A Caveat on the Isomorphism Conjecture in L-theory

Mathematicum (de Gruyter 2002

A Caveat on the Isomorphism Conjecture in L-theory

Tom Farrell, Lowell Jones, Wolfgang LuÈck

(Communicated by Andrew Ranicki)

Abstract. The Isomorphism Conjecture of Farrell and Jones for L-theory [5] has only been formulated forL^ÿy and in this formulation been proved for a large class of groups, for in- stance for discrete cocompact subgroups of a virtually connected Lie group. The question arises whether the corresponding conjecture is true forL^efor the decorationseˆp;h;s. We give examples showing that it fails for any of the decorationseˆp;h;s. The groups involved are of the shapeZ²Ffor ®niteF.

2000 Mathematics Subject Classi®cation: 19G24.

We ®rst summarize the Farrell-Jones-Conjecture forL-theory as far as needed here.

We will follow the notation and setup of [6], which is di¨erent, but equivalent to the original setup in [5], [8], and slightly more convenient for our purposes. Let Gbe a group. A family Fis a class of subgroups of G which is closed under conjugation and taking subgroups. Our main examples will be the familiesVCof virtually cyclic subgroups, i.e. subgroups which are either ®nite or containZas a normal subgroup of ®nite index, and the family ALL of all subgroups. Let L_n^e for eAZ, eU2 and eˆ ÿy be de®ned as in [9, §17]. Notice that with this convention L²_nˆL_n^s, L_n¹ˆL_n^h and L_n⁰ˆL_n^p holds. Denote by L^e the corresponding Or…G†-spectrum, i.e. covariant functor from the orbit category Or…G† to the category of spectra. It has the property thatpn…L^e…G=H†† ˆL_n^e…ZH†. An Or…G†-spaceXis a contravariant functor X from Or…G† to the category of topological spaces. Associated to X and L^e there are homology groups H_n^G…X;L^e† (see [4], [6, Section 1] for more details).

Given a family F, we denote by ?_G;F the Or…G†-space, which sends G=H to the space consisting of one point, ifHAF, and to the empty set, ifH BF. There is an obvious map ?_G;VC!?_G;ALL and an obvious identi®cation of H_n^G…?_G;ALL;L^e† withp_n…L^e…G=G†† ˆL_n…ZG†. Thus we obtain a map, calledassembly map

…1† H_n^G…?_G;VC;L^e† !H_n^G…?_G;ALL;L^e† ˆL_n^e…ZG†:

TheFarrell-Jones-Conjecturesays that (1) is an isomorphism for allnAZ, ifeˆ ÿy.

We want to give examples which show that (1) is not an isomorphism for allnAZ, if

eisp,h ors, or in other words, that the version of the Farrell-Jones-Conjecture for eˆp;h;sis not true.

LetFbe a ®nite group. PutGˆZ²F. LetKbe the set of all subgroupsKHZ² such thatZ²=KGZ. In other words, an element inKis a subgroupKHZ² which is neither trivial nor Z² and has the property that forxAZ² andnAZ, n00 the implicationnxAK)xAK holds. LetSUB…F†andSUB…KF†be the family of all subgroups ofFandKF. Consider the following diagram of Or…G†-spaces

`

KAK?_G;SUB…F† ƒƒƒ! `

KAK?_G;SUB…KF†

??

?y

??

?y

?_G;SUB…F† ƒƒƒ! ?_G;VC

where the horizontal arrows are the obvious inclusions and the vertical arrows are the disjoint unions of the obvious inclusions. Any ®nite subgroup HHG is already contained inF. LetVHGbe an in®nite virtually cyclic subgroup. De®neVHZ²to be the subgroup of elementsxAZ²for which there existsnAZ,n00 withnxAV.

Then V AK and VASUB…VF†. If K AK satis®es VHKF, then VˆK.

Now one easily checks that the diagram of Or…G†-spaces above evaluated at a sub- groupHHGlooks as follows

`

KAK ƒƒƒ!^Id `

KAK

??

?y

??

?y

ƒƒƒ!^Id

if jHj<y;

j ƒƒƒ!

Id

??

?y ^Id

??

?y j ƒƒƒ! ;

j ƒƒƒ!^Id j

Id

??

?y ^Id

??

?y j ƒƒƒ!^Id j;

else;

ifjHj ˆy; HAVC;

where denotes the space consisting of one point. In each case we get a pushout of spaces whose upper horizontal arrow is a co®bration. Hence by excision the inclusion induces an isomorphism [4, Lemma 4.4], [6, Lemma 1.8]

…2† L

KAKH_n^G…?_G;SUB…KF†; ?_G;SUB…F†;L^e† !^G H_n^G…?_G;VC; ?_G;SUB…F†;L^e†:

We get from [6, Lemma 2.7] isomorphisms

H_n^G…?_G;SUB…F†;L^e†GHn…B…G=F†;L^e…ZF††;

H_n^G…?_G;SUB…KF†;L^e†GHn…B…G=KF†;L^e…Z‰KFŠ††;

whereH_n…X;L^e…ZF††denotes the homology of a spaceXassociated to theL-theory spectrum L^e…ZF† and analogously for H_n…X;L^e…Z‰KFŠ††. Since B…G=F† ˆT² andB…G=KF† ˆS¹, we obtain isomorphisms

…3† H_n^G…?_G;SUB…F†;L^e†GL_n^e…ZF†lL_nÿ1^e …ZF†lL_nÿ1^e …ZF†lL_nÿ2^e …ZF†;

…4† H_n^G…?_G;SUB…KF†;L^e†GL_n^e…Z‰KFŠ†lL_nÿ1^e …Z‰KFŠ†:

We claim that the two maps induced by the inclusions

…5† H_n^G…?G;SUB…F†;L^e†‰1=2Š !^G H_n^G…?G;VC;L^e†‰1=2Š !^G H_n^G…?G;ALL;L^e†‰1=2Š

are isomorphism. The ®rst one is an isomorphism by (2), (3) and (4), the target of the second one can be computed directly from the splitting due to Shaneson [10] and Wall [11], which yields the claim. Notice that we have inverted 2 so that we do not have to worry about decorations by the Rothenberg sequence. One can also deduce the bijectivity of the two maps in (5) from more general results, namely, from [6, Theorem 2.4] and [13, Theorem 4.11]. There are isomorphisms coming from the splitting due to Shaneson [10] and Wall [11] for L^s and for other decoration from [9, §16]

…6† L_n^e…ZG†GL_n^e…ZF†lL^eÿ1_nÿ1…ZF†lL^eÿ1_nÿ1…ZF†lL^eÿ2_nÿ2…ZF†;

…7† L_n^e…Z‰ZFŠ†GL_n^e…ZF†lL^eÿ1_nÿ1…ZF†:

Since the L-groups of integral group rings of ®nite groups are ®nitely generated abelian groups [12], we conclude from (3) and (6) that H_n^G…?_G;SUB…F†;L^e† and H_n^G…?_G;ALL;L^e† ˆL_n^e…ZG†are ®nitely generated abelian groups. We conclude from (5) that the inclusion induces a 2-isomorphism, i.e. a map whose kernel and cokernel are ®nite 2-groups.

…8† H_n^G…?_G;SUB…F†;L^e†ƒ!^G2 H_n^G…?_G;ALL;L^e†:

This implies together with the long exact sequence of a pair of Or…G†-spaces

Lemma 9.Suppose that the assembly map(1)forL^eis bijective for all nAZ.Then,for all nAZ,the inclusion induces a2-isomorphism

H_n^G…?_G;SUB…F†;L^e†ƒ!^G2 H_n^G…?G;VC;L^e†

and the group H_n^G…?_G;VC; ?_G;SUB…F†;L^e†is a ®nite2-group.

GivenKHK, there is an isomorphism G!^G G which sends K bijectively to 0Z and leavesF®xed. It induces an isomorphism

H_n^G…?_G;SUB…KF†; ?_G;SUB…F†;L^e† !^G H_n^G…?_G;SUB…0ZF†; ?_G;SUB…F†;L^e†:

Thus we obtain from (2) an isomorphism

…10† L

KAKH_n^G…?_G;SUB…0ZF†; ?_G;SUB…F†;L^e† !^G H_n^G…?G;VC; ?_G;SUB…F†;L^e†:

SinceKis an in®nite set, we conclude from Lemma 9 and (10) together with the long exact sequence of a pair of Or…G†-spaces

Lemma 11. Suppose that the assembly map(1)forL^e is bijective for all nAZ.Then, for all nAZ,

H_n^G…?_G;SUB…0ZF†; ?_G;SUB…F†;L^e† ˆ0 and the inclusion induces an isomorphism

…12† H_n^G…?_G;SUB…F†;L^e† !^G H_n^G…?_G;SUB…0ZF†;L^e†:

Notice that we have identi®ed already the source and the target of the homomor- phism (12) in (3) and (4). Using furthermore the isomorphism (7) the homomorphism (12) can be identi®ed with a homomorphism

…13† L_n^e…ZF†lL_nÿ1^e …ZF†lL_nÿ1^e …ZF†lL_nÿ2^e …ZF†

!L_n^e…ZF†lL^eÿ1_nÿ1…ZF†lL_nÿ1^e …ZF†lL^eÿ1_nÿ2…ZF†:

Example 14.Consider the cyclic groupGˆZ=29 of order 29. Then we get from [1, Theorem 1, 3 and 5] and [2, Corollary 4.3 on page 58].

L_n^e…Z‰Z=29Š† ˆ0 eˆh;s;p;nodd;

L₀^e…Z‰Z=29Š† ˆZ¹⁵ eˆs;p;

L₂^e…Z‰Z=29Š† ˆZ=2lZ¹⁴ eˆs;p;

L₀^h…Z‰Z=29Š† ˆZ¹⁵lH^²…Z=2; ~K0…Z‰Z=29Š††;

L₂^h…Z‰Z=29Š† ˆZ=2lZ¹⁴lH^²…Z=2; ~K₀…Z‰Z=29Š††:

We have K~0…Z‰Z=29Š† ˆZ=2lZ=2lZ=2 [7, page 30]. Hence K~0…Z‰Z=29Š† is Z‰Z=2Š-isomorphic either to Z=2‰Z=2ŠlZ=2 or to Z=2lZ=2lZ=2, where Z=2 carries the trivialZ‰Z=2Š-structure. In both casesH^ⁱ…Z=2; ~K0…Z‰Z=29Š††is non-trivial for all iAZ. This implies that the homomorphism (13) is for no nAZ an isomor- phism, ifeissorh. We conclude from Lemma 11 that the assembly map (1) is not an isomorphism for allnAZ, ifeissorhandGˆZ²Z=29.

Remark 15. With some further e¨ort the homomorphism (13) and hence the homo- morphism (12) can be identi®ed with

…16† …Id;f;Id;f†:L_n^e…ZF†lL_nÿ1^e …ZF†lL_nÿ1^e …ZF†lL_nÿ2^e …ZF†

!L_n^e…ZF†lL^eÿ1_nÿ1…ZF†lL_nÿ1^e …ZF†lL^eÿ1_nÿ2…ZF†;

where f denotes the forgetful map given by lowering the decoration e to eÿ1.

Because of the Rothenberg sequence we see that (12) is bijective for all nAZ if and only if H^ⁿ…Z=2;Wh_eÿ1…F†† ˆ0 for all nAZ. We conclude from Lemma 11 that, given eU2, the assembly map (1) for L^e is bijective for all nAZ, only if H^ⁿ…Z=2;Wh_eÿ1…F†† ˆ0 for all nAZ. Since Wh_ÿ1…Z‰Z=6Š† is isomorphic to Z [3], H^ⁿ…Z=2;Wh_ÿ1…Z=6†† ˆ0 cannot be zero for allnAZ. Hence the assembly map (1) forL^p cannot be an isomorphism for allnAZ, ifGˆZ²Z=6.

Remark 17.One may speculate which family of subgroups one has to choose so that the Isomorphism Conjecture inL-theory has a chance to be true for all decorations e. Of course the family ALL of all subgroups works, but the point is to ®nd an e½cient family. For instance we do not know whether the family of virtually poly- cyclic is enough. The smallest possible candidate is the familyBof subgroupsHofG for which there is a virtually cyclic groupVand an integerkV0 withHHVZ^k. With this choice all the counterexamples above are taken care of, but for the trivial reason that for them BˆALL. The family B at least passes the following test.

Namely, it can be shown that the splitting formula due to Shaneson and Wall is also true for the source of the assembly map

H_n^G…?_G;B;L^e† !H_n^G…?_G;ALL;L^e† ˆL_n^e…ZG†:

Namely, one has a canonical splitting compatible under the assembly map with the splitting formula due to Shaneson and Wall on the target

H_n^GZ…?GZ;B;L^e† ˆH_n^G…?G;B;L^e†lH_nÿ1^G …?G;B;L^eÿ1†;

L_n^e…Z‰GZŠ† ˆL_n^e…ZG†lL^eÿ1_nÿ1…ZG†:

Suppose that Gis an extension 1!Zⁿ!G!F !1 such thatF is ®nite and the conjugation action of F on Zⁿ is free outside the origin or that Gis a cocompact Fuchsian group. Then the Farrell-Jones-Conjecture with respect to the family VC holds forL^efor all decorationseif it holds foreˆ ÿy, as explained in [6, Remark 4.32, Remark 6.5]. This ®ts together with the fact that in the ®rst case Bˆ VCWSUB…Zⁿ†and in the second case BˆVC. This implies in both cases that the obvious map H_n^G…?G;VC;L^e† !H_n^G…?G;B;L^e† is bijective for all nAZ and eU2.

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Received August 22, 2000

T. Farrell, Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13091, U.S.A.

farrell@math.binghamton.edu

L. Jones, Department of Mathematics, State University of New York, Stony Brook, New York 11794, U.S.A.

lejones@math.sunysb.edu

W. LuÈck, Mathematisches Institut, WestfaÈlische Wilhelms-UniversitaÈt, Einsteinstr. 62 48149 MuÈnster, Germany

http://wwwmath.uni-muenster.de/u/lueck, lueck@math.uni-muenster.de