The topological K-theory of certain crystallographic groups

Let p be a prime. Let ρ : Z/p → Aut(Z n ) = GL(n, Z) be a group homomorphism. Throughout this paper we will assume: Condition 0.1 (Free conjugation action). The induced action of Z/p on Z n is free when restricted to Z n -0. Denote by Γ = Z n ⋊ ρ Z/p (0.2) the associated semidirect product. Since Γ has a finitely generated, free abelian subgroup which is normal, maximal abelian, and has finite index, Γ is isomorphic to a crystallographic group. An example of such group Γ is given by Z p-1 ⋊ ρ Z/p where the action ρ is given by the regular representation Z[Z/p] modulo the ideal generated by the norm element. When n = 1 and p = 2, Γ is the infinite dihedral group. Let BΓ := Γ\EΓ be the classifying space of Γ. Denote by EΓ be the classifying space for proper group actions of Γ. Let BΓ = Γ\EΓ. The space BΓ is the quotient of the torus T n under the Z/p-action associated to ρ. It is not a manifold, but an orbifold quotient. To compute the K-theory of the C * -algebra, we will use the Baum-Connes Conjecture which predicts for a group G that the complex and real assembly maps -→ KO n (C * r (G; R)), are bijective for n ∈ Z. The point of the Baum-Connes Conjecture is that it identifies the very hard to compute topological K-theory of the group C * -algebra of G to the better accessible evaluation at EG of the equivariant homology theory given by equivariant topological K-theory. The Baum-Connes Conjecture has been proved for a large class of groups which includes crystallographic groups (and many more) in [19]. We will later use the composite maps, where in each case the second map is induction with the projection Γ → {1}. ← -KO Γ m (EΓ) → KO m (BΓ). Next we describe the main results of this paper. We will show in Lemma 1.9 (i) that k = n/(p -1) is an integer. Let P be the set of conjugacy classes {(P )} of finite non-trivial subgroups of Γ. Theorem 0.3 (Topological K-theory of the complex group C * -algebra). Let Γ = Z n ⋊ ρ Z/p be a group satisfying Condition 0.1. (i where In particular K m (C * r (Γ)) is always a finitely generated free abelian group. (ii) There is an exact sequence 0 → -→ K 1 (BΓ) is an isomorphism. Restricting to the subgroup Z n of Γ induces an isomorphism K 1 (C * r (Γ)) )) Z/p . Remark 0.4 (Twisted group algebras). The computation of Theorem 0.3 has already been carried out in the case p = 2 and in the case n = 2 and p = 3 in [17, Theorem 0.4, Example 3.7]. In view of [17,Theorem 0.3] the computation presented in this paper yields also computations for the topological K-theory K * (C * r (Γ, ω)) of twisted group algebras for appropriate cocycles ω. One may investigate whether the whole program of [17] can be carried over to the more general situation considered in this paper. Remark 0.5 (Computations by Cuntz and Li). Cuntz and Li [13] compute the K-theory of C * -algebras that are associated with rings of integers in number fields. They have to make the assumption that the algebraic number field contains only {±1} as roots of unity. This is related to our computation in the case p = 2. Our results, in particular, if we could handle instead of a prime p any natural number, may be useful to extend their program to the arbitrary case. However, the complexity we already encounter in the case of a prime p shows that this is a difficult task. We are also interested in the slightly more difficult real case because of applications to the question whether a closed smooth spin manifold carries a Riemannian metric with positive scalar curvature (see Theorem 0.7). The numbers r l appearing in the next theorem will be defined in (1.4) and analyzed in Subsection 1.3. Theorem 0.6 (Topological K-theory of the real group C * -algebra). Let p be an odd prime. Let Γ = Z n ⋊ Z/p be a group satisfying Condition 0.1. Then for all m ∈ Z : (i) is an isomorphism. Restricting to the subgroup Z n of Γ induces an isomorphism If M is a closed spin manifold of dimension m with fundamental group G, one can define an invariant α(M ) ∈ KO m (C * r (G; R)) as the index of a Dirac operator. If M admits a metric of positive scalar curvature, then α(M ) = 0. This theory and connections with the Gromov-Lawson-Rosenberg Conjecture will be reviewed in Subsection 12.1. Theorem 0.7 ((Unstable) Gromov-Lawson-Rosenberg Conjecture). Let p be an odd prime. Let M be a closed spin manifold of dimension m ≥ 5 and fundamental group Γ as defined in (0.2). Then M admits a metric of positive scalar curvature if and only if α(M ) is zero. Moreover if m is odd, then M admits a metric of positive scalar curvature if and only if the p-sheeted covering associated to the projection Γ → Z/p does. Example 0.8. Here is an example where the last sentence of Theorem 0.7 applies. Choose an odd integer k > 1. Let M be a balanced product S k × Γ R n where Γ acts on the sphere via the projection Γ → Z/p and a free action of Z/p on the sphere and Γ acts on R n via its crystallographic action. Then its p-fold cover S k × T n admits a metric of positive scalar curvature since it is a spin boundary or since it is a product of a closed manifold with a closed Riemannian manifold with positive scalar curvature, and hence M admits a metric of positive scalar curvature. Remark 0.9. Notice that Theorem 0.7 is not true for Z 4 × Z/3 (see Schick [39]), whereas it is true for Z 4 ⋊ ρ Z/3 for appropriate ρ by Theorem 0.7. The computation of the topological K-theory of the reduced complex group C *algebra C * r (Γ) and of the reduced real group C * -algebra C * r (Γ; R) will be done in a sequence of steps, passing in each step to a more difficult situation. We will first compute the (co-)homology of BΓ and BΓ. A complete answer is given in Theorem 1.7 and Theorem 2.1. Then we will analyze the complex and real topological K-cohomology and Khomology of BΓ and BΓ. A complete answer is given in Theorem 3.3, Theorem 4.3, Theorem 5.1 and Theorem 6.3 except for the exact structure of the p-torsion in K 2m+1 (BΓ), KO 2m+1 (BΓ), K 2m (BΓ), and KO 2m (BΓ). In the third step we will compute the equivariant complex and real topological K-theory of EΓ, and hence the K-theory of the complex and real C * -algebras of Γ. A complete answer is given in Theorem 0.3 and Theorem 0.6. It is rather surprising that we can give a complete answer although we do not know the full answer for BΓ. Finally we use the Baum-Connes Conjecture to prove Theorem 0.3 and Theorem 0.6 in Sections 11. The proof of Theorem 0.7 will be presented in Section 12. Although we are interested in the homological versions, it is important in each step to deal first with the cohomological versions as well since we will make use of the multiplicative structure and the Atiyah-Segal Completion Theorem. This paper was financially supported by the Hausdorff Institute for Mathematics, the Max-Planck-Institut für Mathematik, the Sonderforschungsbereich 478 -Geometrische Strukturen in der Mathematik -, the NSF-grant of the first author, and the Max-Planck-Forschungspreis and the Leibniz-Preis of the second author. We thank the referee for his detailed report. The paper is organized as follows: In this section we compute the cohomology of BΓ and EΓ for the group Γ defined in (0.2). It fits into a split exact sequence We write the group operation in Z/p and Γ multiplicatively and in Z n additively. We fix a generator t ∈ Z/p and denote the value of ρ(t) by ρ : Z n → Z n . When wish to emphasize that Z n is a Z[Z/p]-module, we denote it by Z n ρ . 1.1. Statement of the computation of the cohomology. Notation 1.2 (EG and BG). For a discrete group G we let EG denote the classifying space for proper G-actions. Let BG be the quotient space G\EG. Recall that a model for the classifying space for proper G-actions is a G-CWcomplex EG such that EG H is contractible if H ⊂ G is finite and empty otherwise. Two models are G-homotopy equivalent. There is a G-map EG → EG which is unique up to G-homotopy. Hence there is a map BG → BG, unique up to homotopy. If G is torsion-free, then EG = EG and BG = BG. For more information about EG we refer for instance to the survey article [30]. We will write H m (G) and H m (G) instead of H m (BG) and H m (BG). Example 1.3 (EΓ and BΓ). Since the group Γ is crystallographic and hence acts properly on R n by smooth isometries, a model for EΓ is given by R n with this Γ-action. In particular BΓ is a quotient of the n-torus T n by a Z/p-action. The main result of this section is the computation of the group cohomology of BΓ and BΓ. Most of the calculation for H * (BΓ) has already been carried out by Adem [3] and later, with different methods, by Adem-Ge-Pan-Petrosyan [5]. The computation of H * (BΓ) has recently and independently obtained by different methods by Adem-Duman-Gomez [4]. We include a complete proof since the techniques will be needed later when we compute topological K-theory. Let be the norm element. Denote by I(Z/p) the augmentation ideal, i.e., the kernel of the augmentation homomorphism Z[Z/p] → Z. Let ζ = e 2πi/p ∈ C be a primitive p-th root of unity. We have isomorphisms of Define natural numbers for m, j, k ∈ Z ≥0 . where here and in the sequel Λ m means the m-th exterior power of a Z-module. Notice that these numbers r m , a j and s m depend on k but we omit this from the notation since k will be determined by the equation n = k(p-1) (see Lemma 1.9 (i)) and hence by Γ. Note that r 0 = 1, r 1 = 0, a 0 = 1, a 1 = k, s 0 = 0, s 1 = 1, and s 2 = k + 1. We will give more information about these numbers in Subsection 1.3. (ii) For m ≥ 0 the restriction map The map induced by the various inclusions Remark 1.8 (Multiplicative structure). A transfer argument shows that the kernel of the restriction map H m (Γ) → H m (Z n ) is p-torsion. Theorem 1.7 together with the exact sequence (1.14) implies that the map induced by the restrictions to the various subgroups is injective. The multiplicative structure of the target is obvious. This allows in principle to detect the multiplicative structure on H * (Γ). 1.2. Proof of Theorem 1.7. The proof of Theorem 1.7 needs some preparation. Lemma 1.9. (i) We have an isomorphism of Z[Z/p]-modules, where the We have (ii) Each non-trivial finite subgroup P of Γ is isomorphic to Z/p and its Weyl group W Γ P := N Γ P/P is trivial. (iii) There are isomorphisms and a bijection cok ρ -id : If we fix an element s ∈ Γ of order p, the bijection sends the element u ∈ Z n ρ /(1 -ρ)Z n ρ to the subgroup of order p generated by us. (iv) We have |P| = p k . (v) There is a bijection from the Z/p-fixed set of the Z/p-space Then N •u is fixed by the action of t ∈ Z/p and hence is zero by assumption. Thus Z n ρ is a finitely generated module over the Dedekind domain Any finitely generated torsion-free module over a Dedekind domain is isomorphic to a direct sum of non-zero ideals (see [36, page 11]). Since (ii) This is obvious. (iii) Since the norm element N acts trivially on Z n ρ , we get cok ρ -id : Lemma 1.10 (i). One easily checks that the map cok ρ -id : Z n → Z n → P is bijective. (iv) This follows from assertion (iii). (v) Consider the short exact sequence of Z[Z/p]-modules × and hence we get ker(ρ -id) = ker(ρ i -id) = 0 and im(ρ -id) = im(ρ i -id). This implies Next will analyze the Hochschild-Serre Spectral sequence (see [12, page 171]) (1.1). We say that a spectral sequence collapses if all differentials d i,j r are trivial for r ≥ 2 and all extension problems are trivial. The basic properties of the Tate cohomology H i (G; M ) of a finite group G with coefficients in a Z[G]module M are reviewed in Appendix A. Lemma 1.10. (i) H i+j (Z/p; Z) = (Z/p) aj i + j even; 0 i + j odd. (ii) The Hochschild-Serre spectral sequence associated to the extension (1.1) collapses. There is a sequence of Z[Z/p]-isomorphisms , where ρ(t) * : Z n → Z n for t ∈ Z/p is given by the transpose of the matrix describing ρ(t) : Z n → Z n . The natural map given by the product in cohomology is bijective and hence is a Z[Z/p]-isomorphism by naturality. Thus we obtain a Z[Z/p]-isomorphism is bijective. Hence we obtain an isomorphism and Λ l (Z[ζ]) = 0 for l ≥ p, we get Therefore we obtain an isomorphism Hence it suffices to show for l 1 , . . . , l k in {0, 1, . . . , p -1} This will be done by induction over j = k a=1 l a . The induction beginning j = 0 is trivial, the induction step from j -1 to j ≥ 1 done as follows. We can assume without loss of generality that 1 ≤ l 1 ≤ p -1 otherwise permute the factors. There is an exact sequence of Z[Z/p]-modules where 1 ∈ Z maps to the norm element N ∈ Z[Z/p]. Since this exact sequence splits as an exact sequence of Z-modules, it induces an exact sequence of Z[Z/p]-modules We next show that the middle term of (1.11) is a free Z[Z/p]-module when 1 ≤ l 1 ≤ p -1. Since Z/p = {t 0 , t 1 , . . . , t p-1 } is a Z-basis for Z[Z/p], we obtain a Z-basis for Λ l1 Z[Z/p] by An element s ∈ Z/p acts on Λ l1 Z[Z/p] by sending the basis element t I to ±t s+I . The Z/p action on {I ⊂ Z/p, |I| = l 1 } which sends I to s + I for s ∈ Z/p, is free. Indeed, for s ∈ Z/p -{0}, the permutation of the p-element set Z/p given by a → s + a cannot have any proper invariant sets since the permutation has order p and p is prime. This implies that the Z[Z/p]-module Λ l1 Z[Z/p] is free. We obtain from the exact sequence (1.11) an exact sequence of Z[Z/p]-modules with a free Z[Z/p]-module in the middle Hence we obtain for i ∈ Z an isomorphism Now apply the induction hypothesis. This finishes the proof of assertion (i). (ii) Next we want to show that the differentials d i,j r are zero for all r ≥ 2 and i, j. By the checkerboard pattern of the E 2 -term it suffices to show for r ≥ 2 and that the differentials d 0,j r are trivial for r ≥ 2 and all odd j ≥ 1. This is equivalent to show that for every odd j ≥ 1 the edge homomorphism (see Proposition A.5) is surjective. But H 0 (Z/p, H j (Z n ρ )) = 0 by assertion (i), so the norm map N = ι j •trf j : H j (Z n ρ ) Z/p → H j (Z n ρ ) Z/p is surjective (see TheoremA.3), so ι j is surjective. It remains to show that all extensions are trivial. Since the composite is multiplication with p, the torsion in H i+j (Γ) has exponent p. Since p • E i,j ∞ = p • E i,j 2 = 0 for i > 0, all extensions are trivial and Proof of assertions (i) and (ii) of Theorem 1.7. These are direct consequences of Lemma 1.10. Proof of assertion (iii) of Theorem 1.7. We obtain from [34,Corollary 2.11] together with Lemma 1.9 (ii) a cellular Γ-pushout where i 0 and i 1 are inclusions of Γ-CW -complexes, pr P is the obvious Γ-equivariant projection and P is the set of conjugacy classes of subgroups of Γ of order p. Taking the quotient with respect to the Γ-action we obtain from (1.12) the cellular pushout where j 0 and j 1 are inclusions of CW -complexes, pr P is the obvious projection. It yields the following long exact sequence for m ≥ 0 where ϕ * is the map induced by the various inclusions P ⊂ Γ for (P ) ∈ P. Now assertion (iii) follows from (1.14) since there is a n-dimensional model for BΓ. We still need to prove assertion (iv) of Theorem 1.7. In order to compute H * (BΓ), we need to compute the kernel and image of ϕ 2m . Lemma 1.15. Let m ≥ 1. (i) Let K 2m be the kernel of ϕ 2m . There is a short exact sequence → 0 where the first non-trivial map is the restriction of ι * : H 2m (Γ) → H 2m (Z n ρ ) Z/p to K 2m and the second non-trivial map is given by the quotient map appearing in the definition of Tate cohomology. It follows that We first claim that K 2m = L 2m . Indeed, the following diagram commutes / / (P )∈P H 2m+2n (P ) Since dim(BΓ) ≤ n, we have H i+2n (BΓ) = 0 for i ≥ 1. Hence the lower horizontal arrow is bijective by (1.14). The right vertical arrow is bijective. Thus K 2m = L 2m . Recall that we have an descending filtration ∞ by Lemma 1.10 (ii). From the multiplicative structure of the spectral sequence we see that the image of the map lies in F 2n,2m and the following diagram commutes where the columns are exact. The upper horizontal arrow is bijective. Namely, one shows by induction over r = -1, 0, 1, . . . , 2m -1 that the map is bijective. The induction beginning r = -1 is trivial since then both the source and the target are trivial, and the induction step from r -1 to r follows from the five lemma and the fact that the map The bottom horizontal map in diagram (1.16) can be identified with the composition of the canonical quotient map . So what do we know about diagram (1.16)? The top horizontal map is an isomorphism, the kernel of middle horizontal map is L 2m , and the bottom horizontal map is onto. We conclude from the snake lemma that the middle map is an epimorphism and that we have a short exact sequence The first non-trivial map is the composite of the inclusion induced by the inclusion ι : Z n → Γ. We have already identified the second nontrivial map (up to isomorphism) with the quotient map as desired. Hence the sequence in assertion (i) is exact. Since the middle term is isomorphic to Z rm and the right term is finite, K 2m is also isomorphic to Z rm . (ii) The exact sequence has the property that ι 2m restricted to K 2m is injective. Thus we can quotient by K 2m and ι 2m (K 2m ) in the middle and right hand term respectively and maintain exactness. Hence we have the exact sequence where we used assertion (i) to compute the right hand term. We conclude from Lemma 1.10 Since H 2m (Γ)/K 2m is isomorphic to a subgroup of (P )∈P H 2m (P ) by the long exact cohomology sequence (1.14) it is annihilated by multiplication with p. Hence the short exact sequence (1.17) splits and we conclude from (1.18) and (1.19) This finishes the proof of Lemma 1.15. We conclude from the exact sequence (1.14), Theorem 1.7 (i), Lemma 1.9 (iv), and Lemma 1.15 Corollary 1.20. For m ≥ 1 the long exact sequence (1.14) can be identified with Proof of assertion (iv) of Theorem 1.7. Obviously H 0 (BΓ) ∼ = Z. Since (Z n ) Z/p = 0 by assumption, we get H 1 (Γ) = 0 from assertion (ii) of Theorem 1.7. We conclude H 1 (BΓ) ∼ = 0 from the long exact sequence (1.14). The values of H m (BΓ) for m ≥ 2 have already been determined in Corollary 1.20. Hence assertion (iv) of Theorem 1.7 follows. This finishes the proof of Theorem 1.7. 1.3. On the numbers r m . In this subsection we collect some basic information about the numbers r m , a j and s m introduced in (1.4),(1.5), and (1.6). Since Z n acts freely on EΓ = R n , we conclude from Lemma 1.9 (i) and Proposition A.4 Since Tate cohomology is rationally trivial, the norm map is a rational isomorphism, hence also Proof. In the proof below we write Λ l V instead of Λ l Q V for a Q-vector space V . (i) This follows directly from the definitions. (ii) Suppose 1 ≤ l ≤ p -1. By rationalizing the exact sequence (1.11) we have the short exact sequence of Q[Z/p]-modules We conclude from (1.25) and (1.26) we obtain the following equality in R Q (Z/p) if p is odd: There is a homomorphism of abelian groups and r m respectively. Hence we conclude from (1.27) If X is a finite Z/p-CW-complex with orbit space X, then the Riemann-Hurwitz formula states that One derives this formula by verifying it for both fixed and freely permuted cells. Applying Proposition A.4, the Riemann-Hurwitz formula, and Lemma 1.9 (v) to the Z/p-action on the torus T n , one sees We conclude from (1.28) and (1.30 The first formula follows from (1.21) and applying the homomorphism Φ to (1.24). The rest of (iii) is clear from the definitions. Next we determine the group homology of the group Γ. Recall that for a Z (i) For m ≥ 0, (ii) For m ≥ 0, the inclusion map Z n → Γ induces an isomorphism (iii) The map induced by the various inclusions ϕ m : (v) For m ≥ 1 the long exact homology sequence associated to the pushout (1.13) Proof. (i) (iii) (iv) and (v) Recall there is a exact sequence , Z) → 0 for every CW -complex X with finite skeleta, natural in X. This, Theorem 1.7 and Corollary 1.20 imply (i), (iv), and (v). (ii) Here again we use the Hochschild-Serre spectral sequence Then the Universal Coefficient Theorem, Lemma A.1, and Lemma 1.10 (i) imply that for i + j even, Hence E 2 i,j = 0 when i + j is even and i > 0. Since Next we analyze the values of complex K-theory K * on BΓ and BΓ. Recall that by Bott periodicity K * is 2-periodic, K 0 ( * ) = Z, and K 1 ( * ) = 0. A rational computation of K * (BG) ⊗ Q has been given for groups G with a cocompact G-CW -model for EG in [31, Theorem 0.1], namely where con q (G) is the set of conjugacy classes (g) of elements g ∈ G of order q d for some integer d ≥ 1 and C G g is the centralizer of the cyclic subgroup g . It gives in particular for G = Γ because of Theorem 1.7 (ii) and (i) and Lemma 1.9 Recall that we have computed l∈Z r 2l and l∈Z r 2l+1 in Lemma 1.22 (ii). We are interested in determining the integral structure, namely, we want to show Theorem 3.3 (K-cohomology of BΓ and BΓ). (i) For m ∈ Z, Here Z p is the p-adic integers. (ii) There is a split exact sequence of abelian groups (iii) Restricting to the subgroup Z n of Γ induces an isomorphism for a finite abelian p-group T 1 for which there exists a filtration Its kernel is isomorphic to T 1 and is isomorphic to the cokernel of the map The proof of Theorem 3.3 needs some preparation. We will use two spectral sequences. The Atiyah-Hirzebruch spectral sequence (see [43,Chapter 15]) for topological K-theory E i,j 2 = H i (BΓ; K j ( * )) ⇒ K i+j (BΓ) converges since BΓ has a model which is a finite dimensional CW -complex. We also use the Leray-Serre spectral sequence (see [43,Chapter 15]) of the fibration BZ n → BΓ → BZ/p. Recall that its E 2 -term is E i,j 2 = H i (Z/p; K j (BZ n ρ )) and it converges to K i+j (BΓ). The Leray-Serre spectral sequence converges (with no lim 1 -term) by [32,Theorem 6.5]. Lemma 3.4. In the Atiyah-Hirzebruch spectral sequence converging to K * (BΓ), where Since BΓ has a finite CW -model, all differentials in the Atiyah-Hirzebruch spectral sequence converging to K * (BΓ) are rationally trivial and there exists an N so that for all i, j, E i,j N = E i,j ∞ . The E 2 -term of the Atiyah-Hirzebruch spectral sequence converging to K * (BΓ) is given by Theorem 1.7 (i) A map with a torsion free abelian group as target is already trivial, if it vanishes rationally. Now consider (i, j) such that it is not true that i is odd and j is even. Then one shows by induction over r ≥ 2 that E i,j r is zero for odd j and Z ri for even j, the differential ending at (i, j) in the E r -term is trivial and the image of the differential starting at (i, j) is finite, and E i,j r is an abelian subgroup of E i,j r+1 of finite index. Next consider (i, j) such that i is odd and j is even. Then one shows by induction over r ≥ 2 that the image of the differential ending at (i, j) in the E r -term lies in the torsion subgroup of E i,j r+1 , the differential starting at (i, j) is trivial, the rank of E i,j r+1 is r i and its torsion subgroup is isomorphic to Z/p t for some t with t ≤ p k -s i . This finishes the proof of Lemma 3.4. Lemma 3.5. (i) For every m ∈ Z, there is an isomorphism of Z[Z/p]-modules In particular we get (iii) All differentials in the Leray-Serre spectral sequence are trivial. Proof. (i) Since K * ( * ) is torsion free, Lemma 3.6 below shows that the Chern character gives an isomorphism Since T n is a model for the Z/p-space BZ n ρ and ch m is natural with respect to self-maps of the torus, ch m is an isomorphism of Z[Z/p]-modules. Since H m+2l (Z n ρ ) Z/p ∼ = Z r m+2l by Theorem 1.7 (ii) and (i), assertion (i) follows. (ii) This follows from Lemma 1.10 (i) and assertion (i). (iii) Next we want to show that the differentials d i,j r are zero for all r ≥ 2 and i, j. By the checkerboard pattern of the E 2 -term it suffices to show for r ≥ 2 that the differentials d 0,j r are trivial for r ≥ 2 and all odd j ≥ 1. This is equivalent to showing that for every odd j ≥ 1 the edge homomorphism (see Proposition A.5) is surjective. To show this we use the transfer, whose properties are reviewed in Appendix A. For j odd, H 0 (Z/p, K j (Z n ρ )) = 0 by assertion (ii). Thus the norm map N = ι j • trf j is surjective, and so ι j is surjective as desired. Let H * be a generalized homology theory and H * a generalized cohomology theory. Dold defined (see [16] and [27, Example 4.1]) Chern characters The homological Chern character is a natural transformations of homology theories defined on the category of CW -pairs. When X = * , then after the obvious identification of the targets. Hence these Chern characters are rational isomorphisms for any CW -pair. In cohomology there are parallel statements after restricting oneself to the category of finite CWpairs. (If the disjoint union axiom is fulfilled, finite dimensional suffices). A CW -pair (X, Y ) is is a monomorphism, and ch m gives an isomorphism onto the image of i Q . There is a similar definition of H * -Chern integral for finite CW -pairs. Lemma 3.6 (Chern character). (ii) Consider the following commutative diagram with split exact columns. The columns come from the long exact sequence of a pair where D 1 is included in S 1 as the upper semicircle. The splitting maps are given by a constant map S 1 → D 1 . It is elementary to see that the bottom row is isomorphic to One may also argue by using the fact that stably X ×S 1 agrees with X ∨S 1 ∨ΣX and the property H * -Chern integral is inherited by suspensions and wedges. Proof of Theorem 3.3. (iv) (v) These assertions follow from the Atiyah-Hirzebruch spectral sequence converging to K * (BΓ) using Lemma 3.4. (ii) (iii) (vi) We first claim that for all m ∈ Z, the inclusion ι : Z n → Γ induces an epimorphism ι m : K m (BΓ) → K m (BZ n ρ ) Z/p and K m (BZ n ρ ) Z/p ∼ = Z l∈Z rm+2l . We will also show that for m odd, the map ι m is an isomorphism. By Lemma 3.5 (iii), the Leray-Serre spectral sequence collapses, so E 0,m 2 = E 0,m ∞ . Hence the edge homomorphism ι m is onto (see Proposition A.5). The computation of K m (BZ n ρ ) is given in Lemma 3.5 (i). Now assume m is odd. For any i > 0, E i,m-i 2 = 0 by Lemma 3.5 (ii). Hence H m (BΓ) = E 0,m ∞ , so the edge homomorphism is injective. We have now proved assertion (iii) of our theorem. We next note that for all m ∈ Z, the kernel and cokernel of the composite Z/p are finitely generated abelian p-groups. This follows from Proposition A.4 and the following commutative diagram By Lemma 1.9 (iv), the number of conjugacy classes of order p subgroups of Γ is p k . By the Atiyah-Segal Completion Theorem (see [8]) where I C (Z/p) ⊂ R C (Z/p) is the augmentation ideal. Hence We are now in a position to analyze the long exact sequence associated to the cellular pushout (1.13). We will work from right to left. Since K 1 (BΓ) ∼ = K 1 (BZ n ρ ) Z/p is torsion free, it follows that the kernel of f 1 equals T 1 , the p-torsion subgroup of K 1 (BΓ). By exactness of (3.8), T 1 also equals the cokernel of ϕ 0 . This completes the proof of assertion (vi). We showed above that ker f 1 = im δ 0 is a finite abelian p-group. It follows that ker δ 0 = im ϕ 0 is also isomorphic to (Z p ) (p-1)p k since any finite abelian pgroup A is p-adically complete, and hence a Z p -module, a Z-homomorphism from (Z p ) (p-1)p k → A is automatically a Z p -homomorphism and Z p is a principal ideal domain. Consider the commutative diagram with exact rows We have already seen that the middle vertical map is surjective with free abelian target, hence split surjective. Let K be the kernel of ι 0 . Then by the Snake Lemma, there is a short exact sequence As we noted above, im ϕ 0 ∼ = (Z p ) (p-1)p k and coker (ι 0 ) is a finite abelian p-group. Thus K is also isomorphic to (Z p ) (p-1)p k . This completes the proof of assertion (ii). (i) This follows from assertions (ii) and (iii). This finishes the proof of Theorem 3.3. In this section we compute complex K-homology of BΓ and BΓ. Rationally this can be done using the Chern character of Dold [16] which gives for every CWcomplex a natural isomorphism l∈Z In particular we get from Theorem 2.1 (i) and (iv) We are interested in determining the integral structure, namely, we want to show Theorem 4.3 (K-homology of BΓ and BΓ). (i) For m ∈ Z, and K 0 (BZ n ρ ) Z/p ∼ = Z l∈Z r 2l . (iii) There is a split short exact sequence of abelian groups (iv) We have where T 1 is the finite abelian p-group appearing in Theorem 3.3 (v). (v) We have (vi) The group T 1 is isomorphic to a subgroup of the kernel of Its proof needs some preparation. Theorem 4.4 (Universal Coefficient Theorem for K-theory). For any CW -complex X there is a short exact sequence If X is a finite CW -complex, there is also the K-homological version Proof. A proof for the first short exact sequence can be found in [6] and [46, A reference for the basic properties is [18]. These include: G is also a locally compact abelian group. The natural map from G to its double dual is a isomorphism. Here Z/p ∞ is given the discrete topology and the p-adic integers Z p are given the p-adic topology. This statement is included in [18, paragraph 25.2], but also follows from the following assertion proved in [25, 20.8] if We will now give the computation of K * (BZ/p). The Atiyah-Hirzebruch Spectral Sequence shows that K 0 (BZ/p) = 0. Vick [44] shows that K 1 (BG) is the Pontryagin dual of K 0 (BG) for any finite group G. Applying these facts to G = Z/p we get (see also Knapp [22,Proposition 2.11]) Thus the long exact K-homology sequence associated to the cellular pushout (1.13) reduces to the exact sequence 0 → K 0 (BΓ) Note that im ∂ 0 is a finite abelian p-group, since it is a finitely generated subgroup of the p-torsion group we see that im ϕ 0 has finite p-power index in (Z p ) (p-1)p k hence is itself isomorphic to (Z p ) (p-1)p k (compare the proof of Theorem 3.3 (iv) and (v)). Dualizing again, we see im The map f 1 is split surjective since its target is free abelian by assertion (v). (ii) The Universal Coefficient Theorem in K-theory shows that K 0 (BZ n ρ ) ∼ = K 0 (BZ n ρ ) * . In Lemma 3.5 we showed there is an isomorphism of Z[Z/p]-modules K 0 (BZ n ρ ) ∼ = ⊕ ℓ H 2ℓ (Z n ρ ). Now we proceed exactly as in the proof of Theorem 2.1 (ii), using the Leray-Serre spectral sequence One shows E 2 0,2m = K 2m (BZ n ρ ) Z/p is torsion-free, and for i > 0, E 2 i,j has exponent p and vanishes if i + j is even. Thus By Remark A.2 and the Universal Coefficient Theorem, (K 2m (BZ n ρ ) Z/p ) * ∼ = K 2m (BZ n ρ ) Z/p which is isomorphic to Z l∈Z r 2l by Lemma 3.5 (i). (i) This follows from assertions (ii), (iii), and (v). (vi) This follows from assertion (iv) and the exact sequence (4.6). This finishes the proof of Theorem 4.3. In this section we compute real K-cohomology KO * of BΓ. Recall that by Bott periodicity KO * is 8-periodic, i.e., there is a natural isomorphism KO m (X) ∼ = KO m+8 (X) for every m ∈ Z and CW -complex X, and KO -m ( * ) is given for m = 0, 1, 2, . . . 7 by Z, Z/2, Z/2, 0, Z, 0, 0, 0. We will assume from now on that p is odd in order to avoid the extra difficulties arising from the fact that KO m ( * ) ∼ = Z/2 for m = 1, 2. Theorem 5.1 (KO-cohomology of BΓ and BΓ). Let p be an odd prime and let m be any integer. (ii) There is a split exact sequence of abelian groups and KO 2m (BZ n ρ ) Z/p ∼ = l∈Z KO 2m-l ( * ) r l . (iii) Restricting to the subgroup Z n of Γ induces an isomorphism and KO 2m+1 (BZ n ρ ) Z/p ∼ = l∈Z KO 2m+1-l ( * ) r l . (iv) We have where T O 2m+1 is a finite abelian p-group for which there exists a filtration Its kernel is isomorphic to T O 2m+1 and is isomorphic to the cokernel of the map Lemma 5.2. Let p be an odd prime. In the Atiyah-Hirzebruch spectral sequence converging to K * (BΓ) after localizing at p E i,j i even, j ≡ 0 (mod 4); Z ri (p) ⊕ (Z/p) t ′ i i odd, i ≥ 3, j ≡ 0 (mod 4); 0 i = 1, j ≡ 0 (mod 4); 0 j ≡ 0 (mod 4), where 0 ≤ t ′ i ≤ p k -s i . Proof. Because of Theorem 1.7 (i) the E 2 -term of the spectral sequence converging to K * (BΓ) (p) is given after localization at p by i even, j ≡ 0 (mod 4); Z ri (p) ⊕ (Z/p) p k -si i odd, i ≥ 3, j ≡ 0 (mod 4); 0 i = 1, j ≡ 0 (mod 4); 0 j ≡ 0 (mod 4). The rest of the proof is analogous to the proof of Lemma 3.4. Lemma 5.3. Let p be an odd prime. For every m ∈ Z, there are isomorphisms of ] is a generalized cohomology theory with torsion-free coefficients, the Chern character and Lemma 3.6 give the first isomorphism. One proves that there are isomorphisms of abelian groups by induction on n using excision and the fact that BZ n = S 1 × BZ n-1 . It follows that the Atiyah-Hirzebruch spectral sequence E i,j 2 = H i (BZ n ; K j ( * )[1/p]) ⇒ KO i+j (BZ n )[1/p] collapses. This spectral sequence is natural with respect to automorphisms of Z n . Hence we obtain a descending filtration by Z It thus suffices to show that these exact sequences split over Z[1/p][Z/p] for all i. If m -i ≡ 3, 5, 6, 7 (mod 8), this follows from the fact that KO m-i ( * ) = 0. If m-i ≡ 0, 4 (mod 8), then K m-i ( * ) ∼ = Z and hence is projective. Finally, suppose m -i ≡ 1, 2 (mod 8). Since the Atiyah-Hirzebruch spectral sequence collapses, there is a homomorphism of abelian groups s : Then s is a homomorphism of Z[Z/p]-modules and π • s is multiplication by p and hence is the identity since K m-i ( * ) ∼ = Z/2. Lemma 5.4. Let p be an odd prime. (i) For every m ∈ Z, there is an isomorphism of abelian groups (ii) (iii) All differentials in the Leray-Serre spectral sequence associated to the extension (1.1) converging to KO * (BΓ) vanish. Proof. (i) It suffices to show the isomorphism exists after inverting 2 and after localizing at 2. Furthermore, if since localization is an exact functor. The assertion then follows from Lemma 5.3 and the definition of the numbers r l . The first isomorphism in assertion (ii) then follows since localization is an exact functor and the Tate cohomology groups are p-torsion. The second isomorphism follows from Lemma 1.10 (i). (iii) First note that the Leray-Serre spectral sequence converges with no lim 1 -term, see [32,Theorem 6.5]. It suffices to prove the differentials vanish after inverting p and after localizing at p. If we invert p, the claim follows from If we localize at p, the proof that the differentials vanish is identical to the proof of Lemma 3.5 (iii). Proof of Theorem 5.1. (iv) We first note that Proposition A.4 and Lemma 5.4 (i) imply that for all m ∈ Z, the kernel and cokernel of the composite The cellular pushout (1.13) yields for m ∈ Z a long exact sequence (5.7) 0 → KO 2m (BΓ) Define T O 2m+1 to be the kernel of the surjection f 2m+1 . Since f 2m+1 is an isomorphism after inverting p by (5.5) and assertion (iii), T O 2m+1 is p-torsion. We next claim f 2m+1 is split. We only need verify this after localizing at p in which case it follows since K 2m+1 (BΓ)⊗Z (p) is free over Z (p) by assertion (iii) and Lemma 5.4 (i). Finally, the stated filtration of T O 2m+1 is a consequence of Lemma 5.2. The completes the proof of assertion (v). Assertion (vi) is a consequence. (ii) The proof of this is identical to that of Theorem 3.3 (ii); the only missing part is to show the epimorphism is free over Z (p) . After inverting p, the splitting is provided by composing the inverse of the composite (5.5) with the map KO 2m (BΓ)[1/p] → KO 2m (BΓ)[1/p]. In this section we want to compute the real K-homology KO * of BΓ and BΓ. Rationally this can be done using the Chern character of Dold [16]: for every CWcomplex there is a natural isomorphism In particular we get from Theorem 2.1 (i) and (iv) We are interested in determining the integral structure. Theorem 6.3 (KO-homology of BΓ and BΓ). Let p be an odd prime and m be any integer. (i) and KO 2m (BZ n ρ ) Z/p ∼ = l∈Z KO 2m-l ( * ) r l . (iii) There is a split short exact sequence of abelian groups where T O 2m+5 is the finite abelian p-group appearing in Theorem 5.1 (v). (vi) The group T O 2m+5 is isomorphic to a subgroup of the kernel of (P )∈P KO 2m+1 (BP ) → KO 2m+1 (BΓ). Theorem 6.4 (Universal Coefficient Theorem for KO-theory). For any CWcomplex X there is a short exact sequence If X is a finite CW -complex, there is a short exact sequence Proof. A proof for the first short exact sequence can be found in [6] and [46, (3.1)], the second sequence follows then from [1, Note 9 and 15]. Proof of Theorem 6.3. (iv) (v) These assertions follow from Theorem 5.1 (iv) and (v) and the Universal Coefficient Theorem for KO-theory 6.4. (ii) There are natural transformations of cohomology theories i * : KO * → K * and r * : K * → KO * , induced by sending a real representation V to its complexification C⊗ R V and a complex representation to its restriction as a real representation. The composite r * • i * : KO * → KO * is multiplication by two. Since the map is bijective by Theorem 4.3 (ii), the map is bijective after inverting 2. In order to show that it is itself bijective, it remains to show that it is bijective after inverting p. This follows from Proposition A.4. Since we are dealing with KO-homology, the Atiyah-Hirzebruch spectral sequence converges also for the infinite-dimensional CW -complex BΓ. Because of the existence of Dold's Chern character, all its differentials vanish rationally. For m ∈ Z we have H 2m (BΓ) ∼ = Z r2m by Theorem 2.1. Hence we get for an odd prime p since KO m ( * ) (p) is Z (p) for m ≡ 0 (mod 4) and 0 otherwise holds after localizing at p. It remains to show that it holds after inverting p. This follows from Proposition A.4 and assertion (iv). (iii) The Atiyah-Hirzebruch spectral sequence shows that KO 2m (BZ/p) = 0 for all m ∈ Z. The methods of [44] together with the Universal Coefficient Theorem for KO-theory show that KO 2m+3 (BG) is the Pontryagin dual of KO 2m (BG) for any finite group G. Applying these facts to G = Z/p for an odd prime p, we see Thus the long exact KO-homology sequence associated to the cellular pushout (1.13) reduces to the exact sequence (6.5) 0 → KO 2m (BΓ) Note im ∂ 2m is a finite abelian p-group, since it is a finitely generated subgroup of the p-torsion group Thus im ϕ 2m-1 ∼ = (Z/p ∞ ) (p-1)p k /2 (compare with the proof of Theorem 3.3 (iii)). It remains to see that f 2m-1 splits, which we verify at p and away from p. The target of f 2m-1 is free after localizing at p by assertion (v), so it splits. After inverting p, the exact sequence 6.5 shows that f 2m-1 [1/p] is an isomorphism. (i) This follows from assertions (ii), (iii) and (v). (vi) This follows from assertions (ii) and (iv) and the long exact sequence (6.5). This finishes the proof of Theorem 6.3. In the sequel an equivariant cohomology theory is to be understood in the sense of [29,Section 1]. Equivariant topological complex K-theory K * ? is an example as shown in [29,Example 1.6] based on [32]. This applies also to equivariant topological real K-theory KO * ? . Rationally one obtains , from [32, Theorem 5.5 and Lemma 5.6] using Theorem 1.7 (iv) and Lemma 1.9. We want to get an integral computation. Recall that we have computed l∈Z r 2l and l∈Z r 2l+1 in Lemma 1.22 (ii). (i) (ii) There is an exact sequence where T 1 is the finite abelian p-group appearing in Theorem 3.3 (v). (iii) The canonical maps In the sequel we will often use the following lemma. Lemma 7.2. ? be an equivariant cohomology theory in the sense of [29, Section 1]. Then there is a long exact sequence where H m P ( * ) is the cokernel of the induction map ind P →1 : H m ( * ) → H m P ( * ) and the map ϕ m is induced by the various inclusions P → Γ. The map * be an equivariant homology theory in the sense of [27, Section 1]. Then there is a long exact sequence where H P m ( * ) is the kernel of the induction map ind P →1 : H P m ( * ) → H m ( * ) and the map ϕ m is induced by the various inclusions P → Γ. The map is split surjective. Proof. (i) From the cellular Γ-pushout (1.12) we obtain a long exact sequence From the cellular pushout (1.13) we obtain the long exact sequence Induction with the group homomorphism Γ → 1 yields a map from the long exact sequence (7.4) to the long exact sequence (7.3). Recall that the induction homomorphism H m (Γ\X) → H m Γ (X) is an isomorphism if Γ acts freely on the proper Γ-CW -complex X. Therefore the maps are bijective. Hence one can splice the long exact sequences (7.3) and (7.4) together to obtain the desired long exact sequence, after noting the commutative diagram We have the following commutative diagram, where the vertical arrow are given by induction with the group homomorphism Γ → 1 The upper horizontal arrow is split injective after p by Proposition A.4. The right vertical arrow is bijective since Γ acts freely on Γ × Z n EZ n . Hence H m (BΓ) → H m Γ (EΓ) is injective after inverting p. (ii) The proof is analogous to the one of assertion (i). This finishes the proof of Lemma 7.2. Proof of Theorem 7.1. Recall that K 0 Γ (Γ/P ) ∼ = R C (P ) and K 1 Γ (Γ/P ) ∼ = 0. Hence we obtain from Lemma 7.2 (i) the long exact sequence where R C (P ) is the cokernel of the homomorphism R C (1) → R C (P ) given by restriction with P → 1. Notice that the composite of the augmentation ideal I C (P ) → R C (P ) with the projection R C (P ) → R C (P ) is an isomorphism of finitely generated free abelian groups and that I C (P ) is isomorphic to Z p-1 . (iii) We have already shown in Theorem 3.3 (iii) that the map K 1 (BΓ) is bijective and that K 1 (BΓ) ∼ = Z l∈Z r 2l+1 . Hence it remains to prove that the composite is bijective. We obtain from (3.8) and (7.5) the following commutative diagram with exact rows By the five lemma it suffices to show that the map BΓ) is surjective. We conclude from Theorem 3.3 (vi) that the kernel of K 1 (BΓ) → K 1 (BΓ) is the finite abelian p-group T 1 appearing in Theorem 3.3 (v). Hence it remains to show for every integer l > 0 that the obvious composite is surjective. By the Atiyah-Segal Completion Theorem the map R C (P ) → K 0 (BP ) can be identified with the map id ⊕i : Hence it suffices to show that the composite is surjective. This is true since the latter map can be identified with the canonical epimorphism Z → Z/p l . (ii) This follows from Theorem 3.3 (vi), the long exact sequence (7.5), the isomorphism (7.6) and assertion (iii). (i) We have shown K 0 (BΓ) ∼ = Z l∈Z r 2l in Theorem 3.3 (iv). We have I(Z/p) ∼ = Z (p-1)/2 . The order of P is p k by Lemma 1.9 (iv). Hence we conclude from assertion (ii) Γ (EΓ) follows from Theorem 3.3 (iii) and assertion (iii). Remark 7.7 (Geometric interpretation of T 1 ). The exact sequence appearing in Theorem 7.1 (ii) has the following interpretation in terms of equivariant vector bundles. Since Γ is a crystallographic group, Γ acts properly on R n such that this action reduced to Z n is the free standard action and R n is a model for EΓ. Hence the quotient of Z n \R n is the standard n-torus T n together with a Z/p-action. There is a bijection -→ (T n ) Z/p coming from the fact that (R n ) P consists of exactly one point for (P ) ∈ P. In particular (T n ) Z/p consists of p k points (see Lemma 1.9 (v).) Hence for any complex Z/p-vector bundle ξ we get a collection of complex Z/p-representations sending the class of a Z/p-vector bundle ξ to the collection ) be the homomorphism coming from the pullback construction associated to the projection T n → (Z/p)\T n . We obtain the exact sequence which can be identified with exact sequence of Theorem 7.1 (ii). Thus the group T 1 is related to (stable version of) the question when a collection of Z/p-representations can be realized as the fibers of a Z/p-vector bundle ξ over T n at the points in (T n ) Z/p . Moreover, a Z/p-vector bundle over T n is stably isomorphic to the pullback of a vector bundle over (Z/p)\T n if and only if for every x ∈ (T n ) Z/p the Z/prepresentation ξ x has trivial Z/p-action. In the sequel equivariant homology theory is to be understood in the sense of [27,Section 1]. Equivariant topological complex K-homology K ? * is an example (see [14], [33,Section 6]). The construction there yields the same for proper G-CW -complexes as the construction due to Kasparov [21]. It is two-periodic. For finite groups G the group K G m ( * ) is R C (G) for even m and trivial for odd m. We obtain from [28, Theorem 0.7] using Lemma 1.9 an isomorphism and hence from Theorem 4.3 We want to get an integral computation. Theorem 8.3 (Equivariant K-homology of EΓ). (i) We have (ii) There is a natural isomorphism where R C (P ) is the kernel of the map R C It splits after inverting p. Its proof needs some preparation. Lemma 8.4. Let G be a finite group. Then there is an isomorphism of In particular we get for any [35, 2.5 and 2.10]. Theorem 8.5 (Universal coefficient theorem for equivariant K-theory). Let G be a finite group and X be a finite G-CW -complex. Then there are for n ∈ Z natural exact sequences of R C (G)-modules Proof. The first sequence is proved in [10]. The second sequence follows from the first by equivariant S-duality (see [35], [45]). Proof of Theorem 8.3. (ii) Since Z n acts freely on EΓ, induction with Γ → Z/p induces isomorphisms -→ K n Γ (EΓ). Since Z n \EΓ is a finite Z/p-CW -complex, we obtain from Lemma 8.4 and Theorem 8.5 the exact sequence of R C (Z/p)-modules (Another construction of the sequence above will be given in [20].) Hence we get an exact sequence of R C (Z/p)-modules (see also [35,Proposition 2.8]) is a finitely generated free abelian group for all n ∈ Z by Theorem 7.1, we obtain for n ∈ Z an isomorphism of R C (Z/p)-modules ) Apply Theorem 7.1 (i) and assertion (ii) to get the concrete identification of K Γ n (EΓ). (iii) From Lemma 7.2 (ii) we obtain a long exact sequence Recall that equivariant topological real KO-theory KO * ? is an equivariant cohomology theory in the sense of [29,Section 1]. It is 8-periodic. Recall also that equivariant topological real K-homology KO ? * is an equivariant homology theory in the sense of [27,Section 1]. It is 8-periodic. We first give some information about KO G m ( * ) and KO m G ( * ) for finite G. We have where K m (RG) is the topological K-theory of the real group C * -algebra RG. Let {V i | i = 0, 1, 2, . . . , r} be a complete set of representatives for the RG-isomorphism classes of irreducible real G-representations. By Schur's Lemma the endomorphism ring D i = end RG (V i ) is a skewfield over R and hence isomorphic to R, C or H. There are positive integers k i for i ∈ {0, 1, . . . , r} such that we obtain a splitting Since topological K-theory is compatible with products, by Morita equivalence we obtain for m ∈ Z an isomorphism We conclude from [29, Theorem 5.2] using Lemma 1.9 (i) for m ∈ Z Again we seek an integral computation. Theorem 9.7 (Equivariant KO-cohomology). Let p be an odd prime and let m be any integer. For m ∈ Z the composite is surjective and has a finite abelian p-group as kernel by Theorem 5.1 (vi). Hence the map β is surjective for all m ∈ Z. Since α is surjective by (9.8), the map ker (β • α) → ker (β) is surjective and hence the kernel of β is a finite abelian p-group. The following diagram commutes where the left horizontal maps are given by induction with R → C, the right horizontal maps by restriction with R → C and the middle vertical arrow is an isomorphism by Theorem 7.1. Hence the kernel of the epimorphism KO 2m+1 Γ (EΓ) → KO 2m+1 (BΓ) is an abelian group of exponent 2. We have already shown that its kernel is a finite abelian p-group. Since p is odd, we conclude that The bijectivity of KO 2m+1 (BΓ) -→ KO 2m+1 (BZ n ρ ) Z/p has already been proved in Theorem 5.1 (iii). (i) Since kernel of the epimorphism KO 2m+1 (BΓ) → KO 2m+1 Γ (EΓ) is a finite abelian p-group and (P )∈P KO 2m Z/p ( * ) is isomorphic to Z p k (p-1)/2 by Lemma 1.9 (iv) and by (9.6), we conclude from the exact sequence (9.8) that . Since we have already computed KO 2m (BΓ) and KO 2m+1 (BΓ) in Theorem 5.1, assertion (i) follows using assertion (iii). We obtain from [28, Theorem 0.7] using Lemma 1.9 isomorphisms We want to get an integral computation. Theorem 10.1 (Equivariant KO-homology). Let p be an odd prime and m be any integer. (i) which splits after inverting p. We have proven assertions (ii) and (iii). Since the composite EΓ) is multiplication with 2 and K Γ i (EΓ) is a finitely generated free abelian group by Theorem 8.3, the torsion subgroup of the finitely generated abelian group KO Γ i (EΓ) is annihilated by 2 for i ∈ Z. Since by Theorem 6.3 (iv) for a finite abelian p-group T O 2m+5 and the torsion in n i=0 KO m-i ( * ) ri is annihilated by multiplication with 2, we get from (10.3) an isomorphism of abelian groups This is the even case of assertion (i). The odd case of assertion (i) follows from assertion (ii) and Theorem 6.3 (v). Proposition A.4), the fact that norm map is always bijective after inverting p, and the the isomorphism (11.2) for Z n , the claim holds. 12. The group Γ satisfies the (unstable) Gromov-Lawson-Rosenberg Conjecture In this section we give the proof of Theorem 0.7, after first giving some background. 12.1. The Gromov-Lawson-Rosenberg Conjecture. For a closed, spin manifold M of dimension m with fundamental group G, one can define an invariant which vanishes if M admits a metric of positive scalar curvature (see [37]). The (unstable) Gromov-Lawson-Rosenberg Conjecture for a group G states that if α(M ) = 0 and dim M ≥ 5, then M admits a metric of positive scalar curvature. The (unstable) Gromov-Lawson-Rosenberg Conjecture is known to be valid for some fundamental groups, for example, the trivial group (see [41]), for finite groups with periodic cohomology (see [11] and [23]), some torsion-free infinite groups, for example, when G is fundamental group of a complete Riemannian manifold of non-positive sectional curvature (see [37]), and some infinite groups with torsion, for example, cocompact Fuchsian groups (see [15]), but not in general -there is a counterexample when G = Z 4 × Z/3 due to Schick [39]. There is a weaker version of the conjecture which may be valid for all groups. Let B 8 be a "Bott manifold," a simply-connected spin 8-manifold with A-genus equal to one. We say that a manifold M stably admits a metric of positive scalar curvature if M × (B 8 ) j admits a metric of positive scalar curvature for some j ≥ 0. The stable Gromov-Lawson-Rosenberg Conjecture formulated by Rosenberg-Stolz [38] states that for a closed spin manifold M with fundamental group G, then M stably admits a metric of positive scalar curvature if and only if α(M ) = 0. Since the Baum-Connes Conjecture implies the stable Gromov-Lawson-Rosenberg Conjecture (see [42,Theorem 3.10] for an outline of the proof) and Γ satisfies the Baum-Connes Conjecture, we know already that Γ satisfies the stable Gromov-Lawson-Rosenberg Conjecture. There are two definitions of the invariant α, one topological and one analytic. Let KO be the periodic spectrum underlying real K-theory, and let p : ko → KO be the 0-connective cover, that is, it induces an isomorphism on π i for i ≥ 0 and π i (ko) = 0 for i negative. Then the topological definition of α(M ) is the image of the class [f M : M → BG] where f M induces the identity on the fundamental group under the composite where D is the ko-orientation of spin bordism, p BG is the canonical map from connective to the periodic K-theory, and A is the assembly map. The analytic definition of α(M ) is the index of the Dirac operator. These two definitions agree (see [37]). Furthermore if M has positive scalar curvature, then the Bochner-Lichnerowicz-Weitzenböck formula shows that the index is zero so that α(M ) = 0. Finally, we mention one more result in our quick review, and that is the generalization of the Gromov-Lawson surgery theorem of due to Jung and Stolz [38, 3.7 Proof. If M is a Z[Z/p]-module, then H i (Z/p; M )[1/p] = 0 for i ≥ 1 and hence the canonical maps are bijective for i ≥ 1. We conclude from the Atiyah-Hirzebruch spectral sequences that the vertical maps in the commutative diagram are bijective for m ≥ 0. Hence it suffices to prove the surjectivity of the lower horizontal map. Since p is odd, Ω Spin j ( * ) (p) is zero for j ≡ 0 (mod 4) and Ω Spin j ( * ) (p) is a finitely generated free Z (p) -module for j ≡ 0 (mod 4) (see [7]). The same is true for ko j ( * ) (p) by Bott periodicity. Hence there are no differentials in Atiyah-Hirzebruch spectral sequences converging to Ω Spin i+j (BZ/p) (p) and ko i+j (BZ/p) (p) and we get for the E ∞ -terms It suffices to show that the map on the E ∞ -terms is surjective for all i, j. Hence it is enough to show that the map is surjective for all j. Since ko * ( * ) (p) is a polynomial algebra on a single generator in dimension 4, it suffices to prove D (p) is onto when j = 4. In this case both Ω Spin This fact, the Universal Coefficient Theorem, Lemma A.1, and Lemma 1.10 (i) imply H i+1 (Z/p; ko j (BZ n ρ )) = 0 when i + j is even. Thus E 2 0,2m = ko 2m (BZ n ρ ) Z/p maps injectively to ko 2m (BZ n ρ ) Z/p and hence is p-torsion-free, and for i > 0, E 2 i,j has exponent p and vanishes if i + j is even. Thus Lemma A.1 (Tate duality). Let G be a finite group and M be a finitely generated ZG-module which contains no p-torsion for all primes p dividing the order of G. Then for all integers i there is an isomorphism of abelian groups Hence for all integers i > 0, Proof. For ordinary (co)homology theory, π * • trf * and trf * • π * are both multiplication by q = |G|. This has the consequence that π * and π * are isomorphisms after inverting q. These last composite formulae are no longer true for generalized (co)homology theories, but one can say something. A generalized homology theory is 1/q-local if H * (X) ⊗ Z → H * (X) ⊗ Z[1/q] is an isomorphism for all X and m. For example, for any generalized homology theory, H * (X) ⊗ Z[1/q] is a 1/q-local generalized homology theory. There is an analogous definition and remark for generalized cohomology theories. Proposition A.4. Let G be a finite group of order q. Let H * and H * be 1/q-local (co)homology theories. Let X be a G-CW -complex and π : X → X the quotient map. (i) π m : H m (X) G ∼ = -→ H m (X) is an isomorphism for all m ∈ Z. (ii) If X is a finite CW -complex, then π m : H m (X) ∼ = -→ H m (X) G is an isomorphism for all m ∈ Z. Proof. We give the argument only for homology, the one for cohomology is analogous. Given a G-CW -complex X, we obtain a natural map Since the functor sending a Z[1/q][G]-module M to M G is an exact functor, the assignment sending a G-CW -complex X to H * (X) G and to H * (G\X) are Ghomology theories and j * is a natural transformation of G-homology theories. One easily checks that j * is a bijection when X is G/H for any subgroup H ⊂ G. A Mayer-Vietoris argument implies that j * is a bijection for any finite G-CWcomplex, and, since homology commutes with colimits, j * is a bijection for any G-CW -complex. Atiyah's computation of K 0 (BZ/p) shows that a finiteness hypothesis is necessary for a generalized cohomology theory. At several places in this paper we use a property of edge homomorphisms in spectral sequences and we review this now. Let H * and H * be (co)homology theories. Let F → E → B be a fibration. Assume that B is path-connected with fundamental group G. There are Leray-Serre spectral sequences E 2 i,j = H i (B; H j (F )) ⇒ H i+j (E) E i,j 2 = H i (B; H j (F )) ⇒ H i+j (E). These spectral sequences have coefficients twisted by the action of G on the (co)homology of the fiber, in particular The spectral sequences give maps the composites are called the edge homomorphisms. The proof of the proposition below follows the proof in the untwisted case [43, page 354]. i (EΓ)[1/p] → KO i (BΓ)[1/p] is split surjective for i ∈ Z.