson [Hig00] (compare [HR00 , Section 4]). This also
implies the rational injectivity of (0.4)for R = Z. It should also be noted th*
*at
Ferry-Weinberger [FW91 ] prove the Novikov conjecture and the rational injectiv*
*ity
of (0.2)(with R = Z) for fundamental groups of non-positively curved manifolds
that are not necessarily compact. By [Bar03a], our results are in accordance wi*
*th
the Farrell-Jones conjecture [FJ93]. For more information about the Baum-Connes
and Farrell-Jones conjectures we recommend [LR05 ].
In Sections 8 and 9, Theorems 8.1 and 9.1 are proven which are slightly stro*
*nger
than the theorems stated in this introduction. There we will be considering as-
sembly maps for an additive category with a -action, as introduced in [BR05 ].
For example, this more general setup enables one to study the K- and L-theory
of crossed product rings (see [BR05 , Section 6]). Theorems A and B follow from
Theorems 8.1 and 9.1 respectively, by using the category of finitely generated *
*free
R-modules with the trivial action of .
The authors would like to thank Burkhard Wilking for his help with Section 3.
1. Uniform contractibility
Let A be a subset of a metric space X, and let R > 0. Denote by AR the set of
all points x in X for which d(x, A) R. If A = {x} consists of just one point,*
* then
we will abbreviate xR = {x}R .
Definition 1.1 (Uniformly Fin-contractible). Let act by isometries on a metric
space X. Then X is said to be uniformly Fin-contractible if for every finite su*
*bgroup
G of and every R > 0 there is an S > 0 such that the following holds: If B is
a G-invariant subset of X of diameter less than R, then the inclusion, B ! BS,
is G-equivariantly null homotopic. In particular, BS contains a fixed point for*
* the
action of G.
Remark 1.2 (Uniformly contractible). If is the trivial group, or more general*
*ly if
is torsion-free, then X is uniformly Fin-contractible if and only if X is uni*
*formly
contractible. That is, for every R > 0 there is an S > R such that for every x *
*2 X,
the inclusion, xR ! xS, is null homotopic.
4 ARTHUR BARTELS AND DAVID ROSENTHAL
Let X be a metric space with an isometric action of a finite group G. Let
q :X ! X=G denote the quotient map. Then
dX=G (y, y0) = dX (q-1(y), q-1(y0)), y, y02 X=G
defines a metric on X=G. We will always consider quotients of metric spaces by *
*an
isometric action of a finite group as metric spaces using this metric.
Notation 1.3. Let X be a space with an action of a group G, and let S be a coll*
*ection
of subgroups of G. Define XS to be the union of all fixed sets XH , where H is *
*in S.
If S = {G}, then XS is simply XG . If S is closed under conjugation by elements
of G, then XS is a G-invariant subspace of X.
Lemma 1.4. Let X be a metric space that is uniformly Fin-contractible with re-
spect to an isometric action of a group . If G is a finite subgroup of and S*
* is a
collection of subgroups of G that is closed under conjugation by G, then the qu*
*otient
XS =G is uniformly contractible.
Proof.Let R > 0 be given. Since X is assumed to be uniformly Fin-contractible
and G is finite, there is an S > 0 such that for every subgroup H of G and every
H-invariant subset B X of diameter less than or equal to 2R|G|, the inclusion,
B ! BS, is H-equivariantly null homotopic.
Let y 2 XS =G and x 2 q-1(y), where q :XS ! XS =G is the quotient map. Let
H be the subgroup of G consisting of all g 2 G for which there are g1, . .,.gn *
*2 G
such that g1 = e, gn = g and d(gix, gi+1x) 2R, for i = 1, . .,.n - 1. Then the
diameter of B = HxR X is bounded by 2R|G|. Therefore, the inclusion B ! BS
is H-equivariantly null homotopic. In particular, there is a point z 2 BS that *
*is
fixed by H. For g 2 G - H, gB \ B = ;. Therefore, the inclusion, G . B ! G . BS,
is G-equivariantly homotopic to a map that sends gB to gz. By G-equivariance,
this homotopy can be restricted to XS , which induces the required null homotopy
on the quotient.
Lemma 1.5. Let be a group such that there is a finite -CW -model, X, for E_ .
Let d be a -invariant metric on X. Then X is uniformly Fin-contractible.
Proof.Let X be a finite -CW -model for E_ . That is, there is a proper -CW -
complex X with finitely many -cells such that XG is contractible for every fin*
*ite
subgroup G of and empty otherwise. Note that X is a locally compact space
since the -action is cocompact and proper, and for every finite subgroup G of *
* ,
X is G-equivariantly contractible.
Let R > 0 be given. If B is a finite G-invariant subcomplex of X of diameter*
* less
than R, then there is an S = S(B, G) such that B ! BS is G-equivariantly null
homotopic. If fl 2 , then flB ! flBS is Gfl= flGfl-1-equivariantly null homoto*
*pic
since the metric is -invariant. Consider the set of all pairs (B, G), where G *
*is a
finite subgroup of and B is a G-invariant subcomplex of X whose diameter is l*
*ess
than R. On this set, fl 2 acts by sending (B, G) 7! (flB, Gfl). Since the -a*
*ction
on X is proper, the quotient by this action is finite. Therefore, we can choose*
* S
independent of B. (In fact, S can be chosen independent of both B and G.)
2.Asymptotic dimension
Let U be an open cover of the metric space X. The cover U is called locally
finite if every compact subset of X meets only finitely many members of U. The
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 5
dimension of an open cover U is defined to be the smallest number n such that e*
*ach
x in X is contained in at most n + 1 members of U. This is also the dimension
of the associated simplicial complex |U|. If the diameters of the open sets in*
* U
are uniformly bounded, then U will be called a bounded cover. The asymptotic
dimension [Gro93, p.28] of X is the smallest integer n such that for any R > 0,
there exists an n-dimensional bounded cover U of X whose Lebesgue number is at
least R.
Lemma 2.1. Let X be a proper metric space of finite asymptotic dimension n.
Then for every ff there exists a locally finite n-dimensional bounded cover U o*
*f X
whose Lebesgue number is at least ff.
Proof.Let U be an n-dimensional bounded cover of X such that for every x 2 X
there is U 2 U that contains the closed ball xR . For each U 2 U, let U-ffbe th*
*e open
set consisting of all points x 2 X for which xff U. Then U-ff= {U-ff| U 2 U} is
an open cover of X. Since X is a proper metric space, we can find a subcollecti*
*on
U0 U and an open set U0 U-fffor each U 2 U0 such that {U0 | U 2 U0} is a
locally finite set. For each U 2 U0, let U00be the interior of (U0)ff. The coll*
*ection
U00= {U00| U 2 U} is also locally finite. Since U00 U for U 2 U0, U00is a boun*
*ded
cover of X of dimension at most n. By construction, the Lebesgue number of U00
is at least ff.
Recall the definition of XS in Notation 1.3.
Lemma 2.2. Let X be a metric space with a proper isometric action of the group
. Let G be a finite subgroup of . Let S be a collection of subgroups of G t*
*hat
is closed under conjugation by G. If X has finite asymptotic dimension, then the
quotient XS =G has finite asymptotic dimension.
Proof.Let n be the asymptotic dimension of X, and let R > 0 be given. Because
the asymptotic dimension of a subspace is bounded by the asymptotic dimension
of the ambient space, there exists an n-dimensional bounded cover U of XS whose
Lebesgue number is at least R. Let p: XS ! XS =G be the quotient map. Define
p*U = {p(U) | U 2 U}. It is easy to check that p*U is a bounded cover whose
Lebesgue number is at least R. For every x 2 XS =G, p-1(x) contains no more
than |G| points. Therefore, the dimension of p*U is at most (n + 1)|G| - 1.
3.Discrete subgroups of Lie groups
In this section it is proven that discrete subgroups of virtually connected *
*Lie
groups satisfy the assumptions of Theorems 8.1 and 9.1. Let be a discrete sub-
group of a virtually connected Lie group G. If K is a maximal compact subgroup *
*of
G, then G=K is a finite dimensional -CW complex that is a model for the univer*
*sal
proper -space E_ [L"uc05, Theorem 4.4]. Furthermore, there exists a G-invaria*
*nt
Riemannian metric on G=K (compare 3.4 below). By [CG04b , Section 3] and [Ji04,
Proposition 3.3], G=K has finite asymptotic dimension. 1Therefore, we must prove
the following result.
Proposition 3.1. Let be a discrete subgroup of a Lie group G with finitely ma*
*ny
components. Let K be a maximal compact subgroup of G. Then the -space G=K
equipped with a G-invariant Riemannian metric is uniformly Fin-contractible.
____________
1These references only consider connected Lie groups. If G0 is the component*
* of the identity,
then G0=G0\ K ~=G=K and the general case follows.
6 ARTHUR BARTELS AND DAVID ROSENTHAL
It is easy to see that G=K is uniformly contractible, since G=K is contractib*
*le
and has a transitive action by isometries. Thus, in the torsion-free case, the *
*proof
of Proposition 3.1 is trivial. For the general case, the proof depends on a fix*
*ed point
result (Proposition 3.7), which was explained to the authors by Burkhard Wilkin*
*g.
The following classical facts about Lie groups will be needed.
(3.2) Let G be a Lie group with finitely many components that has a discrete
subgroup D such that the quotient G=D is compact. Then G is isomorphic
to the semidirect product, V o K, of a vector group 2, V , with a compact
group, K, acting on V . This can, for example, be extracted from the pro*
*of
of Lemma XV.3.3 in [Hoc65] (see the end of the first paragraph on p. 183*
*).
(3.3) Let G be a Lie group, and let V be a normal vector subgroup of G such th*
*at
G=V is compact. Then G isomorphic to the semidirect product V o G=V .
This follows from [Hoc65, Theorem III.2.3].
(3.4) Let G be a Lie group. Denote by Ad G the adjoint representation of G
on its Lie algebra g. If H is a closed subgroup of G such that Ad G(H)
is compact, then there exists a positive definite inner product on g that
is invariant under the restriction of Ad G to H. Such an inner product
induces, by translation, a well-defined G-invariant Riemannian metric on
G=H.
(3.5) Let G be a semisimple Lie group with a finite number of components. Let
K be a maximal compact subgroup of AdG (G), and let L be the preimage
of K in G. Then any G-invariant metric on G=L has nonpositive sectional
curvature (see [Hel62] 3.)
We make the following convenient definition.
Definition 3.6. Let H be a closed subgroup of a Lie group G such that AdG (H)
is compact. The pair (G, H) has nearby Fin-fixed points if the following holds:
Let d be a G-invariant Riemannian metric on G=H, F a finite subgroup of G,
and R > 0. Then there exists a T > 0 such that for every F -invariant subset B *
*of
G=H whose diameter is bounded by R, there is a point fixed by F such that the
ball of radius T around this fixed point contains B.
It is not difficult to check that the existence of T , in the above definiti*
*on, does
not depend on the chosen metric.
Proposition 3.7. Let G be a Lie group with finitely many components and K a
maximal compact subgroup of G. Then (G, K) has nearby Fin-fixed points.
The proof of Proposition 3.7 will require the following transitivity result.
Lemma 3.8. Let K H be closed subgroups of the Lie group G, and assume that
Ad G(H) is compact. If (H, K) and (G, H) have nearby Fin-fixed points, then so
does (G, K).
____________
2Lie groups isomorphic to Rn are called vector groups.
3This is well known. Since we did not find the statement in this form, we g*
*ive a proof
with precise references: We may assume that G is connected, since G0=G0 \ L ~=G*
*=L. Let
l V be a Cartan decomposition of the Lie algebra g of G (see [Hel62, III.x7])*
*. Then l is a
maximal compactly imbedded subalgebra of g (see [Hel62, Proposition III.7.4]). *
*This means that
the subgroup of AdG(G) corresponding to l is a maximal compact subgroup. Since *
*all maximal
compact subgroups are conjugated [Hoc65, Theorem XV.3.1.], we may assume that t*
*his subgroup
is K. Thus, the pair (G, L) is of noncompact type [Hel62, p.194/195], and any G*
*-invariant metric
on G=L has nonpositive curvature.
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 7
Proof.Since AdG (H) is compact, there is an H-invariant inner product on the Lie
algebra g of G. The projection G ! G=H induces an isomorphism from h? to
the tangent space TeH(G=H) of G=H at eH, where h is the Lie subalgebra of g
corresponding to H. We use this isomorphism to transport the inner product from
h? to TeH(G=H). Since the inner product on h? is H-invariant, this inner product
extends to a G-invariant Riemannian metric on G=H. (Here, it is important that
the inner product on g is H-invariant, otherwise the inner product on TgH (G=H)
would depend on the choice of g, and not just on gH.) Similarly, we obtain a G-
invariant metric on G=K. Thus, the projection, ss :G=K ! G=H, is a Riemannian
submersion. This implies the following property for ss.
(3.9) If ! is a smooth path in G=H and x is a lift of its the initial point to*
* G=K,
then there is a canonical lift of ! to G=H that begins at x and has the
same arc length as !.
Let F be a finite subgroup of G, and let B be an F -invariant subset of G=K of
diameter bounded by R. Since (G, H) has nearby Fin-fixed points and ss is non-
expanding, there is a fixed point gH in G=H such that ss(B) is contained in gH *
*T,
the ball of radius T around gH. Here, T only depends on F and R (and not on B).
For b in B, choose a geodesic, !b, in G=H from ss(b) to gH. Let '(b) be the end*
*point
of the lift of !b to G=K with initial point b. Let C = {'(b) | b 2 B} ss-1(gH*
*).
Property (3.9)implies B CT and C BT . It follows that the diameter of C
with respect to the metric on G=H is bounded by R + 2T . The diameter with
respect to the restriction of the Riemannian metric to ss-1(gh) may be larger, *
*but
will still be bounded by some number R0depending only on R + 2T . This is true
because Hg = g-1Hg acts transitively and isometrically with respect to either
distance on ss-1(gH). Since (H, K) has nearby Fin-fixed points, there is a fixed
point, hK, for F g= g-1F g, and a T 0> 0, depending0only on R0and F g, such that
g-1(C) ss-1(eK) = H=K is contained0in hKT . Therefore, ghK is a fixed point
for F , and B is contained in ghKT+T .
Proof of Proposition 3.7.We begin by considering two special cases.
Case 1: If G is the semidirect product, V o K, of a vector group V with a
compact group K, then V ~=G=K. That is, G=K is Euclidean space and the result
follows from the Cartan Fixed Point Theorem.
Case 2: Suppose that G semisimple. Let L be a subgroup of G containing K that
maps to a maximal compact subgroup of AdG (G) under the adjoint representation
Ad. By (3.4), there exists a G-invariant metric on G=L, which is nonpositively
curved by (3.5). By the Cartan Fixed Point Theorem, (G, L) has nearby Fin-fixed
points. Since G=L is connected, L will only have a finite number of components.
Since G is semisimple, the kernel, D, of AdG is a discrete subgroup [Hel62, Cor*
*ollar-
ies II.5.2+II.6.2]. Since L=D is compact, the first case applies to L by (3.2).*
* Thus,
(L, K) has nearby Fin-fixed points. By Lemma 3.8, (G, K) has nearby Fin-fixed
points.
For the general case, we proceed by induction on dimG. If G is not semisimple
there is a closed nontrivial normal subgroup that is either a vector group V or*
* a
torus T [Hoc65, Lemma XV.3.6]. If the subgroup in question is a torus, then we
may assume that it contains K. Thus, the result for G=T implies it for G. If it*
* is a
vector group V , then consider the subgroup V K of G. By (3.3), the first case *
*applies
8 ARTHUR BARTELS AND DAVID ROSENTHAL
to V K. By induction, G=V K ~=(G=V )=(V K=V ) and (G=V, V K=V ) have nearby
Fin-fixed points. Once again, the result for (G, K) follows from Lemma 3.8.
Lemma 3.10. Let G be a Lie group with finitely many components and K a max-
imal compact subgroup. Equip G=K with a G-invariant Riemannian metric. Then
for every T > 0 there is an S > 0 such that the ball of radius T around eK is
K-equivariantly contractible inside the ball of radius S.
Proof.By [Hoc65, Theorem XV.3.1], there is a finite dimensional vector space V
with a linear K-action and a K-equivariant homeomorphism G=K ! V sending
eK to 0. Thus, eKT is K-equivariantly contractible in G=K. By compactness, this
contraction happens inside some ball of finite radius.
Proof of Proposition 3.1.Let F be a finite subgroup of G. Let R > 0 be given. By
Proposition 3.7, there is a T > 0 such that for every F -invariant subset B of *
*G=K
whose diameter is bounded by R, there is a point fixed by F such that the ball *
*of
radius T around this fixed point contains B. By Lemma 3.10, there is an S > 0
such that eKT is K-equivariantly contractible inside the ball of radius S.
Let B0 an F -invariant subset of G=K whose diameter is bounded by R. Then
there is an F -fixed point gK in G=K such that B0 is contained in gKT . Since gK
is an F -fixed point, F g= g-1F g is a subgroup of K. Thus, g-1B0 is contained
in eKT and is F g-invariant. Therefore, g-1B0 is F g-equivariantly contractible*
* in
eKS. Applying g, this means that B0 is F -equivariantly contractible in gKS.
4. Open covers and simplicial compexes
A map f :X ! Y between metric spaces is metrically coarse if it is proper and
satisfies the following growth condition: for all R > 0 there is an S > 0 such *
*that
dX (x, y) < R =) dY (f(x), f(y)) < S.
Two such maps, f and g, are said to be bornotopic if there is a constant C > 0
such that dY (f(x), g(x)) < C for all x in X (see [HR95 , Section 2]). A metric*
*ally
coarse homotopy between proper continuous maps is called a metric homotopy. In
particular, a metric homotopy is a bornotopy and a proper homotopy.
For the following lemma compare [HR95 , Lemma 3.3].
Lemma 4.1. Let X be a proper metric space that has a finite dimensional CW -
structure such that the diameters of its cells are uniformly bounded, and let Y*
* be a
uniformly contractible proper metric space. Let A be a subcomplex of X containi*
*ng
the 0-cells of X. Then every continuous metrically coarse map f0: A ! Y can be
extended to a continuous metrically coarse map f :X ! Y .
Proof.Let Xn be the union of A with the n-skeleton of X. Since Xn is obtained
from Xn-1 by attaching n-cells, we extend f0 inductively to fn :Xn ! Y . For
every n-cell, e, not in A, we must extend to all of e the restriction of fn-1 t*
*o the
boundary, @e, of e. Because fn-1 is metrically coarse and cells in X have unifo*
*rmly
bounded diameter, fn-1(@e) has uniformly bounded diameter. Since Y is uniformly
contractible, there is a continuous metrically coarse extension of fn-1 to Xn. *
*This
finishes the construction of f.
It remains to show that f is proper. Since the diameters of cells in X are
uniformly bounded and X is finite dimensional, there is an R > 0 such that X =
(X0)R , where X0 denotes the 0-skeleton of X. Since X is a proper metric space,*
* this
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 9
implies that a subset B of X has compact closure if and only if BR \X0 is finit*
*e. Let
Bffbe a closed ball of radius ff in Y . Because f is metrically coarse, there i*
*s an S > 0
such that (f-1 (Bff))R is contained in f-1 ((Bff)S). Since (Bff)S is compact, f*
*0 is
proper, and X0 is closed in X, it follows that (f-1 (Bff))R \X0 f-1 ((Bff)S)\*
*X0
is finite. Since f-1 (Bff) is closed, this implies that f-1 (Bff) is compact. T*
*herefore,
f is proper.
If U is an open cover of a space X, then the realization of its nerve, |U|, i*
*s a
simplicial complex. Denote by [U] the vertex of |U| corresponding to U in U. A
partition of unity subordinate to U, ('U )U2U , induces a map g :X ! |U| defined
by:
X 'U (x)[U]
(4.2) g(x) = ___________P
U2U V 2U'V (x)
The Euclidean path length metric 4on a simplicial complex is the unique path
length metric that restricts to the standard Euclidean metric on each simplex. *
*For
an open cover U, we will always equip |U| with the Euclidean path length metric.
Proposition 4.3. Let X be a complete path metric space and let U be a locally
finite, bounded, finite dimensional cover of X whose Lebesgue number is positiv*
*e.
If g :X ! |U| is induced by a partition of unity subordinate to U as in (4.2), *
*then
g is a bornotopy equivalence.
Proof.This follows from [Roe91, Section 3] (see also [HR95 , Proposition 3.2]).*
* In
these references the spherical metric rather than the Euclidean metric is used,*
* but
since |U| is finite dimensional, this distinction is not important.
Lemma 4.4. Let X be a uniformly contractible complete path metric space. Assume
that X has the structure of a finite dimensional CW -complex. Let U be a locally
finite, bounded, finite dimensional cover of X whose Lebesgue number is positiv*
*e,
and let g :X ! |U| be induced by a partition of unity subordinate to U as in
(4.2). Then g has a right homotopy inverse up to metric homotopy. That is, there
is a continuous and metrically coarse map f :|U| ! X and a metric homotopy
H :Xx [0, 1] ! X from f O g to idX.
Proof.By refining the CW -structure if necessary, we can assume that the cells *
*in
X have uniformly bounded diameter. By Proposition 4.3, g has a bornotopy inverse
f. Using Lemma 4.1, we can assume that f is also continuous. The existence of H
follows by applying Lemma 4.1 to the subspace Xx {0, 1} Xx [0, 1] and the map
that is f O g on Xx {0} and the identity of X on Xx {1}.
Although the Euclidean path length metric changes under restriction to subco*
*m-
plexes, there is the following estimate.
Lemma 4.5. Let oe and o be intersecting faces of the k-simplex . Denote by dpl
the Euclidean path length metric on the subcomplex of spanned by oe and o, and
denote by d the Euclidean standard metric on . Then for x in oe and y in o
p _____
dpl(x, y) 3 k + 1. d (x, y).
____________
4In [Yu98] and [Bar03b] the spherical metric has been used. Using the Euclid*
*ean metric is
convenient for computation in Proposition 4.6. In any event, the difference bet*
*ween the Euclidean
and the spherical metric is not important for this paper.
10 ARTHUR BARTELS AND DAVID ROSENTHAL
Proof.Let v0 be a vertex that is contained in oe \ o. For a face ae of that c*
*ontains
v0, define the projection pae: ! ae as the simplicial map which is the identit*
*y on
ae and maps all vertices not in ae to v0. Using the Cauchy-Schwarz inequality,
p_____
d (pae(z), pae(z0)) k + 1. d (z, z0) 8z, z02 .
Note that poe(x) = x, po(y) = y, poe\o(x) = po(x) and poe\o(y) = poe(y). Theref*
*ore,
dpl(x, y) dpl(x, poe\o(y)) + dpl(poe\o(y), poe\o(x)) + dpl(poe\o(x), y)
= dpl(poe(x), poe(y)) + dpl(poe\o(y), poe\o(x)) + dpl(po(x), po(y))
= d (poe(x), poe(y)) + d (poe\o(y), poe\o(x)) + d (po(x), po(y))
p __
3 k. d (x, y).
The finite asymptotic dimension of a metric space is equivalent to the existe*
*nce
of certain contracting maps to finite dimensional simplicial complexes (see [Gr*
*o93,
p.30]). We will use the following version of this (compare [Yu98 , Lemma 6.3]).
Lemma 4.6. Let U be an n-dimensional bounded cover of a path length metric
space X. Assume that the Lebesgue number R of U is positive. Then there is a
partition of unity subordinate to U such that for the induced map g :X ! |U| (4*
*.2),
dpl(g(x), g(y)) Cn d(x,_y)_R
for all x, y 2 X. Here, dpldenotes the Euclidean path length metric on |U|, and
Cn is a constant that depends only on n.
Proof.Let VU be the vector space of sequences of real numbers indexed by U.
There is a canonical embedding |U| ! VU. We will use both the l1-norm, k.k1,
and the l2-norm, k.k2, on VU. For x in X and U in U, let xU = d(x, X - U) and
f(x) = (xU )U2U 2 VU. Define a partition of unity subordinate to U by 'U (x) =
__xU__
kf(x)k1. The map g :X ! |U| VU induced by this partition of unity is given by
x 7! _f(x)_kf(x)k1.
Let x, y 2 X be given. Clearly, kf(x)k2 kf(x)k1. Because R is the Lebesgue
number of U, there is at least one U in U for which yU R. Therefore, kf(y)k1 *
* R.
Since U is n-dimensional, there are at most 2(n + 1) membersfofiU for which fi
xU 6= 0 or yU 6= 0. From |xU - yU | pd(x,_y),_we conclude fikf(y)k1- kf(x)k1fi
2(n + 1) d(x, y) and kf(x) - f(y)k2 2(n + 1)d(x, y) 2(n + 1) d(x, y). Using
these estimates, it follows that
flfl flfl
kg(x) - g(y)k2= flflf(x)_kf(x)k- _f(y)__flfl
fl 1 kf(y)k12fl fl fl
flkf(y)kf(x) - kf(x)kf(x)fl flkf(x)kf(x) - kf(x)kf(y)fl
flfl___1____________1___kf(x)kflfl+ flfl___1____________1___flfl
fi 1kf(y)k1fi 2 kf(x)k1kf(y)k1 2
fikf(y)k- kf(x)kfikf(x)k kf(x) - f(y)k
______1________1______2_kf(x)k+ ____________2_
1kf(y)k1 kf(y)k1
4(n_+_1)d(x,_y)R.
Restricted to simplices, the embedding |U| ! VU is an isometry (using the l2-no*
*rm
on VU). If d(x, y) < R, then there is a U in U containing x and y, since R is t*
*he
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 11
Lebesgue number of U. Thus, there are simplices oe and o of |U| such that g(x) *
*2 oe,
g(y) 2 o, and [U] 2 oe \ o. In particular, oe \ o is not empty. Because oe and *
*o have
dimension less or equal to n, they span a simplex in VU whose dimension is at
most 2n. Since the l2-norm gives the Euclidean metric on , we can use Lemma 4.5
to estimate
p ______ p ______ d(x, y)
dpl(g(x), g(y)) 3 2n + 1. kg(x) - g(y)k2 12 2n + 1(n + 1)______R,
whenever d(x, y) R. Since the metric on X is a path length metric, the inequa*
*lity
follows for all x, y 2 X.
5. Controlled algebra
Let X be a proper metric space and A a small additive category. Let be a
group that acts on X by isometries and on A by additive functors. (We will cons*
*ider
only left actions.) Fundamental to this paper will be the additive category A(X)
of certain continuously controlled modules. This is a minor variation of simil*
*ar
constructions from the literature, the first of which appeared in [ACFP94 ].
Let Z = Xx x [0, 1). An object, M, in A(X) is given by a sequence of objects
(Mz)z2Z in A, subject to the conditions:fi
(5.1) The image of suppM = z fiMz 6= 0 under the projection Z ! Xx [0, 1)
is locally finite.
(5.2) For every x 2 X and t 2 [0, 1), suppM \ {x}x x {t} is finite.
A morphism, ': M ! N, in A(X) is given by a sequence of morphisms,
('z,z0:Mz0! Mz)(z,z0)2Z2, in A, subject to the conditions:
fi
(5.3) supp' = (z, z0) fi'z,z06= 0 is continuously controlled at Xx {1}. That
is, for every x 2 X and every open neighborhood U of (x, 1) in Xx [0, 1*
*],
there is a (smaller) open neighborhood V of (x, 1) in Xx [0, 1] such th*
*at
(Xx [0, 1) - U)x V and V x(Xx [0, 1) - U) do not intersect the image of
supp' under the projectionfZi! Xx [0, 1].
(5.4) For a fixed z in Z, z0fi(z, z0) 2 supp' or (z0, z) 2 supp' is finite.
(5.5) There exists an R > 0 such that ((x, fl, t), (x0, fl0, t0)) 2 supp' imp*
*lies
d(x, x0) < R.
Composition of morphisms is given by the usual matrix multiplication.
If X is a collection of subsets of Z, then we define A(X ) as the full subca*
*tegory
of A(X) whose objects, M, satisfy the additional condition:
(5.6) There is an S 2 X such that suppM S.
If X is closed under finite unions, then A(X ) is again an additive category. I*
*f Y is
another collection of subsets of Z such that for every S 2 Y there is a T 2 X s*
*uch
that S T , then A(Y) is a subcategory of A(X ). Furthermore, A(Y) will define*
* a
Karoubi filtration [CP95 , Definition 1.27] of A(X ) if, in addition, the follo*
*wing is
satisfied:
(5.7) For everyfSi2 Y and morphism ' in A(X ), there is a T 2 Y such that
S' = z fi9 z02 S such that'z,z06= 0 is contained in T .
The quotient of this Karoubi filtration will be denoted by A(X , Y).
Clearly, A(X) = A({X}) and A(X ) = A(X , ;). If X and Y are -invariant,
then the formula (g(M))z = g(Mg-1z) defines an action of on A(X , Y). For a
subgroup G of , denote the corresponding fixed point category by AG (X , Y). It
12 ARTHUR BARTELS AND DAVID ROSENTHAL
is not difficult to check that taking Karoubi quotients and taking fixed catego*
*ries
commute. Therefore, AG (X , Y) is the quotient of AG (X ) by AG (Y).
Let p: X ! X0 be a continuous map. For a closed subset Y of X, let X (Y, p)
be the collection of subsets S of Z with the following properties:
(5.8) If (x, 1) is a limit point of the image of S under the projection Z !
Xx [0, 1), then x 2 Y .
(5.9) There is an R > 0 such that the image of S under the projection Z ! X
is contained in Y R.
(5.10) There is a compact set K0 X0 such that the image of S under the
composition of the projection Z ! X with p is contained in K0.
Let X (Y, p)0 be the collection of subsets S of Z that satisfy (5.9), (5.10)and*
* the
following strengthening of (5.8):
(5.11) The set of limit points of the image of S under the projection Z ! Xx [0*
*, 1)
is disjoint from Xx {1}.
Let G be a subgroup of such that Y is G-invariant. The controlled categories
that we will use are:
AGp(Y )= AG (X (Y, p)),
AGp(Y )0= AG (X (Y, p)0),
AGp(Y )1= AG (X (Y, p), X (Y, p)0),
AGp(X, Y )= AG (X (X, p), X (Y, p)).
Note that the definitions of these categories depend on the metric space X, even
though this is not reflected in the notation. However, because X is a proper me*
*tric
space, changing X to a smaller or larger proper metric space that still contain*
*s Y
will only change the categories up to G-equivariant equivalence. Since all of *
*our
metric spaces will be proper metric spaces, we can disregard this dependence on
X. It should also be noted that condition (5.5)is only important in the defini-
tions of AGp(Y ) and AGp(Y )0 and not in the definition of AGp(Y )1 , where it *
*affects
the category only up to equivalence (compare [Bar03b, Lemma 3.15]). Similarly,
our notation is slightly imprecise because even before taking fixed categories *
*these
categories depend on the group we have in mind. But this dependence also only
changes the category up to equivariant equivalence and can safely be ignored.
Let K-1 denote the functor from the category of small additive categories to
the category of spectra, which assigns an additive category to its associated n*
*on-
connective K-theory spectrum [PW85 ]. A crucial fact is that applying K-1 to
a Karoubi filtration produces a fibration of spectra, which induces a long exact
sequence in K-theory [CP95 , Theorem 1.28]. By definition, the two following se-
quences are Karoubi sequences. That is, the final category is a Karoubi quotien*
*t of
the middle category by the first. Therefore, each induces a long exact sequence*
* in
K-theory.
(5.12) AGp(X)0 ! AGp(X) ! AGp(X)1
(5.13) AGp(Y ) ! AGp(X) ! AGp(X, Y )
If, in addition, Y R= X for some R > 0, then
(5.14) AGp(Y )1 ! AGp(X)1 ! AGp(X, Y )
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 13
is also Karoubi sequence. To see this, consider the following commutative diagr*
*am
in which the first two columns and the second two rows are Karoubi filtrations.
~=
AGp(Y )0_____//AGp(X)0
| |
| |
|fflffl fflffl|
AGp(Y )_____//_AGp(X)____//AGp(X, Y )
a|| b|| c ||
|fflffl fflffl| fflffl|
AGp(Y )1____//_AGp(X)1_______//_Q
The assumption Y R = X implies that AGp(Y )0 ! AGp(X)0 is an equivalence of
categories. A short exercise in the definition of Karoubi filtrations shows tha*
*t c is
an equivalence of categories.
In [Wei02], it is proven that X 7! K*(AidX(X)1 ) is a locally finite homology
theory. In particular, it is homotopy invariant. It follows from the definition*
* that
AidX(X)0 is functorial in X for metrically coarse maps and that bornotopic maps
induce the same map in K-theory. These facts imply:
(5.15) The category A(X) is functorial in X for metrically coarse continuous
maps (compare [Bar03b, Remarks 3.5, 3.10]). If f, g :X ! Y are two such
maps that are metrically homotopic (see Section 4), then f and g induce
the same map from K*(A(X)) to K*(A(Y )).
In [BR05 , Definition 2.1], the additive category A * T is constructed for a*
* -set
T . By [BR05 , Proposition 2.8(iii)], there is a canonical equivalence of cate*
*gories
A *H H=H ! A *G G=H, for every subgroup H of . If A is the category of finitely
generated free R-modules for a ring R and the action of H on A is trivial, then
A *H H=H is equivalent to the category of finitely generated free R[H]-modules.
For this reason, we will denote the category A *H H=H by A[H] for an arbitrary
A, and call the objects of this category A[H]-modules. In [BR05 , Section 3], t*
*he
Or( )-spectrum KA is also introduced (using the K-1 -functor). Associated to it
is the assembly map
(5.16) H*(E_ ; KA ) ! H*(pt; KA ) = K*(A[ ]).
If A is the category of finitely generated free R-modules, then this is the ass*
*embly
map (0.1). Let p :E_ ! E_ = denote the quotient map. It is not hard to check
that A[ ] is equivalent to the category Ap (E_ )0. Controlled algebra can be us*
*ed
to describe assembly maps as forget control maps. An instance of this is [HP04 *
*],
where the continuously controlled category B (E_ x [0, 1); R) is used to identi*
*fy
various versions of the assembly map. If one modifies the definition of this ca*
*tegory
to allow for coefficients in A, then, since acts on E_ with finite isotropy,*
* the
fixed point category Ap (E_ )1 can be identified with the continuously control*
*led
category B (E_ x [0, 1); A)>0. The only difference between the two categories *
*is
condition (5.5), but as mentioned before, this changes the category only up to
equivalence. Therefore, [HP04 , Theorem 7.4] implies the following fact.
(5.17) If E_ is equipped with a -invariant metric, then the boundary map in
the long exact sequence associated to (5.12), with X = E_ , is equivalent
to the assembly map (5.16).
14 ARTHUR BARTELS AND DAVID ROSENTHAL
6. A vanishing result
In this section, a key component of the proof of Theorem 8.1 is established,
namely Proposition 6.2 below.
Proposition 6.1. Let X be a proper metric space. For each n 2 N, let Qn be
a simplicial complex equipped with the Euclidean path length metric and gn :X !
Qn a continuous and metrically coarse map satisfying the following: For every
R > 0 there exists an S > 0 such that for all x and y in X with d(x, y) < R,
d(gn(x), gn(y)) < S_n. Then for any a 2 Km (A(X)) there is an n0 = n0(a) such
that (gn)*(a) = 0 2 Km (A(Qn)) for all n n0.
Proof.This is almost [Bar03b, Corollary 4.3]. In [Bar03b], the spherical metric
rather then the Euclidean metric is used, but this does not affect the argument*
*. It
is only important that the metric is the same on every simplex.
Proposition 6.2. Let X be a uniformly contractible, complete, proper, path leng*
*th
metric space of finite asymptotic dimension. Assume that X has the structure of
a finite dimensional CW -complex. Let X be a collection of closed subsets, S, *
*of
Xx [0, 1), where S = Kx [0, 1) for some closed subset K of X, that is closed un*
*der
finite unions, and assume that for every closed subset K of X and every ff > 0,
Kffx[0, 1) is contained in X . Then the K-theory of A(X ) vanishes.
Proof.Since X has finite asymptotic dimension, there is a sequence of bounded
open covers Un of X such that the Lebesgue number of Un exceeds n. By Lemma 2.1,
we can assume that each Un is locally finite. By Lemma 4.6 and Proposition 6.1,
there is a sequence of maps, gn :X ! |Un|, induced by partitions of unity, such
that for every closed subset Y of X and every a 2 K*(A(Y )), there is an n0
such that (gn)*(a) = 0 for n n0. By Lemma 4.4, each gn is invertible up to
metric homotopy. Thus, for each n, there is a continuous metrically coarse map
fn :|Un| ! X and a metric homotopy Hn :Xx [0, 1] ! X from fn O gn to idX.
Let b 2 K*(A(X )) be given. Choose a closed subset K of X such that b is
the image of some a 2 K*(A(K)) under the inclusion 'K :A(K) ! A(X ). Let
Qn be the smallest subcomplex of |Un| that contains gn(K). Since fn and Hn
are metrically coarse, there is an ffn > 0 such that fn(Qn), Hn(Kx [0, 1]) Kf*
*fn.
Consider the restriction maps gn|K :K ! Qn and fn|Qn :Qn ! Kffn. Then the
restriction of Hn to Kx [0, 1] gives a metric homotopy from fn|Qn O gn|K to the
inclusion iffn:K ! Kffn. From (5.15)we conclude that (fn|Qn O gn|K )*(a) =
(iffn)*(a). By Proposition 6.1, eventually (gn|K )*(a) = 0. The conclusion now
follows since iffn(a) also maps to b under the inclusion 'Kffn:A(Kffn) ! A(X ).
7. The Descent Principle and Homotopy Fixed Sets
Let S be a spectrum with -action. For a space X with -action, let Map (X,*
* S)
be the spectrum whose n-th space is Map (X, Sn), where Sn is the n-th space in
the spectrum S. The fixed point spectrum, S , is defined to be Map (pt, S).
Let F be a family of subgroups of that is closed under conjugation and taking
subgroups. If X is a space with -action, then its F-homotopy fixed point set is
defined by XhF = Map (EF , X). In general, spectra can be difficult to work
with, however, many arguments for spaces can be extended to -spectra by applyi*
*ng
them levelwise. A particularly useful property they possess is that the homotopy
groups of a product of -spectra is the product of the homotopy groups. In orde*
*r to
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 15
define the F-homotopy fixed points of a spectrum with -action, recall that the*
*re
is a fibrant replacement functor, R : SPECTRA ! - SPECTRA , that comes
equipped with a natural weak equivalence A ! R(A). For a construction of this
functor, see for example, [LRV03 , Section 2]. If S is a spectrum with -action*
* then
so is R(S), and it is not difficult to check that R commutes with taking fixed *
*points.
That is, (R(S)) = R(S ). The F-homotopy fixed point spectrum 5is defined to
be the spectrum
ShF = Map (EF , R(S)).
Notice that the projection EF ! ptinduces a natural transformation S ! ShF .
Lemma 7.1. Let F be a family of subgroups of .
(i) If F :S ! T is an equivariant map of spectra with -action such that
F G:SG ! TG is a weak homotopy equivalence for every subgroup G in
F, then ShF ' ThF .
(ii)Let B be an -spectrum with a G-action, where G is in F, and let act *
*on
S = Map G( , B) by (flf)(x) = f(fl-1x), where fl 2 . Then S ' ShF .
Proof.(i)The proof of [Ros04, Lemma 4.1] shows that the corresponding result
holds for spaces, but this also implies the result for spectra. For every G 2 F*
* and
n 2 N, (R(S)n)G ! (R(T)n)G is a weak equivalence because R commutes with
fixed points. Thus, (R(S)n)hF ! (R(T)n)hF is weak equivalence for all n 2 N.
(ii)The proof of [Ros04, Lemma 4.2] shows that the corresponding result holds
for spaces. The statement for spectra can be deduced from this as follows. A
product of -spectra is again an -spectrum. In particular, SH is an -spectrum
for every subgroup H of . Thus, (Sn)H ! (R(S)n)H is a weak equivalence for
every n. Therefore, (Sn)hF ! (R(S)n)hF is a weak equivalence for every n. On
the other hand, (Sn) ! (Sn)hF is a weak equivalence by the space version of
(ii).
Lemma 7.2. Let F be the family of finite subgroups of , H a finiteQsubgroup of
, B an additive category with an H-action, and let act on C = =HB as it
does on the product in Lemma 7.1(ii). Then
K-1 (C) ' K-1 (C)hF .
Proof.For B = A(pt)1 this is proven in [Ros04, Theorem 6.3] using Lemma 7.1 (ii)
and the fact that K-1 commutes with infinite products [Car95]. The same proof
works in the general case.
Remark 7.3. In order for the L-theory version of Lemma 7.2 to be true, it must *
*also
be assumed that for sufficiently large i, K-i(B[H]) = 0. This is needed because*
* the
compatibility of L-1 with infinite products is only known if the K-theory vani*
*shes
in degree -i for sufficiently large i (see [CP95 , p.756]).
Proposition 7.4, below, is an important fact needed to prove the Descent Pri*
*n-
ciple (Theorem 7.5). Its proof is based on work of Carlsson and Pedersen [CP95 ,
Theorem 2.11] who proved the result in the case where is torsion-free and X is
____________
5Both authors of this paper are guilty of stating an incorrect definition of*
* the homotopy
fixed point spectrum (forgetting R) [Bar03a], [Ros04]. However, in those papers*
*, homotopy fixed
points are only applied to -spectra. In this case, both definitions yield weak*
*ly equivalent spectra.
Thus, the main results are not affected. We thank Holger Reich for pointing out*
* that the correct
definition of the homotopy fixed point spectrum must involve the functor R.
16 ARTHUR BARTELS AND DAVID ROSENTHAL
a finite -CW complex (meaning that X has finitely many -cells). This was later
generalized by the second author [Ros04] to include groups with torsion. The re*
*sult
proven here relaxes the finiteness condition on X, requiring only that X be a f*
*inite
dimensional -CW complex.
Proposition 7.4. Let be a discrete group, F the family of finite subgroups of
, X a finite dimensional -CW complex with finite isotropy, and p the quotient
map X ! X= . Then, for every -invariant metric on X,
1 -1 1 hF
K-1 Ap (X) ' K Ap (X) .
Proof.Proceed by induction on the skeleta`of X. Let X0 denote the 0-skeleton of
X. For some indexing set J, X0 = j2J =Hj, where Hj 2 F for every j 2 J.
Since we are taking germs away from zero and objects have -compact support,
M i Y j
Ap (X0)1 ~= A(pt)1 ,
j2J =Hj
Q
which is a -equivariant equivalence of categories. Let Cj = =HjA(pt)1 . Sin*
*ce
K-1 (Cj) is -invariant for every j 2 J,
i M j i ` j `
K-1 Cj ' K-1 (Cj) ' K-1 (Cj)
j2J j2J j2J
and i M j
hF i ` jhF `
K-1 Cj ' K-1 (Cj) ' K-1 (Cj)hF ,
j2J j2J j2J
where the last weak equivalence makes use of Lemma 7.1 (i). By Lemma 7.2,
K-1 (Cj) ! K-1 (Cj)hF
is a weak homotopy equivalence. Therefore,
` `
K-1 (Cj) ' K-1 (Cj)hF ,
j2J j2J
which completes the base case of the induction.
Now assume that the proposition holds for the (n - 1)-skeleton Xn-1. Let
A ! B ! C denote the sequence
1 -1 1 -1
K-1 Ap (Xn-1) ! K Ap (Xn) ! K Ap (Xn, Xn-1) .
Consider the following commutative diagram:
A _______//B_______//C
a|| b|| c||
fflffl| fflffl| fflffl|
AhF _____//BhF_____//ChF .
Notice that each row in the diagram is a fibration of spectra. The second row is
a fibration since taking homotopy fixed sets and taking homotopy fibers are both
homotopy limits and therefore commute [BK72 ].
We must show that b is a weak homotopy equivalence. By the induction hypoth-
esis, a is a weak homotopy equivalence. Therefore, by the Five Lemma, it suffic*
*es
to prove that c is a weak homotopy equivalence.
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 17
Since we are taking germs away from Xn-1 (and thus away from zero) and
objects have -compact support,
M i Y j
Ap (Xn, Xn-1) ~= A(Dn, Sn-1) .
i2I =Hi
The proofQis now completed by arguing as in the beginning of the induction with
Ci= =HiA(Dn, Sn-1).
Theorem 7.5 (The Descent Principle). Let be a discrete group, F the fam-
ily of finite subgroups of , X a finite dimensional -CW complex, and p the
quotient map X ! X= . Assume that X admits a -invariant metric such that
Kn AGp(X) = 0 for every integer n.
Then the map H*(X; KA ) ! K*(A[ ]) is a split injection.
Proof.Consider the following commutative diagram of fibration sequences:
K-1 Ap (X)0 _______//K-1 Ap (X) ______//_K-1 Ap (X)1
a|| b|| c ||
fflffl| fflffl| fflffl|
K-1 Ap (X)0 hF _____//K-1 Ap (X) hF ____//_K-1 Ap (X)1 hF .
By Proposition 7.4, c is a weak homotopy equivalence. Since each row in the dia-
gram is a fibration, it suffices to show that K-1 Ap (X) hF is weakly contrac*
*tible.
But this follows from Lemma 7.1 (i)and the assumption that Kn AGp(X) = 0 for
every integer n.
8.Proof of the Main Theorem
Theorem 8.1. Let be discrete group and let A be a small additive category.
Assume that there is a finite dimensional -CW model for the universal space for
proper -actions, E_ , and assume that there is -invariant metric on E_ such *
*that
E_ is a complete proper path metric space that is uniformly Fin-contractible a*
*nd has
finite asymptotic dimension. Then the assembly map, H*(E_ ; K-1A) ! K*(A[ ]),
in algebraic K-theory, is a split injection.
Some preparations must be made before we can give the proof of Theorem 8.1.
Let G be a finite subgroup of . Let
G = H0, H1, . .,.Hm = {e}
contain exactly one subgroup from each conjugacy class of subgroups of G and let
the Hi be ordered by cardinality. That is,f|Hi|i |Hi+1|.
For each k, 0 k m, define Sk = Hgifi0 i k, g 2 G and Zk = E_ Sk.
Clearly, Sk is invariant under conjugation by G. Therefore, Zk is G-invariant f*
*or
every k, 0 k m.
Notation 8.2. For each k, let p: Zk ! E_ = be the restriction of the quotient *
*map
p : E_ ! E_ = , and let pG :Zk=G ! E_ = denote the restriction of the canoni*
*cal
projection E_ =G ! E_ = .
Lemma 8.3. For every k, 0 k m, and every subgroup H of G, the K-theory
of A[H]pG(Zk=G) vanishes.
18 ARTHUR BARTELS AND DAVID ROSENTHAL
Proof.By Lemmas 1.4 and 2.2, Zk=G is uniformly contractible and has finite as-
ymptotic dimension. Let K be a subset of Zk=G whose image, K~, in E_ =G is
compact. If ff > 0, the image of Kffis K~ff, which is compact since E_ =G is a
proper metric space. Therefore, we can apply Proposition 6.2 to conclude that t*
*he
K-theory of A[H]pG(Zk=G) vanishes.
For the following fact, compare [Ros04, Lemma 7.4].
Lemma 8.4. For each k, 1 k m,
AGp(Zk, Zk-1) ~=A[Hk]pG(Zk=G, Zk-1=G).
Proof.Since we are taking germs away from Zk-1 and Zk-1=G, every morphism
has a representative that is zero on Zk-1x[0, 1) and on Zk-1=Gx[0, 1), respecti*
*vely.
Therefore, it is irrelevant what the objects over Zk-1 x [0, 1) and Zk-1=G x [0*
*, 1)
are. By construction, the stabilizer subgroup of any point not in Zk-1 x [0, 1)*
* is
a conjugate of Hk. Since AGp(Zk, Zk-1) is a fixed category, the parts of an obj*
*ect
over points in the same orbit must be isomorphic modules. Thus, the object M
in AGp(Zk, Zk-1) is sent to M0, where M0(y,t)(with (y,ft)i=2Zk-1=G x [0, 1)) is
the A[Hk]-module sitting over the point in (x, t) fix 2 p-1(y) whose stabiliz*
*er
subgroup is Hk. The inverse of this is to take the A[Hk]-modulefoveri(y, t) and
use the G-action to spread it around the orbit (x, t) fix 2 p-1(y) . This expl*
*ains
how the objects in AGp(Zk, Zk-1) and A[Hk]pG(Zk=G, Zk-1=G) are identified. To
verify (5.10), notice that if K0 is a subset of Zk= , then the image of p-1(K0)
under the quotient map Zk ! Zk=G is p-1(K0). Since we are taking germs, the
components of a morphism need to become small. Therefore, non-zero components
of a morphism have the same isotropy, namely a conjugate of Hk. Furthermore, the
equivariance of morphisms in AGp(Zk, Zk-1) implies that there is only one choice
when lifting a morphism from A[Hk]pG(Zk=G, Zk-1=G).
Proof of Theorem 8.1.By the Descent Principle, it suffices to show that the spe*
*c-
trum K-1 AGp(E_ ) is weakly contractible for every finite subgroup G of .
Let G be a finite subgroup of and proceed by induction on the filtration
E_ G = Z0 Z1 ... Zm Zm+1 = E_
defined above. Since G acts trivially on E_ G , AGp(E_ G ) is equivalent to A[G*
*]p(E_ G ).
By Lemma 8.3, K-1 A[G]p(E_ G ) is weakly contractible. This completes the base
case of the induction.
Assume now that K-1 AGp(Zk-1) is weakly contractible. We must show that
K-1 AGp(Zk) is weakly contractible. Consider the following Karoubi filtration
AGp(Zk-1) ! AGp(Zk) ! AGp(Zk, Zk-1)
(see (5.13)), which yields a fibration of spectra after applying K-1 . By using*
* the
induction hypothesis, we need only show that K-1 AGp(Zk, Zk-1) is weakly con-
tractible. By Lemma 8.4, AGp(Zk, Zk-1) is equivalent to A[Hk]pG(Zk=G, Zk-1=G),
which fits into the Karoubi filtration:
A[Hk]pG(Zk-1=G) ! A[Hk]pG(Zk=G) ! A[Hk]pG(Zk=G, Zk-1=G).
Both K-1 A[Hk]pG(Zk-1=G) and K-1 A[Hk]pG(Zk=G) are weakly contractible
by Lemma 8.3. Therefore, K-1 A[Hk]pG(Zk=G, Zk-1=G) is also weakly con-
tractible.
ON THE K-THEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION 19
9. L-theory
If A is an additive category with involution and an action of , then there *
*is an
assembly map
H*(E_ ; LA ) ! H*(pt; LA ) = L<-1>*(A[ ])
(see [BR05 , Section 5]). Here LA is an Or( )-spectrum whose value on =H is
weakly equivalent to L-1 (A[H]), where L-1 (A[H]) is the spectrum whose homo-
topy groups are the ultimate lower quadratic L-groups L<-1>*(A[H]) (see [Ran92 ,
Chapter 17]). If A is the category of finitely generated free R-modules for a r*
*ing,
R, with involution, then L<-1>*(A[H]) = L<-1>*(R[H]), and the above assembly
map is
H*(E_ ; LR ) ! H*(pt; LR ) = L<-1>*(R[ ]).
The following is the L-theory version of Theorem 8.1.
Theorem 9.1. Let be discrete group and let A be a small additive category with
involution. Assume that there is a finite dimensional -CW -model for the unive*
*rsal
space for proper -actions, E_ , and assume that there is a -invariant metric *
*on
E_ such that E_ is uniformly Fin-contractible, is a complete proper path metr*
*ic
space and has finite asymptotic dimension. Assume that for each finite subgroup
G there is an i0 2 N such that for i i0, K-i(A[G]) = 0, where the involution
is forgotten and A is considered only as an additive category. Then the assembly
map, H*(E_ ; LA ) ! L*(A[ ]), in L-theory, is a split injection.
Proof.Everything we did for K-theory also works for L-theory with the exception
of Lemma 7.2. As pointed out in Remark 7.3, this lemma will carry over to L-
theory if the additional assumption about the vanishing of K-i(A[G]) for large i
is made. The rest of the argument proceeds with out further changes. For the
required properties of L-theory, see [CP95 , Section 4].
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Fachbereich Mathematik und Informatik, Westf"alische Wilhelms-Universit"at M"un*
*ster,
Einsteinstr. 62, D-48149 M"unster, Germany
E-mail address: bartelsa@math.uni-muenster.de
Department of Mathematics and Computer Science, St. John's University, 8000 Uto*
*pia
Pkwy, Jamaica, NY 11439, USA
E-mail address: rosenthd@stjohns.edu