On the Structure of Finite Groups with Periodic Cohomology

On the structure of ﬁnite groups with periodic cohomology

C. T. C. Wall December 2, 2010

Introduction

It was shown in 1944 by Smith [36] that a non-cyclic group of order p2 (p prime) cannot act freely on a sphere. Hence if the group G does so act, every subgroup of G of order p2 is cyclic. Equivalent conditions on G are that every abelian subgroup of G is cyclic, and that every Sylow subgroup of G is cyclic or quaternionic: the proof of equivalence is not difficult, see e.g. Wolf [47, 5.3.2]. Another equivalent condition, due to Tate (see Cartan and Eilenberg [10, XII,11.6]) is that the cohomology of G is periodic. We shall call groups G satisfying these conditions P−groups. We will say that a finitely dominated CW complex X with fundamental group G and universal cover homotopy equivalent to SN−1 is a (G, N)-space. In this situation, G is still a P−group. Homotopy types of oriented (G, N)-spaces X correspond N ˜ to cohomology generators g ∈ H (G; Z). There is an obstruction o(g) ∈ K0(ZG) to existence of a finite CW-complex homotopy equivalent to X. Such an X which is a manifold we will call a (topological or smooth) space-form. A representation ρ : G → UN of G is said to be an F−representation if the 2N−1 N induced action of G on the unit sphere S ⊂ C is free. The quotient Xρ := S2N−1/G is thus an example of a smooth space-form. In previous work on the topological spherical space-form problem [41], [29], we cited [47] for the list of P−groups. However, in [45] the need was felt for a sharper account of the classification. It is the object of this paper to provide this. Since this involves some reworking, we go back as far as possible to direct arguments. The group theory of P−groups was considered by Zassenhaus [48], who elucidated their structure in the soluble case. The results in the insoluble case depend on a key paper of Suzuki [37]; the complete classification was described (without full details) in Wolf [47]. The P−group condition is not quite inherited by quotient groups: if we enlarge the class to P0−groups by also permitting the Sylow 2-subgroup to be dihedral, it becomes so. We start with notation and preliminaries. In §2, we quote Suzuki’s result, which 0 implies that the only non-abelian simple P −groups are the PSL2(p)(p > 3 prime), and show how it follows that any P0−group G modulo its (odd) core is either a 2- group or isomorphic to some SL2(p) or TL2(p)(p ≥ 3 prime), where TL2(p) is defined below. This yields the list of types I-VI given by Wolf [47]. In §3 we give further details for P−groups, with explicit presentations in each case. The use of induction theorems (following Swan [38]) showed the need for under- standing hyperelementary subgroups of the given group. We finesse the problem of listing all such subgroups by listing in §4 subgroups which are maximal subject to having type I or II: since such groups have a non-trivial cyclic normal subgroup, this can be achieved by studying normalisers of cyclic subgroups. Those type II cases when there is not a cyclic group of index 2 in G present a more complicated representation theory, and cause certain difficulties in the study

1 of free group actions. In §4, we also analyse the general P−group G according to the existence or not of such subgroups: this refines types II, IV and VI into rtypes. A first use of our list is to calculate the period and the Artin exponent. In §5 we re-state, with a sketch of proof, Wolf’s classification of free orthogonal actions. The rtypes help clarify the case distinctions in [47, Theorem 7.2.18]. We also calculate the homotopy type of each quotient space. The finiteness obstruction is discussed in §6: most of this follows [46], but I have added references to subsequent results. The application of these results to the problem of existence of space-forms is given in §7: following the earlier papers [41], [29], [45] we are able to determine in all except certain type II cases the dimensions of the spheres on which G can act freely. Most of this is now 30 years old: we also include a survey of later results, which are mostly due to Hambleton, Madsen and Milgram. In §8 we give a corresponding discussion of classification of space-forms. I am indebted to Ian Hambleton and to Ib Madsen for helpful comments on these sections. A version of the first 4 sections of this paper, written jointly with my late friend Charles Thomas, appeared as a preprint 30 years ago. As was kindly pointed out to me, it was badly presented (my fault, not Charles’), with several minor errors; it was also interdependent with other papers. Here I have taken the opportunity to polish the original (also, the list of maximal type I and II subgroups is new, and leads to simpler proofs of several results), to include related material to make a coherent narrative, and to include the survey in the two final sections. In the course of our earlier work on this topic, we referred to Wolf’s book [47] for the necessary group theory: indeed, that book was a constant companion. It is thus a pleasure to dedicate this paper to Joe Wolf on the occasion of his 75th birthday.

1 Notation and preliminaries

We will use the following notation for (isomorphism classes of) groups: × Fn denotes the ring Z/nZ for any n ∈ N, and Fn its multiplicative group of invertible elements. n Cn: cyclic of order n: hx | x = 1i. n 2 −1 −1 D2n: dihedral of order 2n: hx, y | x = y = 1, y xy = x i. n 2 4 −1 −1 Q4n: quaternionic of order 4n: hx, y | x = y , y = 1, y xy = x i. Groups Q(2k`, m, n) are deﬁned below. GL2(p): group of invertible 2 × 2 matrices over the Galois ﬁeld Fp; throughout this paper, p will be an odd prime. SL2(p): matrices in GL2(p) with determinant 1. P GL2(p): quotient of GL2(p) by its centre (the group of scalar matrices). PSL2(p): the image of SL2(p) in P GL2(p). TL2(p): let ζ be the outer automorphism of SL2(p) induced by conjugation by ω 0 w := , where ω generates ×. Then 0 1 Fp

ω 0 TL (p) = SL (p), z | z−1gz = gζ for all g ∈ SL (p), z2 = . (1) 2 2 2 0 ω−1

The four latter groups are all P0−groups and form a diagram

SL2(p) −→ TL2(p) ↓ ↓ , (2) PSL2(p) −→ P GL2(p)

2 where the horizontal arrows are inclusions of subgroups of index 2 and the verticals epimorphisms with kernel of order 2 (the second vertical arrow is deﬁned by sending z to the class of w). ∗ ∗ ∗ We write T = SL2(3), O = TL2(3) and I = SL2(5) for the binary tetrahe- ∗ ∗ dral, octahedral and icosahedral groups; variants Tv and Ov are deﬁned below. We will refer to a group C with normal subgroup A and quotient C/A ≡ B as an extension of A by B. We write Aut(G) for the group of automorphisms of a group G, Inn(G) for the (normal) subgroup of inner automorphisms, Out(G) for the quotient group. Write Z(G) for the centre of G. The maximal normal subgroup of odd order of G will be termed the core of G, and denoted O(G). We denote the order of a group G by |G|, and write Gp for a Sylow p−subgroup of G. A group is said to be p−hyperelementary if it is an extension of a cyclic group of order prime to p by a p−group. r For p prime, write νp(n) for the largest integer r with p | n. × For r, n ∈ N, write ordn(r) for the order of r in Fn .

The classification of group extensions by the method of Eilenberg & MacLane [26, IV,8] proceeds as follows. Any extension C of A by B determines a homomorphism h : B → Out(A). There is a natural restriction map r : Out(A) → Aut(Z(A)): regard Z(A) as B−module via r ◦ h. There exist extensions corresponding to (A, B, h) iff a certain obstruction in H3(B; Z(A)) vanishes; these are then classified by H2(B; Z(A)). If A is abelian, then Z(A) = A and the zero element of H2(B; A) corresponds to the split extension of A by B. For a split extension, the set of splittings up to conjugacy can be identified with H1(B; A). Recall that if B and A have coprime orders, all groups Hr(B; A) vanish. We apply this method to study extensions with one of the groups cyclic and the other one in (2). In each case, we need to know the centre Z(A) and the outer automorphism group Out(A). We see easily that the centre of each of SL2(p) and TL2(p) is the group {±I} of order 2; the groups PSL2(p) and P GL2(p) have trivial centre.

Lemma 1.1 The groups Out(PSL2(p)) and Out(SL2(p)) have order 2, all automorphisms being induced by P GL2(p), resp. GL2(p).

1 1 0 1 Proof Write u := and t := , both lying in SL (p). Every 0 1 −1 0 2 element of order p in GL2(p) is conjugate to u, but in SL2(p) these fall into two conjugacy classes. Let α be an automorphism of SL2(p) leaving u fixed. Then α preserves the normaliser of U := hui, the group of upper triangular matrices, which is a semi-direct product of U and the group SD of diagonal matrices in SL2(p). Since the orders of these two are coprime, the splitting is unique up to conjugacy, so adjusting α by an inner automorphism, we may suppose that α preserves SD. Hence it also preserves the normaliser of SD, which is an extension of D by hti. 0 a Write α(t) = . Since this lies in SL (p), ab = 1. Since the relation b 0 2 (tu)3 = I is preserved under α, we find a = 1 and b = −1, thus α(t) = t. Since SL2(p) is generated by t, u and SD, if α fixes SD it is the identity. Now α induces an automorphism α of PSL2(p). This fixes the images t, u of t, u and respects the image SD of SD: since SD acts faithfully on U, α fixes SD pointwise. Hence α is the identity. Thus for each g ∈ SL2(p) we have α(g) = (g)g, with (g) = ±I. Since α is a homomorphism and ±I is central, is a homomorphism, hence is trivial.

3 This proves the result for SL2(p); that for PSL2(p) follows easily.

Proposition 1.2 (i) Any extension of PSL2(p), SL2(p), P GL2(p) or TL2(p) by a group of odd order splits as a direct product. 0 (ii) No extension of P GL2(p) or TL2(p) by C2 is a P −group. 0 (iii) Any extension of PSL2(p) resp. SL2(p) by C2 which is a P −group is isomorphic to P GL2(p) resp. TL2(p).

Proof (i) In this case, h is trivial and the cohomology groups vanish (in this case, the orders of Z(A) and B are coprime): the result follows. (ii) For P GL2(p), there are no outer automorphisms, so h is trivial. The induced extension on a Sylow subgroup now cannot be dihedral. And if TL2(p) had an appropriate extension, we could factor out the centre and ﬁnd one for P GL2(p). (iii) There are just two isomorphism classes of extensions of PSL2(p) by C2, according as h : C2 → Out(PSL2(p)) is trivial or not, so any extension is either trivial or isomorphic to P GL2(p). There are four classes of extensions of SL2(p) by C2: h : C2 → Out(SL2(p)) may 2 be trivial or not; in either case, C2 acts trivially on the centre C2 and H (C2; C2) has order 2. When h is trivial, the corresponding extensions have Sylow subgroup containing a direct product of two cyclic groups, so are not P0−groups. For the others, the outer automorphism of SL2(p) was denoted ζ above, and 2 we must adjoin to SL2(p) an element z which induces ζ and has z equal to an ω 0 element of SL (p) inducing ζ2, hence to ± . Taking the + sign gives 2 0 ω−1 TL2(p), with quaternionic Sylow 2-subgroups; the minus sign gives a group whose Sylow 2-subgroups are semi-dihedral. We turn now to extensions of cyclic groups by the groups of (2). Since the automorphism group of a cyclic group is abelian, h factors through the commutator quotient group, which is trivial for PSL2(p) and SL2(p), except if p = 3, when it has order 3, and has order 2 for P GL2(p) and TL2(p).

Proposition 1.3 (i) Any extension by PSL2(p) (p 6= 3), SL2(p) (p 6= 3), P GL2(p) or TL2(p) of a cyclic group of odd order is split. 0 (ii) An extension of C2 by PSL2(p) or P GL2(p) which is a P −group is isomorphic to SL2(p) or TL2(p) respectively. 0 (iii) No extension of C2 by SL2(p) or TL2(p) is a P −group.

2 Proof (i) We must show that H (G; C`) vanishes for G one of the above and ` odd; we may suppose ` prime. The result is clear if ` is prime to |G|. Otherwise, 2 ∼ ∼ H (G`; C`) = Hom(G`; C`) = C`, and the normaliser of G` in SL2(p) acts non- 2 trivially on G`, so has no invariants in H (G`; C`). 2 (ii) Since Out(C2) is trivial, extensions of C2 by G are classiﬁed by H (G; C2). 2 If G has Sylow 2-subgroup G2, this maps injectively to H (G2; C2), so the extension of G2 is enough to determine that of G. First suppose G = PSL2(p). Then G2 is dihedral, so has a presentation of the form k hx, y | x2 = y2 = (xy)2 = 1i. An extension of C2 by G2 is thus of the form k hx, y, z | z2 = 1, xz = zx, yz = zy, x2 = za, y2 = zb, (xy)2 = zci, where the parameters a, b, c ∈ {0, 1} determine the diﬀerent extensions. But the 2k−1 involutions x , y, xy in PSL2(p) are all conjugate, so the induced extensions of C2 by them are all isomorphic. Hence a = b = c. If a = 0, we have the trivial extension, which is not a P0−group. Thus a = 1, there is a unique extension, and this must be SL2(p).

4 An extension of C2 by P GL2(p) has a subgroup of index 2 which is an extension 0 by PSL2(p). If it is a P −group, so is this subgroup, which must be isomorphic to SL2(p) by the above. By (iii) of Proposition 1.2, our group is isomorphic to TL2(p). This proves (ii). For (iii) we note that any extension of C2 by SL2(p) is also an extension by 0 PSL2(p) of a group T of order 4. If it is a P −group, by Lemma 2.4, T is cyclic. Now (as above) a Sylow 2-subgroup of G has the form k hx, y, z | z4 = 1, xz = zx, yz = zy, x2 = za, y2 = zb, (xy)2 = zci, on noting that T must be central since any map from PSL2(p) to Aut(T ) is trivial. Since the quotient by hz2i is as in (ii) above, a, b and c are odd. But k k k k k k za = y−1zay = y−1x2 y = (y−1xy)2 = (z−bx−1zc)2 = z2 (c−b)x−2 = z2 (c−b)−a, so 2k(c − b) − 2a is divisible by 4, the order of z. As k ≥ 1 and a, b and c are odd, we have a contradiction. Any extension of C2 by TL2(p) has a subgroup of index 2 which is an extension 0 of C2 by SL2(p): since the latter cannot be a P −group, nor can the former.

2 Structure of P0−groups

Theorem 2.1 Any group whose Sylow subgroups are all cyclic is soluble. This result was proved by Burnside [9] over a century ago. A modern account of the proof is given in [47, 5.4.3]. The key result for us is the following, due to Suzuki [37]: Theorem 2.2 Let G be a simple group whose odd Sylow subgroups are cyclic and whose even ones are (i) dihedral or (ii) quaternionic. Then in case (i), G is isomorphic to PSL2(p) for some odd prime p > 3. Case (ii) cannot occur. The proof of (i) is contained in [37, §§4,5]. No simpler proof is known, though the Brauer-Suzuki-Wall theorem [5], which is an essential tool, has been considerably simplified by Bender [1]. The result is, of course, contained in the Gorenstein-Walter classification [16] of groups with dihedral 2-subgroup, but this – even in the much reduced version of Bender [2] – can hardly be considered a simplification. The proof of (ii) is given in §6 of Suzuki’s paper. It also follows at once from (i) and the following. Theorem 2.3 Let G be a finite group with a quaternionic Sylow 2-subgroup S. Then G has a normal subgroup Z such that |Z| is twice an odd number. The proof is given in the case |S| > 8 in [15, §12] and in case |S| = 8 in [14]. These arguments do not use block theory, so are technically simpler than Suzuki’s original proof. For the above critique I am indebted to George Glauberman.

Suzuki himself deduced that any non-solvable P0−group contains a normal subgroup G1 = Z×L of index ≤ 2, where Z is a (solvable) group whose Sylow subgroups are all cyclic, and L is isomorphic to P GL2(p) or SL2(p) for some p ≥ 5. To obtain a more precise result, we follow the method of Zassenhaus [48], and analyse the structure of a general P0−group by induction. First observe Lemma 2.4 Suppose G a P0−group, N normal in G. Then either N or G/N has cyclic Sylow 2−subgroups. Proof A Sylow 2-subgroup T of N lies in some Sylow 2-subgroup S of G. Then T = S ∩ N is normal in S, and S/T =∼ NS/N is a Sylow 2-subgroup of G/N. If C is cyclic of index 2 in S, either T ⊆ C is cyclic or T.C = S, so S/T = C/(C ∩ T ) is cyclic.

5 Theorem 2.5 If G is a P0−group, then G/O(G) is either a 2-group or isomorphic to PSL2(p), SL2(p), P GL2(p) or TL2(p) for some odd prime p.

Proof By induction, we may suppose that G has a normal subgroup H satisfying the conclusion and such that G/H is a simple P0−group. Since H is normal in G and O(H) is characteristic in H, O(H) is normal in G. Factoring out O(H), we may now suppose O(H) = 1. First consider the case when H is isomorphic to PSL2(p), SL2(p), P GL2(p) or TL2(p). By Lemma 2.4, G/H has cyclic Sylow 2-subgroup; as it is simple, by Theorem 2.1 it is cyclic of prime order. If this order is odd, then by Proposition 1.2(i), the extension is trivial: G = H × ∼ ∼ C, C = O(G) and G/O(G) = H. If it is even, G/H = C2. By Proposition 1.2(ii), ∼ ∼ if H = PSL2(p) or SL2(p), then G = P GL2(p) or TL2(p). By Proposition 1.2(iii), ∼ we cannot have H = P GL2(p) or TL2(p) in this case. In the remaining case, H is a 2-group, and G/H is simple. According to Theo- ∼ rem 2.2, either G/H is cyclic of prime order or G/H = PSL2(p) with p ≥ 5. In the latter case, by Lemma 2.4, H has cyclic Sylow 2-subgroup, so is cyclic. If |H| = 2, ∼ it follows from Proposition 1.3(ii) that G = SL2(p). If |H| = 4, it follows from the same result that G is an extension of C2 by SL2(p): by Proposition 1.3(iii), this cannot occur. The case |H| > 4 is likewise excluded by considering the quotient of G by a subgroup of index 4 in H. Otherwise, H is a 2-group and G/H is cyclic of prime order p. If p = 2, G is a 2-group. If p is odd, the extension is split, hence trivial (so G/O(G) =∼ H) except when H has an automorphism of odd order, which occurs only when H is the four group or quaternion of order 8. Here G/H must have order 3, and the corresponding G is isomorphic to PSL2(3) resp. SL2(3). Corollary 2.6 For any P−group G, G/O(G) is either a 2-group or isomorphic to SL2(p) or TL2(p) for some odd prime p.

For as O(G) has odd order, the Sylow 2-subgroups of G/O(G) are isomorphic to those of G, hence are cyclic or quaternionic. Following Wolf [47], we deﬁne the type of G in terms of G/O(G) as follows

G/O(G) C k Q k SL (3) TL (3) SL (p) TL (p) 2 2 2 2 2 2 , T ype IIIIIIIVVVI where V , VI are the non-soluble cases p > 3.

3 Presentations of P−groups

The group G has type I iﬀ all its Sylow subgroups are cyclic. The structure of such groups was elucidated by Burnside [8]. A careful treatment is given by Wolf [47, 5.4.1] following Burnside and Zassenhaus [48]. Lemma 3.1 [47, 5.4.5] If G has cyclic Sylow subgroups, its commutator subgroup G0 is abelian, and its order is prime to its index. It follows that there is a presentation

G = hu, v | um = vn = 1, v−1uv = uri. (3)

For consistency we need n prime to m and rn ≡ 1 (mod m). Since G0 = hui,(r − 1) is prime to m. Set d := ordm(r): thus d | n.

6 a b c Any automorphism of G must be given by ψa,b,c(u) = u and ψa,b,c(v) = v u for some a, b, c. For consistency we need (a, m) = 1, (b, n) = 1 and b ≡ 1 (mod d); conversely [47, 5.5.6(ii)] these conditions suﬃce. Since no power of u commutes with v, the map from G0 to Inn(G) is injective; its image consists of the ψ1,1,c, while inner automorphism by v is ψr,1,0. Decompose the cyclic group hvi of order n as a product of groups hv1i, hv2i of coprime orders n1, n2, where each prime factor of n1 divides d but prime factors of n2 do not. Thus hv2i belongs to the centre of G and U := hu, v2i is cyclic. a b1 c b2 Any automorphism is given by (u, v1, v2) 7→ (u , v1 u , v2 ) with (a, m) = 1, (b1, n1) = 1, (b2, n2) = 1 and b1 ≡ 1 (mod d); and the condition b1 ≡ 1 (mod d) implies (b1, n1) = 1. n1 Hence |Aut(G)| = φ(m). d .m.φ(n2). Since |Z(G)| = n/d, we have |Inn(G)| = φ(m) n1 md, so |Out(G)| = d . d .φ(n2). The above is not the only way to present G as metacyclic. We will ﬁnd it more convenient to use presentations of the form (3) where u generates the subgroup U. Thus the condition that (r − 1) is prime to m is replaced by the condition that every prime factor of n divides d. This determines the subgroup hui uniquely. We refer to these presentations as standard. Lemma 3.2 If G, of type I, has a standard presentation and |H| is prime to |G|, then any homomorphism h : H → Out(G) has a unique lift to a homomorphism h˜ : H → Aut(G) such that, for all x ∈ H, h˜(v) = v.

Proof Since U is cyclic, Aut(U) is abelian: write it as X ⊕ Y , where |X| is prime to |G| and every prime divisor of |Y | divides |G|. Since U is characteristic in G, we have natural surjections Aut(G) → Aut(U) → X, whose composite factors through Out(G). Since the kernel has order prime to |X|, the surjection splits, and the splitting map is unique to conjugacy. By inspection, we see it can be taken to ﬁx v. We next sharpen our description of Types III, IV. Following Wolf [47], we intro- duce the groups

∗ 4 2 2 −1 −1 3v −1 −1 Tv := hx, y, w | x = 1, y = x , y xy = x , w = 1, w xw = y, w yw = xyi, (4) ∗ ∗ 2 −1 −1 −1 −1 Ov := hTv , z | z = x, z yz = x y, z wz = w xi. (5) ∗ Thus Tv is the split extension of the quaternion group Q8 by C3v acting non-trivially. ∗ 3 ∗ ∗ ∼ Tv has centre C3v−1 generated by w , and the quotient by this is T1 = T = SL2(3), ∗ ∗ the binary tetrahedral group. Also, O1 = O . We now treat the case omitted in Proposition 1.3(i).

Lemma 3.3 A P−group which is an extension of C3v−1 by SL2(3) is isomorphic ∗ to Tv .

Proof The induced extension by the Sylow 2-subgroup Q8 of SL2(3) is trivial (coprime orders) so Q8 is normal in G. The quotient is isomorphic to a Sylow 3- subgroup, which is cyclic since we have a P−group. The extension of Q8 by C3v is split, and the action of C3v on Q8 is determined since it factors through the non-trivial action of C3.

∗ Lemma 3.4 Any extension G of Tv by C2 which is a P−group is isomorphic to ∗ Ov.

7 Proof The extension must have Sylow subgroup Q16, which is not a central extension of Q8 by C2. Thus the quotient G of G by Q8 maps onto the outer auto- 3v morphism group (dihedral of order 6) of Q8, thus admits a presentation hw, z | w = 2 −1 −1 z = 1, z wz = w i. This gives the structure of G and the map G → Out(Q8): ∼ 2 since Z(Q8) = C2, the extension is determined by a class in H (G; C2), hence in turn by the extension of the Sylow 2-subgroup of G. To ﬁnd a presentation, we may suppose z2 = x; then z has order 8, so to have a quaternion group we need y−1zy = z−1, so z−1yz = x−1y. Now z−1wz = w−1q for ∗ some q ∈ Q8, and since conjugation by z must induce an automorphism of Tv we ﬁnd we must have q = x.

Theorem 3.5 Any P−group G is an extension of a group G0 of odd order by a ∗ ∗ group G1 isomorphic to one of C2k (k ≥ 0), Q2k (k ≥ 3), Tv or Ov (v ≥ 1), SL2(p) or TL2(p) (p a prime ≥ 5). Moreover, we may suppose the orders of G0 and G1 coprime.

Proof If G has type I or II, this follows from the deﬁnition of the type, with G0 = O(G). If G has type V or VI, we again take G0 = O(G). By Proposition 1.3, any extension by SL2(p)(p 6= 3) or TL2(p) of a cyclic group of odd order is split. Now if ` is a prime (necessarily odd) dividing both |G0| and |G/G0|, a Sylow `−subgroup of G is a split extension of non-trivial `−groups, hence is not cyclic, contradicting our hypothesis. If G has type III, it is an extension by SL2(3) of O(G). As 3 is the smallest prime dividing |O(G)| it follows from [9, Art.128] that O(G) has a normal 3-complement ∗ G0. Applying Lemma 3.3 to G/G0, we can identify it with Tv for some v ≥ 1. By construction of G0, its order is prime to 2 and 3, hence to |G/G0|. If G has type IV, it has a subgroup of index 2 of type III, which by the above, ∗ ∼ ∗ we can write as an extension of G0 by Tv . By Lemma 3.4, G/G0 = Ov. Since the extension given by Theorem 3.5 has coprime orders, it is split, and hence classiﬁed by h : G1 → Out(G0). Now Out(G0) is abelian, so G1 maps via its 1 commutator quotient, which is: C2k (type I), C2 × C2 (type II), C3v for type III, C2 for types IV, VI, and 1 for type V. We can now write down an explicit presentation for each type. For type I, this was already given in (3). For the rest, we already have a presentation of G1, and take a standard presentation of G0 (which has type I). By Lemma 3.2, the map G1 → Out(G0) factors through the automorphism group of hui. Hence we have Theorem 3.6 A P−group G has a presentation as follows. If G has type I, use (3). Otherwise, take a standard presentation of the subgroup G0, and a presentation 2k−1 2 2k−2 −1 −1 of G1 given by hx, z | x = 1, z = x , z xz = x i for type II, (4) for type III, (5) for type IV, and (1) for type VI. Then add relations that all the generators of G1 commute with v, and commute with u except as follows: II: x−1ux = ua, z−1uz = ub with a2 ≡ b2 ≡ 1 (mod m), v III: w−1uw = ua with a3 ≡ 1 (mod m), IV: z−1uz = ua with a2 ≡ 1 (mod m), V: no exceptions, VI: z−1uz = ua with a2 ≡ 1 (mod m).

Corollary 3.7 A non-cyclic P−group G is p−hyperelementary if and only if either G has type I and n is a power of p or p = 2, G has type II, and n = 1.

This follows by inspection. 1incorrectly given in [41] as 3

8 4 Subgroups and reﬁnement of type classiﬁcation

Suppose G is 2-hyperelementary of type II, hence an extension of a cyclic group G0 of odd order by a quaternionic 2-group G1 = Q. The extension is determined by the action of Q on G0; since Aut(G0) is abelian, Q acts through its commutator quotient group, which is a four group. As G0 has odd order, so has unique square roots, any involution of G0 determines a direct sum splitting, where the involution ﬁxes one summand and inverts the other. A second involution commuting with the ﬁrst preserves each summand and induce a further splitting of each. We thus have a direct product splitting

G0 = U1 × Ui × Uj × Uk, (6) where x centralises U1 and Ui and inverts Uj and Uk, and z centralises U1 and Uj and inverts Ui and Uk. The notation is intended to suggest a representation 3 Q8 → S with x → i, z → j. Write m∗ := |U∗| for ∗ = 1, i, j, k. Here the mi are odd and mutually coprime. k In the notation of Milnor [33], this is denoted Cm1 × Q(2 mi, mj, mk). We note with Milnor that replacing y by xy will interchange the roles of mj and mk; and that if Q has order 8, we can permute all of mi, mj and mk. For a general group G of type II, we use the presentation of Theorem 3.6 and apply the above splitting to the subgroup hu, x, zi to deﬁne parameters m∗.

We will need information about hyperelementary subgroups of P−groups. The next result will be the key to this. Theorem 4.1 For G of one of the types III-VI, any subgroup H of G of type I or II is, up to conjugacy, contained in one of the pre-images H0, H00 in G of the 0 00 ∼ subgroups H1, H1 of G1 = G/G0 indicated in the following table: 0 00 G1 H1 H1 SL2(p) Q2(p±1) Cp.Cp−1 TL2(p) Q4(p±1) Cp.C2(p−1) ∗ Tv Q8 × C3v−1 C2.3v ∗ v−1 Ov Q(16, 3 , 1) Q4.3v ∼ Proof If H is a subgroup of G of type I or II, its image in G/G0 = G1 has the same type, so is contained in a maximal such subgroup H1. The pre-image of H1 in G has the same type as H1. Thus it suffices to consider subgroups of G1. Since a group of type I or II normalises a non-trivial cyclic subgroup, it suffices to list such normalisers in G1. Define, for g ∈ GL2(p), −1 r × M(g) := {x ∈ GL2(p): x gx = λg for some λ ∈ Fp , r ∈ Z}, with image PM(g) in P GL2(p), and SM(g) := M(g) ∩ SL2(p). × ∼ If g has distinct eigenvalues in Fp, M(g) is a wreath product C2 o Fp , SM(g) = ∼ Q2(p−1) is quaternionic and PM(g) = D2(p−1) is dihedral. The case when the eigenvalues of g do not lie in Fp is dealt with by considering subgroups of GL2(Fp2 ) and then taking invariants under the Galois group. We find ∼ ∼ that SM(g) = Q2(p+1) is quaternionic and PM(g) = D2(p+1). If g is not semi-simple but not central, M(g) is conjugate to the group of upper + ∼ × ∼ triangular matrices, SM(g) is the split extension of Fp = Cp by Fp = Cp−1 with the square of the natural action, and PM(g) the corresponding extension with the natural action. The list of normalisers of cyclic subgroups of TL2(p) is obtained from the list for P GL2(p) by lifting.

9 ∗ Given H ⊂ Tv , if the projection H onto the quotient group C3v is not surjective, 3 H is a subgroup of hx, y, w i ≡ Q8 × C3v−1 . Otherwise H contains an element wq 2 2 with q ∈ Q8: up to conjugacy, this is either w or wx , hence H ⊆ hw, x i ≡ C2.3v , which is a maximal proper subgroup. ∗ ∗ For H ⊆ Ov, the intersection H ∩ Tv must be contained in one of the above. In the ﬁrst case, H is contained in the pre-image of a subgroup of order 2 of Out(Q8), 3 ∼ 3 conjugate to hx, y, z, w i: hx, y, zi = Q16 and w is centralised by y and inverted by z. In the second, hwi must be normal in H; we check that ζ = zxy satisﬁes −1 −1 2 2 ∼ ζ wζ = w and ζ = x so hζ, wi = Q4.3v . It was shown over a century ago by Dixon that this gives all maximal subgroups of the SL2(p) except for binary tetrahedral, octahedral and icosahedral groups.

We next determine, for each G of type III-VI, presented as in Theorem 3.6, the structure of the groups H0 and H00 just defined. The presentation (3) of a group of type I is defined by parameters (m, n, r); we will list (m, n; d) where, as above, d = ordm(r). The presentation of a group of type II is defined by parameters (m, n, r, k, a, b); we again list d rather than r and list the quadruple (m1, mi, mj, mk) introduced in k (6) rather than (m, a, b), so give (m1, mi, mj, mk; n, 2 ; d). 0 0 For SL2(p) and TL2(p), denote by Hη (η = ±1) the cases for H corresponding 0 to Q2(p+η) and Q4(p+η); also set = ±1 with p ≡ (mod 4); and write p − = qp with q a power of 2 and p0 odd. ∗ ∗ For G1 = Tv or Ov we need to distinguish according as the parameter a in the s presentation takes the value 1 or not; in the former case, write 3 = ordm(a). For types IV, VI we have a relation z−1uz = ua and use this to split U (as in (6) with factors of orders m+, m−. Theorem 4.2 The subgroups listed in Theorem 4.1 have the following invariants. The groups H00 all have type I, with parameters

00 00 00 G1 H1 H (a 6= 1) H (a = 1) 1 SL2(p) Cp.Cp−1 (pm, (p − 1)n; 2 (p − 1)d) TL2(p) Cp.C2(p−1) (pm, 2(p − 1)n;(p − 1)d) . ∗ v s v Tv C2.3v (2m, 3 n; 3 d) (2.3 m, n; d) ∗ v v Ov Q4.3v (3 m, 4n; 2d) (4.3 m, n; d)

0 0 The groups H all have type II except H in the SL2(p) case, which has type I with 1 parameters ( 2 m(p + ), 4n, 2d). Their parameters are given by

0 G1 H1 m1 mi mj mk n q d 0 SL2(p) Q2(p−) m p 1 1 n 2q d 0 TL2(p) Q4(p−) m+ p m− 1 n 4q d 1 TL2(p) Q4(p+) m+ 2 (p + ) m− 1 n 8 d ∗ v−1 s−1 Tv (s > 1) Q8 × C3v−1 m 1 1 1 3 n 8 3 d ∗ v−1 Tv (s = 0, 1) Q8 × C3v−1 3 m 1 1 1 n 8 d ∗ v−1 v−1 Ov Q(16, 3 , 1) m+ 1 3 m− 1 n 16 d

Proof The type of H is the same as that of H1. In each case, H is an extension of G0 by H1, the two having co-prime orders. Recall that all generators of G1 commute −1 a −1 a with those of G0 except for: (type III) w uw = u , (types IV,VI) z uz = u . The results for H of type I now follow by inspection. 0 For G of type III, H has Q8 as a direct factor; the other factor has type I and invariants (m, 3v−1n, 3s−1d) if s > 1 or (3v−1m, n, d) if s = 0, 1.

10 In the remaining cases, no element of odd order in G1 acts non-trivially on a cyclic subgroup of G0, so the parameters n and d for H are the same as those for k G0. The parameter 2 is the highest power of 2 dividing |H|. It remains only to find the m∗: recall that for Ca × Q2kb with a, b odd, these are (a, b, 1, 1). For SL2(p) and H1 = Q2(p−), it suffices to note that the quaternion subgroup centralises u. For the case TL2(p), however, it is the generator z of highest order which inverts the summand Cm− , giving the result in that case. ∼ 3 In the final case, we had hx, y, zi = Q16 and w is centralised by y and inverted by z; the same holds for Cm− . For G of type II, with the notation of (6), we say that G is of rtype IIK if we can arrange that mj = mk = 1, rtype IIL if |Q| ≥ 16 and mjmk 6= 1, rtype IIM if |Q| = 8 and two of mi, mj and mk are 6= 1. Thus G has rtype IIK if and only if it has a cyclic subgroup of index 2. We will write IILM for a group whose rtype may be IIL or IIM. If G has type II, and is presented as in Theorem 3.6, the subgroup H := hu, x, yi is 2-hyperelementary and has the same parameters m∗, hence the same rtype, as G. Proposition 4.3 For G of type II, every subgroup H of G of type II has rtype IIK or has the same rtype as G. Proof We may replace H by a 2-hyperelementary subgroup of the same rtype, and so suppose H 2-hyperelementary. Then H normalises a cyclic subgroup Z of odd order. Replacing Z by a conjugate, we may suppose Z ⊆ G0, hence that Z is the direct product of a subgroup of hui and a subgroup of hvi. The Sylow 2- subgroup S of H is conjugate in the normaliser of Z to a subgroup of hx, zi, so may be supposed a subgroup of this. Now if S = hx, zi, the parameters m∗(H) ≤ m∗(G), so the result follows. Oth- erwise S ⊆ hx2, zi (or hx2, xzi), and since x2 centralises Z, H has type IIK in this case. Theorem 4.4 (i) If G has type III or V, G has no subgroup of rtype IIL or IIM. (ii) If G has type VI, then either (a) z commutes with u, G = G0 × TL2(p) is a product (coprime orders), and every type II subgroup has rtype IIK, or (b) z and u do not commute and G has subgroups of rtypes IIK, IIL and IIM. (iii) If G has type IV, then either ∗ (a) v = 1, z commutes with u, G = G0 × O is a product (coprime orders), and every type II subgroup has rtype IIK, or (b) G has subgroups of rtypes IIK and IIL, but none of rtype IIM. Proof By Theorem 4.1, every hyperelementary subgroup H of G is contained in one of the subgroups H0, H00. These are described in Theorem 4.2; in particular, H00 has type I. It follows from Proposition 4.3 that if G has a subgroup of rtype IIL or IIM, then one of the subgroups H0 has that rtype. Now (i) follows by inspection of the list. If G has type VI, then if m− = 1 we only obtain type IIK; if m− 6= 1, the pre-image of Q4(p−) has rtype IIL; the pre-image of Q4(p+) has rtype IIM. This implies (ii). v−1 0 If G has type IV, then if 3 m− = 1 we only obtain type IIK, otherwise H has type IIL. Hence (iii) holds. We define G to have rtype VIK resp. VIL in cases (iia), (iib) in the above The- orem, and to have rtype IVK resp. IVL in cases (iiia), (iiib).

11 The p−period of a ﬁnite group G is the period (if any) for the p−primary part of its cohomology; we denote it by 2Pp(G): equivalently, 2Pp(G) is the least period ˆ for projective resolutions of Zp(G). Thus for G a P−group, its cohomology has period 2P (G) with P (G) the least common multiple of the Pp(G). We compare this with the Artin exponent e(G) [22], the least positive integer such that e(G)1 belongs to the ideal of the rational representation ring generated by representations induced from cyclic subgroups.

Lemma 4.5 (di) P2(G) = 1 (resp. 2) if the Sylow 2-subgroup of G is cyclic resp. quaternionic. (dii) If p is odd, and Gp is a cyclic Sylow p−subgroup of G with normaliser Np, then Pp(G) equals the order of the group of automorphisms induced on Gp by Np. (diii) For a P−group G, the period 2P (G) is the least common multiple of the periods of hyperelementary subgroups. (ei) If H is a hyperelementary P−group of type I, we have e(H) = P (H). (eii) If H is a 2-hyperelementary P−group, we have e(H) = 2 for H of rtype IIK, e(H) = 4 for H of rtype IIL or IIM. (eiii) For any G, e(G) is the least common multiple of the e(H), H a hyperelementary subgroup of G. Here (di)-(diii) are due to Swan [39], (ei)-(eiii) to Lam [22]. We now calculate P (G) and e(G) in terms of the notation of Theorem 3.6. We see at once that for G of type I, we have e(G) = P (G) = d; also that for type IIK, e(G) = P (G) = 2d, while for type IILM we have P (G) = 2d, e(G) = 4d. By (diii) (resp. (eiii)), P (G) (resp. e(G)) is the least common multiple of the P (H) (resp. e(H)) for H a subgroup of G of type I or II, and so it follows from Theorem 4.1 that for G of type III-VI it suﬃces to consider the subgroups there listed. Their parameters are given in Theorem 4.2, from which the results can be read oﬀ. T ype IIIIVKIVLVVIKVIL 00 00 s 1 P (H ) = e(H ) 3 d d 2d 2 (p − 1)d (p − 1)d (p − 1)d 0 s−1 (7) P (H±1) 2.3 d 2d 2d 2d 2d 2d 0 s−1 e(H±1) 2.3 d 2d 4d 2d 2d 4d Corollary 4.6 The period of a P−group G is given by

T ype of G IIIIIIIVV ( = 1) V ( = −1) VI s 1 P (G) d 2d 2.3 .d 2d 2 (p − 1)d (p − 1)d (p − 1)d

For G a P−group, e(G) = P (G) except when ν2(P (G)) = 1 and G has a subgroup of rtype IIL or IIM; equivalently, G has rtype IIL, IIM, IVL or VIL with = −1; and in these cases, e(G) = 2P (G).

5 Free orthogonal actions

We call a representation ρ : G → UN of G an F−representation if the induced action of G on the unit sphere S2N−1 ⊂ CN is free. We next give Wolf’s [47] classification of F−representations. For p, q primes (not necessarily distinct), one says that G satisfies the pq condition if every subgroup of G of order pq is cyclic. Thus G is a P−group if and only if it satisfies all p2 conditions.

Theorem 5.1 [47, 6.1.11, 6.3.1]. The following are equivalent: (i) G has an F−representation,

12 (ii) G satisﬁes all pq conditions and has no subgroup PSL2(p) with p > 5, (iii) G is a P−group, such that in the notation of Theorem 3.6, n/d is divisible by every prime divisor of d, and G1 is not SL2(p) or TL2(p) with p > 5.

To show that (i) implies (ii) it suffices to verify that a non-cyclic group of order pq, or a group SL2(p) with p > 5, has no F−representation: it is elementary to construct all irreducible representations ρ and check that in each case, for some element g 6= 1, ρ(g) has 1 as an eigenvalue. To see that (ii) implies (iii) observe first, that since G satisfies all p2 conditions, it is a P−group, so can be put in our normal form and second, that if p | d but p - n, there is some prime q | m on which the action of Cp is non-trivial, so G has a non-cyclic subgroup of order pq. That (iii) implies (i) follows from the explicit construction of F−representations, which we give next, again following Wolf. Lemma 5.2 (i) If G is cyclic, the irreducible F−representations are the faithful 1-dimensional representations. ∗ ∗ v−1 (ii) T has a unique irreducible F−representation; for v > 1, Tv has 2.3 irreducible F−representations; all have degree 2. (iii) O∗ has 2 irreducible F−representations, both of degree 2. (iv) I∗ has 2 irreducible F−representations, both of degree 2. (v) The irreducible F−representations of a direct product of groups of coprime orders are the (external) tensor products of those of the factors. Here (i) is elementary, the rest obtained by standard representation theory in [47, 7.1.3, 7.1.5, 7.1.7, 6.3.2] respectively. In (iii) and (iv), the two representations are equivalent under the outer automorphism of G; the images of T ∗,O∗,I∗ are, of course, the binary polyhedral groups. In the general case, the result can be stated as follows. Theorem 5.3 The irreducible F−representations of G are induced from those of the subgroup R(G) defined as follows. In each case write R0 for the cyclic subgroup hu, vdi. If G has type I, take R(G) = R0. ∗ If G has type III, write R1 := Ker(G1 → Out(G0)), so R1 = Tv if s = 0 and ∼ R1 = Q8 × C3v−s if s > 0. Take R(G) = R0 × R1. If G has type IVK or V, take R(G) = R0 × G1. If G has type II, IVL or VI, the subgroup G+ of index 2 defined by omitting z from the list of generators has type I, III or V respectively. Take R(G) = R(G+). The proof occupies [47, §7.2]; the result is tabulated in Wolf’s (very different) notation in [47, 7.2.18]. In each case, R(G) C G; the irreducible F−representations of R(G) are given by Lemma 5.2 and have degree δ = 1 if G has type I or II, and 2 otherwise. Note that k + if G has type IILM with parameters (m1, mi, mj, mk; n, 2 , d), G has parameters k−1 (m1mimjmk, 2 n; 2d). The quotient G/R(G) is cyclic except if G has type IILM, and has order rtype of G I IIK IILM III IV K IV L V V I |G : R(G)| d 2d 4d 3s.d d 2d d 2d

It follows that the irreducible F−representations of G all have the same degree δ|G : R(G)|. On comparing with Corollary 4.6, we see that, in each case, this degree equals e(G) (since p = 5 here, the case p ≡ −1 (mod 4) does not arise). To see that the induced representations are indeed ﬁxed-point free, it is necessary to check that, for each g ∈ G \ R(G) with class of order b, say, in G/R(G), we

13 have gb 6= 1: it suﬃces to consider the cases where b is prime.

Each F−representation ρ of G (of degree N) defines a free action of G on S2N−1, 2N−1 and hence a quotient manifold Xρ := S /G. The first Postnikov invariant of 2N th Xρ in H (G; Z) can be identified with the N Chern class cN (ρ). We calculated these classes for G hyperelementary in [46, Theorem 11.1]: as the calculation is delicate, we now repeat it for the somewhat more general case of all groups of type I or II. We first consider the eigenvalues of each ρ(g). Given representations ρ of G and σ of H, if ρ(g) has eigenvalues αi and σ(h) has eigenvalues βj, then the eigenvalues of the external tensor product (ρ ⊗ σ)(g, h) are the αiβj. Now suppose that H CG has quotient Cd and ρ is a representation of H. Choose S˙ a left transversal {gi} for H in G (i.e. G = iHgi). Then for g ∈ G, the eigenvalues G of IndH ρ(g) are: −1 if g ∈ H, the union of the sets of eigenvalues of the ρ(gi ggi); if g 6∈ H, the dth roots of the eigenvalues of ρ(gd). G In particular, as noted above, IndH ρ is a F−representation if and only if ρ is an F−representation and gd 6= 1 for g 6∈ H.

We need a precise notation for cohomology classes. Consider the cyclic group ∼ 1 G = Cm generated by x. There is a natural isomorphism iG : Hom(G, S ) → H2(G; Z), the boundary map in the exact sequence H1(G; R) → H1(G; S1) → H2(G; Z) → H2(G; R), whose extreme terms vanish. For each character ρ, iG(ρ) is the Chern class c1(ρ), and is a cohomology generator. The irreducible F−representations of G have di- 2πik/m mension 1, and are given by ρk(x) = e with k prime to m. Setx ˆ := iG(ρ1) = t c1(ρ1): then c1(ρk) = kxˆ. A direct sum ρ := ⊕i=1ρkb of representations gives an Q action of G on the join of the corresponding spheres, and ct(ρ) = i c1(ρkb ) = Q t Q P ( kb)ˆx . It is important to note that here we have kb, not kb, which is what i P appears in det(ρ(x)) = e2πi kb/m. 4n Next let G = Q8n be a quaternionic 2-group, with presentation hx, y | x = 1, y2 = x2n, y−1xy = x−1i, and cyclic subgroup G+ = hxi of index 2. The irre- + ducible F−representations of G are induced from those of G , which are the ρr G + with r odd: set σr := IndG+ ρr. The restriction of σr to G is ρr ⊕ ρ−r, so has 2 2 Chern class c2(σr) = −r xˆ . Up to isomorphism, there are 2n different ρr and n ˆ 4 2 ˆ different σr. We define X ∈ H (G; Z) to be c2(σ1). Then we have c2(σr) = r X; this is less trivial than the corresponding result in the cyclic case: one proof was given in [46, 11.2]; we can also write down an equivariant map of degree r2 between the representation spaces. Note that H4(G; Z) is cyclic of order 8n: the value of r2 modulo 8n is determined by that of r modulo 4n, and is ≡ 1 (mod 8); the value of c2(σr) determines σr up to isomorphism. Theorem 5.4 Let G be of type I, admitting F−representations; use the above notation. Then the irreducible F−representations σs,t of G have degree d and their th Chern classes cd are the cohomology generators which restrict to each G` as d powers multiplied by λ`(G), where d/2 for ` | m, λ`(G) = 1 for d odd, λ`(G) = r for d even, for ` | n, λ`(G) = 1 for ` or d odd; if ν2(n/d) ≥ 2, λ2(G) = 1+n/2; if ν2(n/d) = 1 and ν2(n) ≥ 3, λ2(G) = 1; if ν2(d) = 1 and ν2(n) = 2, λ2(G) = −1. Proof We have G = hu, v | um = vn = 1, v−1uv = uri, with n prime to m and n r ≡ 1 (mod m), d = ordm(r) and each prime divisor of n divides both d and d d n/d. Then K := hu, v i is cyclic, with F−representations given by σs,t(uv ) = e2πi(s/m+td/n) with s prime to m and t prime to n. The irreducible F−represen- G tations of G are the πs,t = IndK σs,t. The eigenvalues of πs,t(z) are as follows:

14 j if z = u, the e2πisr /m, (0 ≤ j < d); if z = vd, e2πitd/n (d times); if z = v, the e2πi(t/n+j/d), (0 ≤ j < d). d−1 1 Q j d d 2 d(d−1) d Thus cd(πs,t), restricted to hui, is j=0 (sr )û = s r uˆ ; the restriction to Qd−1 d hvi is j=0 (t + jn/d)ˆv . 1 2 d(d−1) d 1 First consider r modulo m. Since r ≡ 1 (mod m), if d is odd, 2 d(d − 1) 1 d(d−1) is divisible by d and we have r 2 ≡ +1 (mod m). If d is even, we see similarly 1 d(d−1) 1 d that r 2 = r 2 (mod m). In the latter case, n is even, so m is odd. Qd−1 Secondly, we need j=0 (t + jn/d) (mod n). We argue as follows, after [46, p × 538]. The residues (mod n) of the 1 + jn/d (0 ≤ j ≤ d − 1) form a subgroup of Fn , since d and n/d have the same prime divisors. Most elements of the subgroup cancel (mod n) with their inverses when we multiply, so we must find the elements of order 2. Now the group is the direct product of its primary subgroups, corresponding to prime factors p|n, so elements of order 2 come from p = 2. If ν2(n/d) ≥ 2, the Sylow 2-subgroup is cyclic and the only element of order 2 is 1+n/2; if ν2(n/d) = 1 and ν2(n) ≥ 3, we have the 4 elements ±1, ±(1 + n/2), whose product is 1 (mod n); if ν2(d) = 1 and ν2(n) = 2, we have the two elements ±1, with product -1. The conclusion can be re-stated as follows. For any G of type I, define

P (G)/P (G) λ`(G) = (−1) ` for ` odd, ν2(|G|)−1 λ2(G) = 1 + 2 if ν2(|G|) − 2 ≥ ν2(P (G)) ≥ 1 or (ν2(|G|), ν2(P (G))) = (2, 1), λ2(G) = 1 otherwise. (8) Corollary 5.5 The theorem also holds with this notation.

1 d Proof The factor r 2 can only take the values ±1 (mod |G`|), and takes the value

−1 only if (` is odd and) d/ord|G`|(r) is odd. But P (G) = d and by Lemma 4.5,

P`(G) = ord|G`|(r). If d is odd, −1 is a dth power, so the sign is irrelevant. If `|n and d is even, P (G)/P (G) P`(G) is 1, so (−1) ` = 1. For G of type II, by Theorem 5.1, G has F−representations if and only if each prime divisor of n divides both d and n/d. We have the presentation of Theorem 3.6 but, as in §4, we split U := hui as a direct product U1 × Ui × Uj × Uk. For any group G of type II, deﬁne parameters for ` odd, λ`(G) = −1 if G has rtype IIK and ` | mi and λ`(G) = +1 otherwise, k−2 2 λ2(G) = 1 for G of rtype IIK, λ2(G) = (1 + 2 ) for G of rtype IILM. Recall that e(G) = 2d for G of rtype IIK and 4d for G of rtype IIKL. Theorem 5.6 Let G be a P−group of type II admitting F−representations. Then the irreducible F−representations of G have degree e(G), and their top Chern classes th are the cohomology generators which restrict to each G` (` odd) as e(G) powers e(G)/2 th multiplied by λ`(G); and to G2 as λ2(G)Xˆ multiplied by an e(G) power.

Proof It follows from Theorem 5.3 that the F−representations of G are obtained + as follows. The group G := hu, v, xi has type I. If G has rtype IIK, Uj = Uk = {1}, and R(G+) = hu, vd, xi; for rtype IILM, R(G+) = hu, vd, x2i. The irreducible F−representations of G+ are induced from the irreducible F−representations of R(G+), and in turn induce the irreducible F−representations of G, which thus have degree e(G), which is 2d, 4d in the two cases. For G of rtype IIK, the parameter d for G+ is odd. Applying Theorem 5.4 to + + G , we ﬁnd that the Chern classes cd of the irreducible F−representations ρ of G

15 are the generators which restrict to `−subgroups as dth powers. Let g ∈ G+ have a 2πik/à order ` with ` odd. Write the eigenvalues of ρ(g) as e for k = k1, . . . , kd, G then if ` | m1n those of IndG+ (ρ)(g) are the same, each repeated; if ` | mi, we must Q th add the values k = −k1,..., −kd. Multiplying up, since b kb is a d power we obtain, in the first case a (2d)th power, and in the second case, the negative of 2 ˆ ˆ d one. For the Sylow 2-subgroup, inducing up σkb leads to kb X, so we obtain X multiplied by a (2d)th power. If G has rtype IILM, the parameter ‘d’ for the group G+ is 2d in our notation, so is even. For g ∈ G+ of order à with ` odd with the eigenvalues of ρ(g) 2πik/à G {e | k = k1, . . . , k2d}, then if ` divides m1min those of IndG+ (ρ)(g) are the same, each counted twice; if ` | mjmk, we must add the values k = −k1,..., −k2d, Q 2 Q th thus the product in each case is ( kb) . Since b kb is λ`(G) times a (2d) power 2 th and λ`(G) ≡ 1, in each case we obtain an arbitrary (4d) power. For ` = 2, k−2 each eigenvalue e2πikb/2 of x2 in the representation of R(G) gives eigenvalues 2πik /2k−1 + k−2 k−2 ±e b of x ∈ G when we induce up. Since kb + 2 ≡ kb(1 + 2 ) (mod k−1 Qd 2 ˆ 2 k−2 2 ˆ k−2 4 2 ), the class is b=1(kb X.kb (1 + 2 ) X). Since d is odd and (1 + 2 ) ≡ 1 k k−2 2 ˆ 4d th (mod 2 ), we have (1 + 2 ) X multiplied by a (4d) power.

th k We can state the condition for G2 more explicitly: the (2d) powers modulo 2 are the same as the squares, and give all numbers ≡ 1 (mod 8); the (4d)th powers give all numbers ≡ 1 (mod 16); and the class of (1 + 2k−2)2 (mod (4d)th powers) is 1 if k = 3, 1 + 2k−1 if k ≥ 4.

6 The ﬁniteness obstruction

A finitely dominated CW complex X with fundamental group G and universal cover homotopy equivalent to SN−1 is said to be a (G, N)-space. The first Postnikov invariant g ∈ HN (G; Z) of X has additive order |G|, and multiplication by g induces isomorphisms of cohomology groups of G in positive dimensions; if we use complete (Tate) cohomology (see [10, Chapter XII]), this holds in all dimensions. An element g with this property is called a (cohomology) generator. It is known (see [40] and [41, Theorem 2.2]) that homotopy types of oriented (G, N)-spaces X correspond bijectively to cohomology generators g. Denote by Xg a (G, N)-space corresponding N to g ∈ H (G; Z); then Xg is a Poincarécomplex. The chain complex of the universal cover X˜g is chain homotopy equivalent to a sequence P∗ of finitely generated projective ZG−modules yielding an exact sequence 0 → Z → PN−1 → ... → P1 → P0 → Z → 0. The class of this sequence in ExtN ( , ) =∼ HN (G; ) is g. Swan’s finiteness obstruction o(g) ∈ K˜ ( G) is ZG Z Z Z 0 Z Pn−1 i defined as the class of 0 (−1) [Pi]: it was shown in [40] that it vanishes if and only if there is a finite CW-complex homotopy equivalent to X. We have

o(g1g2) = o(g1) + o(g2). (9)

The generators g in complete cohomology form a multiplicative group Gen(G). By (9), we have a homomorphism o : Gen(G) → K˜0(G). Taking degree gives a homomorphism Gen(G) → Z with image 2P (G)Z. Its kernel, the torsion subgroup ˆ 0 × ∼ × of Gen(G), is the group H (G; Z) = FM , where M = |G|. We denote the re- 0 × ˜ striction by o : FM → K0(G), and its image, the so-called Swan subgroup, by Sw(G) ⊆ K˜0(G).

0 Lemma 6.1 (i) If r ∈ Z is prime to M, o (g) is the class of hr, IGi C ZG. (ii) o is natural for restriction to subgroups. L (iii) The map K0(ZG) → {K0(ZH) | H ⊂ G hyperelementary} is injective.

16 These results are due to Swan [38], [40]; see also [46, §10]. It follows from (i) that o0(−1) = 0: geometrically, changing the orientation of Xg gives an X−g. If g is the class of an F−representation ρ, then a triangulation of the quotient N−1 manifold Xρ = S /G gives a resolution by free modules, so o(g) = 0. To obtain further examples we use Proposition 6.2 Let G, G∗ be p−hyperelementary P−groups of Type I, φ : G∗ → G an epimorphism with kernel K a p−group. Suppose g∗ ∈ H2N (G∗; Z) a generator ∗ ∗ ∗ th 2N such that (i) o(g ) = 0, (ii) g | Gp is α times a d power; and that g ∈ H (G; Z) ∗ ∗ ∗ th satisfies (iii) φ (g | G`) = (g | G` ) for ` 6= p, (iv) g | Gp is α times a d power. Then g is a generator and o(g) = 0. The proof in [46, 12.1] depends on the representation of o(g) by an adèlewhose construction involves Reidemeister torsions, and a simple calculation. Corollary 6.3 Let G be p−hyperelementary with p odd, and g ∈ H2N (G; Z) a th generator whose restriction to each G` is a d power. Then o(g) = 0. Proof The group G has type I: suppose it given by (3) with n = pk. Define G∗ by the same presentation, but with n = pk+1. Then G∗ admits F−representations, ∗ ∗ and by Theorem 5.4, since each λ`(G ) = 1, we can choose the Chern class g to th be any generator whose restriction to each G` is a d power. Thus the hypotheses of Proposition 6.2 hold, and by that result, o(g) = 0. We recall that for G of type I, P (G) = d. For p = 2, the results are more subtle. Corollary 6.4 Let G be 2−hyperelementary of type I, and g ∈ H2N (G; Z) a gen- N/d th erator which restricts to each G` (` odd) as λ`(G) times a d power and to G2 as a dth power. Then o(g) = 0.

∗ ∗ Proof We can again deﬁne G as above. We have λ`(G) = λ`(G ) for ` odd, ∗ k and λ2(G ) = 1 + 2 or 1. It follows from Proposition 6.2 that if the restriction of 2d ∗ N/d th g ∈ H (G; Z) to each G` is λ`(G ) times a d power, we have o(g) = 0. Since k 1 + 2 ≡ 1 (mod |G2|), the result follows. If G itself has an F−representation, comparing this class g with the Chern classes of representations, we deduce

Corollary 6.5 Let G be 2-hyperelementary of type I, with k − 2 ≥ ν2(d) ≥ 1 or 0 1 (k, ν2(d)) = (2, 1). Then o (1 + 2 M) = 0. We define g ∈ H∗(G; Z) to be h-linear if its restriction to each hyperelementary subgroup is the class of an F−representation; to be w-linear if its restriction to each 2-hyperelementary subgroup is the class of an F−representation and to each `−hyperelementary subgroup (` odd) satisfies the conditions of Corollary 6.3; and to be f-linear if its restriction to each 2-hyperelementary subgroup is either the class of an F−representation or as in Corollary 6.4 and to each `−hyperelementary subgroup (` odd) satisfies the conditions of Corollary 6.3. It follows from Lemma 6.1(iii) together with the cited results that each of f-linearity and w-linearity is sufficient for o(g) = 0. We now discuss these conditions more explicitly: we will show that any P−group G admits an f-linear generator; and G admits a w-linear generator if and only if G satisfies all 2p−conditions. For an `−hyperelementary subgroup (` odd) which satisfies all pq−conditions, any g which satisfies the conditions of Corollary 6.3 is the class of an F−representation. Hence for any P−group G which satisfies all pq−conditions, any w-linear generator is h-linear. For a 2-hyperelementary group of type I which satisfies all 2p−conditions, any g with N/d even which satisfies the conditions of Corollary 6.4 is the class of an F−representation.

17 Proposition 6.6 Let G have type I. Suppose g ∈ H2N (G; Z), where d | N, a gen- N/d th erator whose restriction to each G` (` odd) is λ`(G) times a d power. Then g th is f-linear if its restriction to G2 is a d power. If G satisfies all 2p conditions, g N/d th is w-linear if its restriction to G2 is λ2(G) times a d power. Proof We must show that the restriction of g to each p−hyperelementary subgroup G0 satisfies the stated conditions. Note that here d = P (G) = e(G). First con- 0 sider an odd prime `: then (5.4) we need to show that the restriction of g to each G` 0 th 0 N/P (G0) 0 P (G0)/P (G0) is a P (G ) power multiplied by λ`(G ) , where (8) λ`(G ) = (−1) ` 0 for ` odd if p = 2; also (6.3) if p is odd (but here λ`(G ) = 1). Thus we must show that, for each hyperelementary G0 ⊂ G, 0 (*) (−1)P (G)/P`(G) is (−1)P (G)/P`(G ) times a P (G0)th power. This is clear if either P (G0) is odd, so that −1 is a P (G0)th power, or if 0 P`(G)/P`(G ) is odd, so the powers of −1 are the same. 0 0 But if P`(G)/P`(G ) is even, G cannot contain a Sylow 2-subgroup of G. Then 0 0 for each odd p we either have Pp(G ) odd or ν2(Pp(G )) < ν2(Pp(G)); it follows that 0 ν2(P (G )) < ν2(P (G)). Thus either ν2(P`(G)) < ν2(P (G)), when both the signs in (*) are +1, or the order of the group of inner automorphisms of G0 is divisible by 0 21+ν2(P (G )), and hence −1 is a P (G0)th power mod `. So (*) holds in all cases. For ` = 2, the f-linearity result follows since here (by Corollary 6.4) all the 0 parameters corresponding to λ2 are equal to 1. As to w-linearity, if G does 0 0 not contain a Sylow 2-subgroup, λ2(G) is 1 mod |G | but P (G)/P (G ) is even, 0 P (G)/P (G0) 0 so λ2(G ) ≡ 1. If G does contain a Sylow 2-subgroup, then either 0 0 k−1 0 th 0 λ2(G ) = λ2(G) or ν2(P (G )) = 0, so 1 + 2 is a P (G ) power mod |G |. Proposition 6.7 Let G have type II. Suppose g ∈ H2N (G; Z), where e(G) | N, a N/e(G) generator which restricts to each Sylow subgroup G` (` odd) as λ`(G) times th N/e(G) N/2 th an e(G) power, and to G2 as λ2(G) Xˆ times an e(G) power. Then g is w-linear and f-linear. Proof It suffices to show that if G0 is a p−hyperelementary subgroup, then for 0 e(G)/e(G0) 0 th 0 any `, λ`(G) is (λ`(G )) times an e(G ) power. If p is odd, e(G ) is odd so the condition holds; thus it suffices to take p = 2. 0 For ` odd, if G has rtype IILM, then λ`(G) = 1. Either G also has rtype IILM or e(G)/e(G0) is even: the result holds in either case. If G has rtype IIK, 0 we have λ`(G) = −1 and `|mi and +1 otherwise. If G also has type IIK, we have 0 0 0 λ`(G ) = λ`(G) and mi is the same for G and G . If the image of G in G/G0 is 0 0 contained in hxi, then λ`(G ) = 1 and e(G)/e(G ) is even. Otherwise this image b 0 b 0 has order 4, generated by x z for some b, P (G ) = 2d = P (G), and x z inverts G` 0 only if `|mi, so λ`(G ) = λ`(G). k−2 2 For ` = 2 we have λ2(G) = (1 + 2 ) if G has type IILM and k = ν2(|G|). If 0 0 0 G has type IILM, it has odd index in G, e(G)/e(G ) is odd and λ2(G ) = λ2(G). 0 0 0 e(G)/e(G0) If G does not have type IILM, e(G)/e(G ) is even, so λ2(G ) ≡ 1; but k−1 2 th k (1 + 2 ) is a square, hence a (2d) power (mod 2 ). We now spell out the details for groups of the remaining types. The following is the central result of this paper. It is a version of [46, Theorem 12.5], whose proof referred to three clumsy Lemmas A, B and C, which were proved in the old version of this paper. Here we use instead the above, together with Theorem 4.1. Define parameters λ for G of type III, IV, V or VI by: For ` odd, λ` = +1 except for the cases in the following table, when λ` = −1: rtype p ≡ −1 (mod 4) p ≡ 5 (mod 8) p ≡ 1 (mod 8) V ` | (p2 − 1) ` | p(p2 − 1) ` = p V IK ` | p(p2 − 1) ` = p ` = p V IL ` = p ` = p

18 k−2 2 For G of rtype IVL or VIL, λ2(G) = (1 + 2 ) ; otherwise, λ2(G) = 1. Theorem 6.8 Let G be a P−group of type III-VI, and let g ∈ H2N (G; Z) be a N/e(G) generator whose restriction to the Sylow subgroup G` (` odd) is λ`(G) times N/e(G) N/2 an e(G)th power, and to G2 is λ2(G) Xˆ times an e(G)th power. Then g is w-linear, so o(g) = 0. Proof It follows from Theorem 4.1 that it suffices to check that the restriction of g to each of the subgroups H0 and H00 described in the Theorem satisfies the hypothesis of Proposition 6.6 or Proposition 6.7 according as it has type I or II. The parameters of H0 and H00 are given in Theorem 4.2, and the corresponding values of e in (7); the λ`(H) are given in terms of the parameters of H for H of type I in (8) and for type II in the preamble to Theorem 5.6. There are numerous cases, as we have to distinguish according to the values of s := ν2(p − 1). 0 00 00 rtype s ν2e(G) ν2e(Hη) ν2e(H ) λ2H III 1 1 0 1 IVK 1 1 0 1 IVL 2 2 1 −1 V 1 −1 1 1 0 1 V 2 1 1 1 1 −1 V ≥ 3 1 s − 1 1 s − 1 1 VIK 1 −1 1 1 1 −1 VIK 2 1 2 1 2 1 VIK ≥ 3 1 s 1 s 1 VIL 1 −1 2 2 1 −1 VIL 2 1 2 2 2 1 VIL ≥ 3 1 s 2 s 1 rtype IIIIVKIVLVVIKVIL 0 k−1 λ2(H) 1 1 9 −1 1 1 + 2 0 λ2(H−) − − − 1 1 1 00 For ` an odd prime, we have λ`(H ) = −1 if G has type V or VI and ` = p and 0 = +1 otherwise, and λ`(Hη) = −1 if G has type V or rtype VIK and ` | (p + η) and = +1 otherwise. (These calculations can be clarified on noting that the H of type 0 I admit a group with invariants (m, n; d) as a direct factor; while λ`(H−) = −1 if 0 0 and only if ` | mi(H−) = p ). e(G)/e(H) In most cases, it suffices to check that (λ`(H)) = λ`(G), or at least is congruent (mod |H|). The checking is trivial, but we must remember that if H has type I, Xˆ restricts as −xˆ2, and this will change the sign if e(G)/e(H) is odd and 0 ν2(e(G)) = 1: this situation arises for H with G of type V and ν2(p − 1) ≤ 2, for 00 00 H with G of type V and ν2(p − 1) = 2, and for H with G of rtype VIK and ν2(p − 1) = 1. We have found, for any N divisible by 2e(G), explicit f-linear generators g ∈ H2N (G; Z) with o(g) = 0. There remains the question whether, in the cases where e(G) = 2P (G), there exist g ∈ H2P (G)(G; Z) with o(g) = 0. The class of o(g) ˜ in K0(ZG)/Sw(G) again defines a homomorphism of the group of generators g, and vanishes if the degree of g is divisible by 2e(G). Thus the class of o(g) for g ∈ H2P (G)(G; Z) is independent of g and defines an element o(G) of order 2. Apart from this, the calculation of o is reduced to that of its restriction o0. We know necessary conditions for this to vanish. By a result of [12] (see also [46, 13.1]) ∼ × 0 If G = Q2k , then for s ∈ F2k , o (s) = 0 ⇔ s ≡ ±1 (mod 8). Next, it follows as in [46, 13.3] from a result of [13] that × If G has type I with m = p an odd prime and r = n, then for s ∈ Fmn, 0 th 1 o (s) = 0 ⇔ s is an n power (mod p) (( 2 n)th if n is even). Hence

19 × 0 Lemma 6.9 [46, 13.4] Let G be a P−group of order M, u ∈ FM and o (u) = 0. th 1 th Then (i) for p odd, u is a dp(G) power mod p (a 2 dp(G) power if dp(G) is even), and (ii) if G does not have type I, u ≡ ±1 (mod 8).

There is still quite a gap between this and the above suﬃcient condition, and not much has been written in the interim: e.g. the major survey by Oliver [34] deals mainly with class groups of p−groups. A test case could be the direct product of C13 with a non-abelian group of order 21: we know that for o0(s) = 0 it is necessary that s is a cube mod 7, and suﬃ- cient that s is a cube mod 7.13. We could perhaps narrow the gap if an induction theorem could be established using a smaller class of subgroups than that of all hyperelementary groups.

The only further calculations I can trace refer to the 2-hyperelementary groups Q(8, p, q) of type IIM with p, q distinct odd primes. The ﬁrst results were obtained by Milgram [32]. Building on his work, a detailed study was made by Bentzen and Madsen [3]. ∼ Their ﬁrst result is that for p an odd prime, Sw(Q8p) = S(1) ⊕ S(p) where ∼ 0 S(1) = Sw(Q8) = C2 and S(p) has order 1 or 2. To describe o , write qp for the r quadratic residue symbol qp(r) = p and set q4(r) = ±1 according as r ≡ ±1 (mod 4) and q8(r) = ±1 according as r ≡ ±1 or r ≡ ±3 (mod 8). We have to distinguish cases according to the class of p (mod 8) and, if p ≡ 1 (mod 8), whether 2 has odd × o e 0 or even order in Fp : denote these cases by 1 and 1 . Then [3, (3.6)] the map o is given by the characters

p (mod 8) 1o 1e 3 5 7 0 o q8 (q8, qp)(q8, q4.qp)(q8, qp) q8

× ×2 Next, Sw(Q(8, p, q)) ≡ S(1) ⊕ S(p) ⊕ S(q) ⊕ S(pq). Here write ψp : Fp /Fp → S(p) × for the surjection, ηp = 2 cos 2π/p, and Φ0 for the reduction map Φ0 : Z[ηp, ηq] → × × (Fp ⊗ Z[ηq]) ⊕ (Fq ⊗ Z[ηp]) ; then S(pq) = (ker ψp ⊕ ker ψq) in coker Φ0/squares. In [3, (4.6)], they show that Sw(Q(8, p, q)) is isomorphic to 3C2 or to 2C2, and 0 calculate o in most cases. To save space in the table, we omit the component q8 in each case. (p, q) (1, 3) (1, 7) (3, 3) (3, 5) (3, 7) (5, 5) (5, 7) (7, 7) 0 o qp, q4.qq qp q4.qp, q4.qq q4.qp, qq q4.qp, q4.qq qp, qq qp qp.qq

In [3, (5.3)], Bentzen and Madsen determine whether o(Q(8, p, q)) = 0; this p depends whether the quadratic residue symbol R = q is ±1.

(p, q) (1, 1) (1, 3) (1, 5) (1, 7) (3, 3) (3, 5) (3, 7) (5, 5) (5, 7) (7, 7) R = 1 0 0 ?? 0 6= 0 0 0 0 0 0 R = −1 0 0 0 0 6= 0 6= 0 0 0 6= 0 0

7 Application to the space-form problem

The problem in question is that of describing free actions of finite groups G on spheres SN−1, or equivalently, manifolds X with fundamental group G and universal cover homeomorphic to SN−1. If N is odd, it follows from the Lefschetz fixed point theorem that a non-trivial free diffeomorphism reverses orientation. Hence only the ∼ case G = C2 arises. If N = 2, elementary arguments show that G must be cyclic, and the action is equivalent to that by rotations. If N = 4, it follows from Perelman’s results that any action is equivalent to a free orthogonal action. Otherwise, N ≥ 6, and the problem can be studied by the methods of surgery [42].

20 It was shown (using a direct geometrical argument) by Milnor [33] that a necessary condition for G to act freely on a sphere is that G has at most one element of order 2, or equivalently, that G satisfies all 2p conditions. In fact, any P−group not of type I satisfies all 2p conditions, since by inspection this is the case for G1, and the group generated by G0 and the involution in G1 is a direct product. A different proof of Milnor’s result was given by Lee [23]. Lee introduced a semicharacteristic invariant, recovered Milnor’s result, and also showed [23, 4.15] The group Q(8n, k, `) with n even and k > 1 cannot act freely on any sphere SN−1 with N ≡ 4 (mod 8). This implies that no group of rtype IIL, and hence also no group of rtype IVL or VIL, can act freely on any sphere SN−1 with N ≡ 4 (mod 8) (Lee’s paper also explicitly excludes groups of rtype IVL). These results do not apply to rtype IIM, which is the reason for making the distinction between rtypes IIL and IIM. Although Lee’s invariant can be regarded as part of the surgery obstruction — this relation is generalised and made more explicit in [11] — its formulation is more robust. Lee himself observed that his argument, like Milnor’s, applies not merely to actions on spheres, but also to F2−homology spheres, and it was shown by Ham- bleton and Madsen [17] that it also covers the case of semi-free actions on RN .

For our positive result, first recall Theorem 7.1 [29, Theorem 4.1] Given a normal map of a finite Poincarécomplex of odd formal dimension, surgery to obtain a homotopy equivalence is possible if and only if (i) for each 2-hyperelementary subgroup H ⊂ G, the covering space Xg(H) is homotopy equivalent to a manifold, and (ii) surgery is possible for the covering Xg(G2). This depends on the induction theorem of Dress, which reduces to calculating the surgery obstructions for p−hyperelementary subgroups H of G, for the corresponding covering maps; the calculation that if p is odd, the surgery obstruction for H reduces to that of (its direct factor) H2; and if p = 2 that the same follows (by the formula for surgery obstructions) for maps between closed manifolds. The main existence result is as follows.

Theorem 7.2 Let G be P−group satisfying all 2p conditions, g ∈ H2N (G; Z) (N ≥ 3) a w-linear cohomology generator. Then Xg is homotopy equivalent to a manifold. Moreover, the manifold can be taken smooth, with universal cover S2N−1. We recall the steps in the proof. First apply Theorem 7.1. In our case, (i) holds since (as G satisﬁes all 2p conditions) each 2-hyperelementary subgroup H satisﬁes all pq conditions, so admits F−representations, and as g is w-linear, its lift to H comes from an F−representation. To achieve (ii), we need to construct a suitable normal invariant. A homotopy- theoretic argument from [41] and extended in [29, Lemma 3.2] shows that any normal invariant for G1 extends to one for G. In Cases I-IV, we choose a normal invariant for G1 coming from an F−representation and extend that. For Cases V and VI we use [29, Lemma 3.3] which states that a normal invariant for G2 extends ∗ to one for G if and only if, for each diagram (Q8 ⊂ G2) ⊂ (T ⊂ G) of subgroups, ∗ the restriction of the normal invariant to Q8 extends to a normal invariant of T . As there, this holds automatically if we choose a normal invariant for G2 coming from an F−representation. To obtain a smooth manifold, we use [29, Theorem 4.2]: if Xg is a manifold, it is homotopy equivalent to a smooth manifold if and only if (i) it has a smooth normal invariant and (ii) the covering space Xg(G2) is smoothable.

21 The proof of this, and the proof of the existence of a smooth normal invariant in [29, Lemma 3.1] use transfer techniques, so the smooth normal invariant chosen restricts to G2 as one coming from an F−representation, hence (ii) also is achieved. Finally, it follows from this that Xg(G2) is normally cobordant to a smooth space-form, and the surgery argument of [30, Lemma 4, Theorem 5] yields the ﬁnal conclusion.

This shows that G can act freely on any sphere S2N−1 with N divisible by e(G). That N is divisible by P (G) is necessary for the existence of such actions, and Lee’s result implies that, except for groups of type IIM, divisibility by e(G) also is necessary. The question whether groups of rtype IIM, particularly the groups Q(8, p, q) with p and q prime, can ever act freely on spheres S4N−1 with N odd has attracted a great deal of attention. After careful analysis by Madsen [27] (a companion paper to [3]), detailed results were obtained by Bentzen [4], which include the following: If p ≡ 3 (mod 4), then necessary conditions are that q ≡ 1 (mod 8) and the × order ordq(p) of p in Fq is odd. If, in addition, ordp(q) is maximal odd, such an action exists. The only pairs (p, q) of primes with pq < 2000 for which such an action exists are (3,313), (3,433), (17,103) and (3,601).

It may be that the vanishing of o(g) ∈ K˜0(G) is the main obstruction to Xg being a manifold. By (the proof of) [29, Theorem 3.1], Xg admits a smooth normal invariant provided its covering space with fundamental group G2 does: this holds for all g if G2 is cyclic; and if G2 is quaternionic for all g such that o(g|G2) = 0. For each g and each normal invariant, Theorem 7.1 shows that surgery is possible if and only if this is the case for each 2-hyperelementary subgroup. There are extensive partial calculations of the surgery obstruction groups in [44] and [21], and in the case when one already has a manifold of the same homotopy type, of the surgery obstruction itself in [19], but these cannot be used directly: a direct attack again leads to arithmetic complications. We have a complete result if n is odd. Here P (G) must be odd, and it follows from Corollary 4.6 that G is the direct product of a cyclic 2-group and a group of 0 odd order. In this case, the surgery obstruction lies in L1(G), which vanishes by [44, 2.4.2,3.3.3]. Thus here o(g) = 0 suﬃces for existence of a manifold. Otherwise we can obtain partial results. First, for G 2-hyperelementary, it may be that ‘almost’ any g with o(g) = 0 already comes from an F−representation, at least for G of type I or IIK. Certainly we need not require the restrictions of g to p−hyperelementary subgroups for p odd to come from F−representations. Then the surgery obstruction for G2 can be approached as follows. Given a manifold M and a normal invariant given by f : M → G/T op, it is 0 shown in [19, 0.3] that the obstruction in L3(G) to performing surgery vanishes 2 if and only if κ1c∗(ARF1(f)) does, where ARF1(f) is the component of (vM ∪ ∗ f k) ∩ [M] in H1(M; Z/2), c : M → K(G, 1) induces an isomorphism of π1, and 0 κ1 : H1(G; Z/2) → L3(ZG) is deﬁned there. Since (loc. cit.) κ1 is injective, and for space forms c∗ is an isomorphism, the condition is equivalent to the vanishing of ARF1(f). For G a P−group, H1(G; Z/2) = G ⊗ C2 is trivial if |G| is odd or G has type III,V, is isomorphic to C2 × C2 for G of type II, and otherwise has order 2. Only in a few other cases are existence results known which go beyond Theo- rem 7.2. An explicit result was given by Hambleton and Madsen [18]: k−2 Let G be a 2-hyperelementary type I group with n = 2k, d = 2k−1 and r2 ≡ 2k+1−1 −1 (mod m): then G acts freely on S with homotopy type rg0 if and only if k−1 r is a 2 th power modulo m (so rg0 is w-linear).

22 8 Space-forms: classiﬁcation

For oriented (topological) manifolds M with covering homeomorphic to S2N−1 and a fixed isomorphism π1(M) → G, there are two principal invariants. In [42, 13B2] I defined (using the multi-signature of a manifold bounding some multiple of M) an invariant ρ(G, M): G \{1} → C. The functions ρ(G, M) are class functions, taking real (imaginary) values for n even (odd), and the group of such functions contains the 16-fold multiples of group characters as a subgroup of finite index. We may thus also write ρ ∈ RO˜ (G) ⊗ Q. In [46, §4] I defined a torsion invariant of periodic projective resolutions. In the first place, in the present context, this lies in K1(QG), though as the component in ∼ × K1(Q) = Q takes a known value, this component can be discarded. To make the invariant more concrete, we may replace it by its image in the centre Z(QG) under the reduced norm, or even, taking logarithms, in the additive group of Z(RG). I denote this invariant by ∆(M). For the case when G is cyclic of odd order, these two invariants suffice for the classification: details, e.g. saying which values they can take, were given in [42, 14D,14E]; see also [6]. We now consider how much we can say in general. The most convenient flavour of surgery obstruction groups is given by the inter- 0 mediate L groups Lk(G) of [44]. We recall that according to [43, 6.5] we can write 0 0 T ors(K1(ZG)) = {±1} ⊕ (G/G ) ⊕ SK1(ZG); the quotient K1(ZG) of K1(ZG) by this is thus torsion-free, and the standard involution acts trivially on it. Hence any finite CW complex which is a Poincarécomplex with fundamental group G of odd formal dimension is weakly simple. 0 We have a (non-canonical) splitting W h(G) = K1(ZG) ⊕ SK1(ZG), and the Rothenberg exact sequences relating the L0 groups and the Ls groups have the other term derived from SK1(ZG). Calculations of SK1(ZG) have been given for many cases in papers by Bob Oliver, and surveyed by him in [35]. For G a P−group, it is shown there that the odd torsion in SK1(ZG) vanishes, and that the even torsion is a sum of κG copies of C2, and a recipe is given for κG. In particular, SK1(ZG) vanishes for G of type I and for the groups G1 in the notation of §3; however for ∼ G = Cm × Q2k , κ(G) is 1 less than the number of divisors of m. 0 0 If M → M is a homotopy equivalence defining δ ∈ K1(ZG), we will have ∆(M) = ∆(M 0).δ2. It remains to consider the case when M and M 0 are normally cobordant. The normal cobordism gives a homotopy class of maps M → G/T op. Such classes form a finite group, and it is convenient to consider the even and odd localisations separately. The p−torsion is detected on restriction to a Sylow p−subgroup. If p is odd, Gp is cyclic, so by the results for cyclic groups, the normal cobordism class is detected by ρ(M). The 2-localisation of G/T op is homotopy equivalent to a product of Eilenberg-MacLane spaces, with fundamental classes 4i ˆ 4i+2 `4i ∈ H (G/T op ; Z(2)), k4i+2 ∈ H (G/T op ; Z/2). Thus if `4i or k4i+2 lifts to a class in the cohomology of BT op it determines a characteristic class for topological manifolds which detects the corresponding normal invariant. It was shown in [7, Theorem 9.9] that if i is not a power of 2, k4i−2 does so 4i lift and so does the image of `4i in H (G/T op ; Z/2); and perhaps these classes `4i themselves lift. According to [28, 10.13] the map j∗ : F∗(BO) ⊗ F∗(G/T op) → F∗(BT op), where F∗(?) denotes the quotient of H∗(? : Z) by its torsion subgroup, is ˆ surjective; and by [28, 10.9], F∗(BT op)⊗Z(2) is the tensor product of the polynomial algebra on classes y4n (α(n) − 4 < ν2(n)) generating the image of F∗(BSO(2)) in these dimensions and a divided polynomial algebra on the spherical classes x4n in the remaining dimensions: here α(i) counts the number of non-zero terms in the dyadic expansion of i. This in turn suggests that `4i lifts to BT op if and only if α(i)−4 ≥ ν2(i) (thus for i = 15, 23, 27, 29, 31,...). But we also have the calculations in [28, §13]; in particular Theorem 13.20 shows that the map E∞(BO × G/T op) →

23 E∞(BT op) of the limit term of the mod 2 Bockstein spectral sequences is surjective. Another approach to defining invariants is to note that the 2-localisation is detected on the Sylow subgroup G2, and compare with a model F−representation of G2 to define a map φ : Xg(G2) → G/T op. One ambiguity here is the choice ∗ of model, and even for G2 cyclic I do not know a computation of the classes φ `4i ∗ and φ k4i+2 arising from comparing two different homotopy equivalent F−representations: only the comparison of maps to BO is easy. k For G2 ≡ C2k , choices were made in [42, §13E] giving invariants t4i ∈ Z/2 N for 1 ≤ i < 2 . As in the case of the odd torsion, however, these invariants are not entirely independent of the values of ρ(G, M). It has been shown recently by Macko and Wegner [24] that (for G = C2k ) in addition to ρ(G, M) invariants ∗ min(k,2i) N ti ∈ Z/2 are required for 1 ≤ i < 2 ; in [25] they extend the result to cyclic groups of arbitrary order, and make some progress towards an explicit choice of invariants. We have already considered the effect on our invariants of changing the finite complex within its homotopy type and of changing the normal invariant. It remains to consider the next term in the surgery sequence. For space-forms of dimension 0 ∼ ≡ 1 (mod 4) the groups G are easy to handle and we have L2(G) = Z/2, given by the Arf invariant. Otherwise we have L0(G), with torsion-free part detected by the multisignature, and (by definition) this acts by adding to the ρ invariant. Thus only the torsion term T ors(L0(G)) remains. ∼ The case G = SL2(p) was studied by Laitinen and Madsen [20]. They show along these lines that the number of manifolds with given ρ and N is bounded by s the cardinality of T orL0(G), and proceed to make explicit calculations. They show that if p < 47 and p 6= 17, 41, the weak simple h-cobordism type of X is determined by ∆, N and ρ, but for p = 17, 41 further invariants are needed.

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