WHITEHEAD GROUPS AND THE BASS CONJECTURE 3 0 F.THOMASFARRELLANDPETERA.LINNELL 0 2 Abstract. This paper will be concerned with proving that certain White- n head groups of torsion-free elementary amenable groups are torsion groups a and related results, and then applying these results to the Bass conjecture. J In particular we shall establish the strong Bass conjecture for an arbitrary 9 elementaryamenablegroup. 1 ] T K 1. Introduction . h t Let k be a field, let Γ be a group, and let Units∗(kΓ) denote the subgroup of a Units(kΓ) consisting of those units in kΓ of the form aγ with a∈k\0 and γ ∈Γ. m Thenwe shalluse the notationWhk(Γ) for the quotientof K (kΓ)by the image of 1 [ Units∗(kΓ) under the naturalhomomorphismUnits∗(kΓ)→K1(kΓ). All ringswill 1 have a unity element 1, and a ring of prime characteristic will mean a ring such v that p1 = 0 for some prime p. Let T indicate the class of torsion abelian groups. 5 Recall that the class of elementary amenable groups is the smallest class of groups 0 which contains Z and all finite groups, and is closed under taking extensions and 2 1 directed unions. Our first result is 0 3 Theorem 1.1. Let k be a field of prime characteristic and let Γ be a torsion-free 0 elementary amenable group. Then Whk(Γ)∈T. / h Let K˜ (R) denote the reduced projective class group of the ring R, that is t 0 a K (R)/h[R]i. Then in the situation of Theorem 1.1, it now follows from Bass’s m 0 contractedfunctorargument[5,ChapterXII,§7]thatK˜ (kΓ)∈T andK (kΓ)∈T 0 i : for all i < 0, where k and Γ are as in Theorem 1.1. However we shall prove the v i following more general result. X r Theorem 1.2. Let k be a field of prime characteristic and let Γ be an elementary a amenable group. Then K (kΓ)⊗Q is generatedby the images of K (kG) as Gruns 0 0 over the finite subgroups of Γ. It is likely that Theorems 1.1 and 1.2 are also true for fields k of characteristic 0. However our proofs depend on Lemma 2.2, which shows that for rings of prime characteristic, certain nil groups are torsion groups. This is certainly false for arbitrary rings, so our proof does not cover the case when k has characteristic 0. Date:SatSep1419:11:23EDT2002. 1991 MathematicsSubject Classification. Primary: 19A31,19B28;Secondary: 16A27,16E20, 20C07. Key words and phrases. Bassconjecture, Whitehead group,nilgroup. ThefirstauthorwassupportedinpartbytheNational Science Foundation. 1 2 F.T.FARRELLANDP.A.LINNELL Given a prime power q, we shall let k indicate the field with q elements. In the q case k = k , we shall give a second proof of Theorem 1.2 in Section 7. This alter- 2 native proof will yield other results; for example we obtain the following variation of [20, Theorem 1.1]. Theorem1.3. LetGbeatorsion-freevirtuallysolvablesubgroupofGL (C). Then n Whk2(G)=0. We shallalsoprove(a2-groupis a groupin whicheveryelementhas ordera power of 2) Proposition 1.4. Let G be an abelian 2-group. Then K˜ (k [G≀Z])=0. 0 2 For the special case G = C in Proposition 1.4, the group is often called the 2 lamplighter group. Many interesting properties of this group are established in [23]. We shall apply Theorem 1.2 to obtain results on Bass’s strong conjecture [6, 4.5]. For convenience, we will restate the conjecture here (see Section 2 for the explanation of some standard notation used below). Conjecture 1.5 (The strong Bass conjecture). Let k be an integral domain, let Γ be a group, let g ∈ Γ, and let P be a finitely generated projective kΓ-module. Suppose o(g) is not invertible in k. Then r (g)=0. P Some cases for which Conjecture 1.5 is known to be true are (1) k =C and G a linear group [6, Proposition 6.2], (2) k =Z and o(g)<∞ [27, Lemma 4.1], (3) k =QandGhascohomologicaldimensionoverQatmosttwo[10,Theorem 3.3]. We shall prove Theorem1.6. ThestrongBass conjectureis trueforelementaryamenable groups. More precisely, let Γ be an elementary amenable group, let k be an integral domain, and let P be a finitely generated projective kΓ-module. If g ∈ Γ and o(g) is not invertible in k, then r (g)=0. P While revisingthis paper,wehavelearnedthatBerrick,ChatterjiandMislin[7] have proved the strong Bass conjecture for amenable groups in the case k =C. In fact they prove a rather stronger version for the Banach space ℓ1(G); on the other handtheirresultsdonotapplytothecasewhenkhasnonzerocharacteristic. Their techniques are rather different from ours, and depend on recent work of Vincent Lafforgue [26]. OneofmanyinterestingresultswhichGeraldCliffprovedinhisimportantpaper [8] was [8, Theorem 1], that if k is a field of nonzero characteristic p and Γ is a polycyclic-by-finite groupwith the property that all finite subgroups havep-power order, then kΓ has no nontrivial idempotents. We will use Theorem 1.2 and the techniques of Section 5 to extend Cliff’s result to elementary amenable groups. Theorem 1.7. Let p be a prime, let k be an integral domain of characteristic p, and let Γ be an elementary amenable group. Suppose every finite subgroup of Γ has p-power order. Then kΓ has no nontrivial idempotents. This work was carried out while we were at the Sonderforschungsbereich in Mu¨nster. We would like to thank Wolfgang Lu¨ck for organizing our visits to Mu¨nster, and the Sonderforschungsbereichfor financial support. WHITEHEAD GROUPS AND THE BASS CONJECTURE 3 2. Preliminary results Notation. Allmodules willbe rightmodules andmappings willbe writtenonthe left. Foreachpositiveintegern,weshallusethenotationC forthecyclicgroupof n ∞ ordern,Mat (R)forthen×nmatricesovertheringR,Mat(R)for Mat (R), n n=1 n and A for the class of finitely generated virtually abelian groups. Also G≀A will S denote the restricted wreath product of the groups G and A; thus G≀Z will have basegroup GandquotientgroupZ. We shallleto(g)denotetheorderofthe i∈Z element g ∈ G. In the case o(g) = ∞, we shall adopt the convention that o(g) is L not invertibleinanyring. AT-exactsequence willmeanasequence whichis exact modulotorsionabeliangroups;inotherwordseveryelementofthehomologygroup at each stage of the sequence has finite order. Similarly a T-epimorphism means a homomorphism which is onto modulo torsion. Suppose α is an automorphism of the ring R. Then R [t] will denote the twisted polynomial ring over R, and α R [t,t−1] will denote the twisted Laurent polynomial ring over R. Following [14, α §2],wedefineC(R,α)tobethecategorywhoseobjectsarepairs(P,φ)whereP isa finitelygeneratedprojectiveR-moduleandφisanα-linearnilpotentendomorphism of P, and whose morphisms g: (P ,φ ) → (P ,φ ) are R-linear homomorphisms 1 1 2 2 g: P → P with gφ = φ g. Then we shall let C(R,α) = K (C(R,α))/h[(R,0)]i, 1 2 1 2 0 and C˜(R,α) denote the subgroup of C(R,α) generated by elements of the form [(Rn,φ)]. We remark that C˜(R,α) is isomorphic to the subgroup of K (R [t]) 1 α generatedbyelementswhicharerepresentedbymatricesinMat(R [t])ofthe form α I+Nt, where I is the identity matrix and N ∈Mat(R) such that Nt is nilpotent; this can be seen from [14, proof of Theorem 13 and Proposition 20]. If β is an automorphism of the group Γ, then we shall let β also indicate the automorphism of kΓ induced by β. We need a standard description (cf. [25, §3]) of elementary amenable groups in order to carry out induction arguments. If X and Y are classes of groups, then G ∈ XY will mean that G has a normal subgroup H such that H ∈ X and G/H ∈ Y, and G ∈ LX will mean that every finitely generated subgroup of G is containedinaX-group(ifX isclosedundertakingsubgroups,thisisequivalentto sayingthateveryfinitelygeneratedsubgroupofGisanX-group). Foreachordinal α, we define X inductively as follows. α X is the class of finite groups; 0 X =(LX )A if α is a successor ordinal; α α−1 X = X if α is a limit ordinal. α β<α β Then the proof of [25, Lemma 3.1] (see also [28, Lemma 4.9]) yields the following. S Lemma 2.1. (i) The class of elementary amenable groups is X ; α≥0 α (ii) Each X is closed under taking subgroups; α S (iii) If H(cid:1)G with G/H finite and H ∈X or LX , then G∈X or LX respec- α α α α tively. Lemma 2.2. Let p be a prime, let R be a ring such that p1 = 0, and let α be an automorphism of R. Then C˜(R,α) is an abelian p-group. Proof. Let t be an indeterminant and let N ∈ Mat(R) such that (Nt)n = 0 for some positive integer n. Then I +Nt represents an element of K (R [t]) and we 1 α need to prove that this element has finite order. Now for M ∈ Mat(R [t]), the α 4 F.T.FARRELLANDP.A.LINNELL binomial formula shows that (I +M)p = I +Mp because pM = 0 and p divides p when 0<i<p. This equation by repetition yields i (cid:0) (cid:1) (I+M)pn =I +Mpn. Substituting into this equation M = Nt and observing that (Nt)pn = 0 because pn ≥ n, we obtain (I +Nt)pn = I. Hence C˜(R,α) is generated by elements of p-power order and the result follows. (cid:3) Lemma2.3. Let Rbea ringof prime characteristic and let αbean automorphism of R. Then there is a natural T-exact sequence K (R)−→K (R [t,t−1])−→K (R)1−−→α∗ K (R). 1 1 α 0 0 Proof. This follows immediately from [14, Theorem 19c] and Lemma 2.2. (cid:3) Remark 2.4. The constantgrouphomomorphismZn →1induces aringhomomor- phism R[Zn]→R[1]=R which splits the inclusion ring homomorphism R → R[Zn]. Hence K (R[Zn]) is i naturallyadirectsumofK (R)andthecomplementarysummandKˆ (R[Zn])which i i isthekernelofthehomomorphisminducedbytheabovesplitting. TheBass-Heller- Swanformula,whichisthefundamentaltheoremofalgebraicK-theory,assertsthat Kˆ (R[Z])∼=K (R)⊕C(R,id)⊕C(R,id). 1 0 InhiscontractedfunctortheoryBassusedthisformulatodefinetheloweralgebraic K-groups K (R) for n > 0, so that they are direct summands of Kˆ (R[Zn]); cf. −n 0 [5, Chapter XII, §7]. Corollary 2.5. Let k be a field of prime characteristic and let Γ = π ⋊Z be a group such that Whk(π×Zn) ∈ T for all n ≥ 0. Then Whk(Γ×Zn) ∈ T for all n≥0. Proof. Note that [14, Theorem 21d] remains true with k in place of Z and Whk in placeofWh. Soapplyingthisresult,Lemma2.2andtheBass-Heller-Swanformula, we obtain the exact sequence 0=Whk(π×Zn)⊗Q−→Whk(Γ×Zn)⊗Q−→K˜ (k[π×Zn])⊗Q 0 ⊆Whk(π×Zn+1)⊗Q=0. This proves Whk(Γ×Zn)∈T for all n≥0, as required. (cid:3) Corollary 2.6. Let R be a ring of prime characteristic and let α be an automor- phism of R. Suppose the natural map K (R)→K (R[s,s−1]) is a T-epimorphism. 0 0 Then the natural map K (R)→K (R [t,t−1]) is also a T-epimorphism. 0 0 α WHITEHEAD GROUPS AND THE BASS CONJECTURE 5 Proof. Consider the following commutative diagram K (R) −−−−→ K (R [t,t−1]) 0 0 α ↑ ↑ p p p p   K (R[s,s−1]) −−−−→ K (R [s,s−1,t,t−1]) −−−−→ K (R[s,s−1]) −−1−−α−→∗ K (R[s,s−1]) 1 y 1 α y 0 0 p T-epi p ↓ x x x x K1(R) −−−−→ K1(Rα[t,t−1]) −−−−→ K0(R) −−1−−α−→∗ K0(R) The squares in this diagram all commute and the 3 pairs of vertical arrows going in the opposite directions are splittings. The two horizontalsequences are T-exact by Lemma 2.3. Also in each of the first two columns, the composite of the two up vertical arrows is 0. A simple diagram chase using the fact that the marked up vertical arrow is a T-epi yields the result. (cid:3) Corollary 2.7. Let R be a ring of prime characteristic, let α be an automorphism of R, and suppose that the natural map K (R) → K (R[Zn]) is a T-epimorphism 0 0 for all n. Then K (R)∈T and K (R [t,t−1][Zn])∈T for all i<0, and K (R)→ i i α 0 K (R [t,t−1][Zn]) is a T-epimorphism, for all n. 0 α Proof. We use Bass’s contracted functor theory to deduce this; in particular, we use Remark 2.4. By our assumption Kˆ (R[Zn]) ∈ T and hence K (R) ∈ T for all 0 i i<0 because K (R)⊆Kˆ (R[Zn]) where n=−i. Since i 0 K (R[Zn])−→K (R[Zn+1])=K (R[Zn][s,s−1]) 0 0 0 isclearlyaT-epimorphismforalln≥0,f : K (R[Zn])→K (R [t,t−1][Zn])isalso 1 0 0 α aT-epimorphismbyCorollary2.6. Consequentlysoisf : K (R)→K (R [t,t−1][Zn]) 2 0 0 α since it is the composite of the two T-epimorphisms K (R) → K (R[Zn]) and f . 0 0 1 Setting n=m+(−i), where m≥0 and 0>i are given, we see that f : K (R [t,t−1][Zm])−→K ((R [t,t−1][Zm])[Z−i]), 3 0 α 0 α is also a T-epimorphismsince f =f ◦f where f : K (R)→K (R [t,t−1][Zm]). 2 3 4 4 0 0 α Therefore Kˆ (S[Z−i]) ∈ T where S = R [t,t−1][Zm]. But K (S) is a direct sum- 0 α i mand of Kˆ (S[Zi]), consequently K (R [t,t−1][Zm])∈T. (cid:3) 0 i α Corollary 2.8. Let k be a field of prime characteristic, let Γ = π⋊Z be a group such that for all n ≥ 0, K (k[π×Zn])⊗Q is generated by the images of K (kG) 0 0 as G varies over the finite subgroups of π. Then for all i<0 and n≥0, it follows that K (kπ) ∈ T and K (k[Γ×Zn]) ∈ T, and K (k[Γ×Zn])⊗Q is generated by i i 0 the images of K (kG) as G varies over the finite subgroups of Γ. 0 Proof. Let α denote the automorphism of π determined by the conjugation action of Z on π, and let R=kπ. Note that R[Zn]=k[π×Zn] and R [t,t−1][Zn]=k[Γ×Zn]. α Thusthe naturalmapK (R)→K (R[Zn])is T-surjective,andwe cannow obtain 0 0 the result from Corollary 2.7. (cid:3) 6 F.T.FARRELLANDP.A.LINNELL Lemma 2.9. Let n ≥ 0, let G be a finite group, and let k be a field. Then K (kG) = 0 for all i < 0 and the inclusion G ֒→ G×Zn induces an isomorphism i K (kG)→K (k[G×Zn]). 0 0 Proof. Let J denote the Jacobson radical of kG. Since kG is an Artinian ring, J is nilpotent [24, Corollary 1, p. 39] and kG/J is a semisimple Artinian ring [24, §III.3], and in particular kG/J is a regular ring. Then we have a commutative diagram K (kG) −−−−→ K (k[G×Zn]) 0 0 ∼= ∼= K0(kG/J) −−−∼=−→ K0(k[G×Zn]/J[Zn]). y y The two vertical arrows are isomorphisms by [40, Lemma II.2.2] and the bottom horizontal line is an isomorphism by the Bass-Heller-Swan formula, hence the top arrow is also an isomorphism as required and the second statement is proven. A consequence of the second statement is that Kˆ (S[Z−i]) = 0 where S = kG 0 and i<0. Hence K (kG)=0 since it is a direct summand of Kˆ (kG[Z−i]). (cid:3) i 0 Lemma 2.10. Let L∈A. Then L is isomorphic to a discrete cocompact subgroup of a virtually connected Lie group. Proof. There clearly exists a non-negative integer n such that L is an extension of afinite groupF bythe freeabeliangroupZn,whereZn denotesthe integrallattice points in the additive group Rn. This extension determines an action of F on Zn and hence on Rn, and a cohomology class θ ∈ H2(F,Zn). Let θ′ ∈ H2(F,Rn) be the image of θ and let G be the extension of F by Rn determined by θ′. Then G is a virtually connected Lie group containing L as a discrete cocompact subgroup. (Note that θ′ =0 and hence G=Rn⋊F.) (cid:3) The subclass of A consisting of the virtually cyclic groups is of particular im- portance to us. There is fortunately the following quite useful structure theorem for this subclass due to Scott and Wall [36]; cf. [19, Lemma 2.5] for another proof. Proposition 2.11. A virtually cyclic group Γ contains a finite normal subgroup F such that Γ/F is either trivial, infinite cyclic, of infinite dihedral. 3. Whitehead groups of elementary amenable groups Theorem 3.1. Letk be afield of prime characteristic, and let π(cid:1)Γ begroups such thatΓ/π isacrystallographic group. SupposeWhk(π˜×Zn)∈T forallnon-negative integers n whenever π˜/π is a finite subgroup of Γ/π. Then Whk(Γ)∈T. Proof. Our proof of Theorem 3.1 follows the pattern established in [16, 33]. In these papers it was shown that WhR(Γ) = 0 for Γ a torsion-free virtually poly-Z group and R any subring of Q. The case R = Z was done in [16] and the general case in [33]. Quinn had to develop important new geometric algebra concepts to do the general case, and these concepts are crucial in our proof of Theorem 3.1. Notation 3.2. Let L be a crystallographic group. Then L is isomorphic to a discrete cocompact subgroup of the group of all rigid motions Iso(En)∼=Orthog(n)⋉Rn WHITEHEAD GROUPS AND THE BASS CONJECTURE 7 of some Euclidean space En. The number n is the rank of a torsion-free abelian subgroup of finite index in L, and is called the dimension of L or dimL. Also the image of L in Orthog(n) is a finite group G, called the holonomy group of L, and its isomorphism class is determined by L. The order of G is called the holonomy number of L and is denoted by #(L). If S is a subgroup of finite index in L, then S isalsocrystallographic. FurthermoredimS =dimLandtheholonomygroupG 1 of S is isomorphic to a subgroup of G, hence • If G is cyclic, then so is G . 1 • #(S)≤#(L). Let L = Γ/π where Γ and π come from the statement of Theorem 3.1, let φ: Γ ։ L denote the natural epimorphism, and let G denote the holonomy group of L. Frobenius induction relative to G reduces the proof of Theorem 3.1 to the case G is cyclic. This follows from [37, Corollary 2.12] (see [16] for more details). It is also recommended that the reader now glance at the summary of Swan’s “Frobenius induction theory” given later in this paper in the paragraph following (3.2); in particular, see the important fact (3.5) mentioned there. We proceed to prove Theorem 3.1 by simultaneous induction on dim(L) and #(L), where we alwaysassumethatGiscyclic. Ourexplicitinductiveassumptionisthatwhenever L is a crystallographic group such that either 0 dim(L )<dim(L) or 0 dim(L )=dim(L) and #(L )<#(L), 0 0 then the theorem is true for L . So primary induction on dim(L) and secondary 0 induction on #(L). To startthe nth secondaryinduction, we need to show that Theorem3.1 is true when L = Zn (i.e. when #(L) = 1). Clearly we may assume that n > 0. In this case Γ/π ∼= Zn, so there exists a normal subgroup Γ of Γ containing π such that 0 Γ/Γ ∼=ZandΓ /π ∼=Zn−1. ThenΓ∼=Γ ⋊Z,andbyinductionWhk(Γ ×Zm)∈T 0 0 0 0 for all m≥0. We may now apply Corollary 2.5 with π =Γ . 0 Remark 3.3. LetT betheabeliannormalsubgroupofthecrystallographicgroupL consistingofallpuretranslations,soT =L∩Rn(seethenotationabove). ThenLis anextensionofGbyT. Thisextensiondeterminesbyconjugationarepresentation of G on T, called the holonomy representation of L. It is well known that L maps epimorphically onto Z if and only if TG 6= 0, where TG denotes the subgroup fixed by G: cf. [15, Lemma 1.4] and [12]. This fact can be proven by applying the Lyndon-Hochschild-Serre spectral sequence to the group extension 1 → T → L → G → 1. Its Epq term is Hp(G;Hq(T,Z)) and it converges to Hp+q(L,Z). Using 2 that E2,0 = H2(G,Z) is finite, one sees easily that E0,1 = (Hom(L,Z))G vanishes 2 2 if and only if H1(L,Z) = Hom(L,Z) vanishes. But (Hom(T,Z))G vanishes if and only if TG vanishes. We now consider the general inductive step, which is divided into two cases according to whether TG 6=0 or TG =0. Case 1. TG 6=0. By Remark 3.3, we may write L = L⋊Z for some L(cid:1)L. Let Γ = φ−1(L), so 0 Γ /π ∼= L. Since an A-group is crystallographic if and only if its unique maximal 0 finite normalsubgroupis 1,weseethatL isacrystallographicgroupwithdimL= 8 F.T.FARRELLANDP.A.LINNELL dimL−1. Therefore Whk(Γ ×Zm) ∈ T for all m ≥ 0 because of our inductive 0 assumption. We may now apply Corollary 2.5 to complete the inductive step in Case 1. Case 2. TG =0. The geometric algebra developed by Quinn [31, 32, 33, 34] is used crucially here, replacing the h-cobordisms used in [16]. A relatively simple example which con- cretely illustrates the terminology used in the remainder of the proof of Theorem 3.1 is worked out in detain in Section 7; cf. Case 2 of the proof of Corollary 7.3 where L is the infinite dihedral group C ∗C . It is recommended that the reader 2 2 keepthis example in mind while perusing the rest of the proof of Theorem 3.1. He would also see [32, Appendix] for details about stratified systems of fibrations. Topology now enters into our proof. Let M be a connected smooth manifold with π (M) = Γ and denote its universal cover by M˜. Also identify Γ with the 1 group of all deck transformations of M˜ → M. Projection onto the first factor of E :=Rn× M˜ induces a map Γ q: E −→Rn/L, where Γ acts on Rn via Γ ։ L ⊆ Iso(Rn). This map is a stratified system of fibrationsonRn/L in the sense of [32, Definition 8.2],whose strataaredetermined in the standard way by the holonomy groupaction of G on the n-torus Rn/T. Let s be any prime congruentto 1 mod #(L); recall that there is an infinitude of such primes by Dirichlet’s theorem. For each such s, there is an endomorphism ψ of L and a ψ-equivariant diffeomorphism f: Rn → Rn (relative to L ⊆ Iso(Rn), so fh=ψ(h)f for all h∈L) such that • |df(x)|=s|x| for each vector x tangent to Rn; • ψ(T)⊆T; • ψ induces id on G=L/T and G • ψ| is multiplication by s. T This is a restating of [16, Theorem 2.2] which is itself an immediate extension of the classical Epstein-Shub result [11]. Notation 3.4. WedenotebyL andT thefinitequotientgroupsL/sT andT/sT s s respectively. Note that the exact sequence 1−→T −→L −→G−→1 s s splits,anddoessouniquelyuptoconjugacysince(s,|G|)=1. Onesplitting,which we shall denote by G , is given by the image of ψ(L) in L under the projection s s L։L . Let η: Γ→L denote the composite epimorphism s s φ Γ։L։L , s andlet Γ ⊆Γ indicate the inverseimage ofψ(L) with respectto the epimorphism s φ: Γ ։ L, so Γ /π = ψ(L). Observe that Γ = η−1(G ). Furthermore, let E s s s s denote Rn× M˜, Γs let fˆ: Rn/L → Rn/ψ(L) denote the map induced by f, and let q : E → Rn/L s s be the composite of the map E → Rn/ψ(L) induced by projection onto the first s factor of Rn×M˜ with the homeomorphism fˆ−1. WHITEHEAD GROUPS AND THE BASS CONJECTURE 9 The map q is also a stratified system of fibrations with the same strata as q. s The following is an important observation. Remark 3.5. Let α be a smooth curve in E and let α˜ be a lift of α to the covering spaceE . Then|q ◦α˜|≤|q◦α|/s,where|.|denotesarclength(measuredviaRn). s s Proof. Let αˆ be a lift of α˜ to Rn×M˜ which is the universal cover of both E and s E. Then αˆ =(αˆ ,αˆ ) where αˆ is a smooth curve in Rn and αˆ is a smooth curve 1 2 1 2 inM˜. Then|q ◦α˜|isby definitionthearclengthoff−1◦α while |q◦α|is thearc s 1 length of α . Now the inequality (in fact equality) asserted in Remark 3.5 follows 1 from the fact (noted above) that df stretches tangent vectors by a factor of s. (cid:3) Another important observation is the following. Remark 3.6. Let S be a cyclic subgroup of L such that S projects onto G under s the second map in the short exact sequence 1−→T −→L −→G−→1. s s Then S∩T =1; i.e., S splits this sequence and is consequently conjugate to G . s s This observationis a consequence of our assumption that TG =0 together with [15, Lemmas 1.2 and 1.4]. Remark 3.7. So far mostofthe proofofTheorem3.1canbe repeatedverbatimfor the proofof Theorem4.1 in the next section. Howeverat this point the two proofs diverge somewhat. WenowrecallsomebasicfactsaboutQuinnassembly;cf.[32,Appendix]formore details. These facts will also be used in Sections 4 and 7. Let S be a homotopy invariant (covariant) functor from the category of topological spaces to Ω-spectra. Important examples of such functors are: X 7→ K(Rπ X), X 7→ Wh(π X), X 7→ 1 1 Whk(π X), where R is a ring with 1. Here Wh(π) is the cofiber of the standard 1 map of spectra H(Bπ;K(Z))−→K(Zπ) defined by Loday [29] and likewise Whk(π) is the cofiber of H(Bπ;K(k))−→K(kπ) defined also in [29]. Let M denote the category of continuous surjective maps; i.e. an object in M is a continuous surjective map p: E → B between topological spaces E and B, while a morphism from p : E → B to p : E → B is a pair 1 1 1 2 2 2 of continuous maps f: E → E , g: B → B making the following diagram a 1 2 1 2 commutative square of maps: f E −−−−→ E 1 2 p1 p2   B1 −−−−→ B2. y g y Quinn [32, Appendix] constructed a functor from M to the category of Ω-spectra whichassociatestothemappthespectrumH(B;S(p))insuchawaythatH(B;S(p))= 10 F.T.FARRELLANDP.A.LINNELL S(E)inthespecialcasewhereBisasinglepoint*. Furthermorethemapofspectra a: H(B;S(p))→S(E) functorially associated to the commutative square id E −−−−→ E p   B −−−−→ ∗ y y is calledthe (Quinn) assemblymap. The mapinduced byabetween the ithhomo- topy groups of these spectra is also called the assembly map and is denoted by the same symbol; i.e. a: H (B;S(p))→π (S(p)). i i Let x be an arbitrary (but fixed) element of Whk(Γ). We need to show that x has finite order to prove Theorem 3.1. For this purpose, recall that Quinn showed that x can be represented by a geometric isomorphism (denoted h) of geometric k-modulesonthespaceE [34,§3.2]. Thetransferofh(denotedbyh )tothe finite s sheeted cover E → E clearly represents the transfer of x to Whk(Γ ) denoted by s s x . But the radii ǫ of these geometric isomorphisms h measured via q in Rn/L s s s s go to zero as s→∞; i.e. (3.1) lim ǫ =0. s s→∞ In fact h represents an element x¯ ∈ Whk(Rn/L,q ,ǫ ) and x¯ maps to x under s s s s s s the natural homomorphism Whk(Rn/L,q ,ǫ ) → Whk(Γ ). Since the strata in s s s Rn/L of the stratified systems q are independent of s, Quinn’s stability theorem s [32, §4] together with equation (3.1) and [33, §3] yield that x is contained in s the image of H (Rn/L;Whk(q )) under the Quinn assembly map provided s is 1 s sufficiently large. Now there is an Atiyah-Hirzebruch-Quinn spectral sequence Et ij converging to H (Rn/L;Whk(q )) such that i+j s E2 =H (Rn/L;Whk(q )) ij i j s [32, Theorem 8.7]. Here Whk(.)=Whk(.), Whk(.)=K˜ (.), Whk(.)=K (.) if j <0. 1 0 0 j j The stalk ofthe coefficientsheafWhk(q ) overy ∈Rn/Lis Whk π (q−1(y)) . But j s j 1 s π is a subgroup of finite index in π (q−1(y)). Using the given hypotheses with 1 s (cid:0) (cid:1) π˜ = π (q−1(y)) and Bass’s contracted functor theory [5, Chapter XII, §7], we see 1 s that Whk π (q−1(y)) ∈T if j ≤1. j 1 s Since Rn/L is a finite polyhed(cid:0)ron, we con(cid:1)clude that H1(Rn/L;Whk(qs)) ∈ T. So this discussion yields that (3.2) x has finite order s provided s is sufficiently large. We now fix, for the remainder of the proof, a sufficiently large prime s such that • s≡1 mod |G| and • x has finite order. s We proceed to apply Frobenius induction to Whk(Γ) relative to the factor group L . Let η: Γ→L denote the composite epimorphism s s φ Γ։L։L . s