THE SPHERICAL SPACE FORM PROBLEM by DAVID WILLIAM MURRAY B.Sc, Memorial U n i v e r s i t y of Newfoundland, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1978 (c) David William Murray, 1978 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M A T H E M A T I C S The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date A P R I L 19, l__^ 2, and l e t K be a r e a l number. Then M n i s complete, connected, and of constant curvature K i f f M n i s isometric to a quotient S n / iT i f K > 0 R n/ir i f K = 0 Hn/-rr i f K < 0 where S n denotes the n-sphere, R n Euclidean ri-space, H n hyperbolic n-space, and where TT i s a proper group of isometries of the host manifold. R e c a l l that IT acts properly on a space X means that each point of X has a neighbourhood U s.t. for a l l f e Tr, f =f 1 , we have f(U) n U = 0 . Remark: An obvious consequence of the statement of the H o p f - K i l l i n g theorem i s that TT = Tr^(Mn) . Here, of course, the condition n >_ 2 i s e s s e n t i a l . - 3 -Evidently, the theorem of H o p f - K i l l i n g reduces the f i n d i n g of a l l space forms to a group-theoretic problem, the determination of a l l TT which can act properly on S n, Rn, or H n by isometries. A group-theoretic approach to the s p h e r i c a l space form problem, i . e . the case K > 0, was formulated i n 1947 by Vincent, who obtained a p a r t i a l s o l u t i o n . A complete, e x p l i c i t s o l u t i o n to the C l i f f o r d -K l e i n s p h e r i c a l space form problem was f i r s t obtained by Wolf who e s s e n t i a l l y undertook Vincent's general program, supplemented by c e r t a i n group-theoretic r e s u l t s due to Zassenhaus and Suzuki. By achieving a synthesis of these r e s u l t s , Wolf solved the C l i f f o r d -K l e i n s p h e r i c a l space form problem. In t h i s t h e s i s , we s h a l l be concerned only with the sphe r i c a l space form problem. The Euclidean and hyperbolic space form problems are at present unsolved; however, a compendium of known fact s concerning these problems, as well as some "constant curvature" r e s u l t s f o r pseudo-riemannian manifolds and c e r t a i n symmetric spaces may be found i n the book by Wolf [23]. Note that, since S n i s compact, a group rr acts property on S n i f f TT i s f i n i t e and acts fixed point f r e e l y . Bearing t h i s i n mind, we see that the theorem of H o p f - K i l l i n g reduces the f i n d i n g of a l l s p h e r i c a l space forms to the question of fin d i n g a l l f i n i t e groups which can act f i x e d point f r e e l y on S n by isometries. Suppose, however, that we wish to consider f i n i t e groups which act fixed point f r e e l y on S n, but which do not act by isometries. Accordingly, we can generalize the C l i f f o r d - K l e i n s p h e r i c a l space form problem to ask which f i n i t e groups can act fixed point f r e e l y on S n by homeomorphisms - 4 -(top o l o g i c a l s p h e r i c a l space form problem), by piecewise l i n e a r homeo-morphisms (PL sphe r i c a l space form problem), by diffeomorphisms (smooth spherical space form problem), or, more generally, which f i n i t e groups can act fixed point f r e e l y on a space having the homotopy type of a sphere (homotopy sp h e r i c a l space form problem). In t h i s way, we obtain a hierarchy of d i f f e r e n t s p h e r i c a l space form problems, each admitting, a p r i o r i at l e a s t , d i f f e r e n t c l a s s i f i c a t i o n s of "space forms" as s o l u t i o n . Concerning t h i s more general question of the free action of a f i n i t e group TT on a sphere: Certain necessary conditions f or free actions were determined i n 1944 by Smith [11] and i n 1957 by Milnor [5]. These may be stated as conditions on the structure of c e r t a i n subgroups of TT . D e t a i l s s h a l l be given i n l a t e r chapters. Recently, Madsen, Thomas, and Wall [4] have shown, using techniques of surgery, that these necessary conditions taken together are i n fact s u f f i c i e n t , thus solving a long-standing problem. Our in t e n t i o n , i n t h i s t h e s i s , i s to study aspects of both the " c l a s s i c a l " and the "generalized" sp h e r i c a l space form problem, d i s -cussing the major r e s u l t s , as well as the diverse techniques used to obtain them. This should i l l u s t r a t e one of the most noticeable features of t h i s problem, i t s heterogeneous nature, bringing together as i t does d i f f e r e n t i a l geometry, c l a s s i c a l group theory, homological algebra, as well as algebraic and d i f f e r e n t i a l topology. It i s hoped that the end r e s u l t w i l l be a c l e a r , systematic presentation of much of the current knowledge of the s p h e r i c a l space form problem. We mention i n passing that, j u s t as i n the sph e r i c a l case, the C l i f f o r d - K l e i n Euclidean and hyperbolic space form problems may be - 5 -generalized to the t o p o l o g i c a l case, and that, since R and H are homeomorphic, the top o l o g i c a l Euclidean space form problem and the topo l o g i c a l hyperbolic space form problem are one and the same. We s h a l l make no further discussion of t h i s problem, but for a survey of r e s u l t s i n t h i s area, one may r e f e r to Wall [20]. - 6 -. 'CHAPTER 1: GROUP ACTIONS AND COHOMOLOGY I . BASIC DEFINITIONS In i n v e s t i g a t i n g the a c t i o n of f i n i t e groups on spheres, notions of group cohomology play a c e n t r a l r o l e . In t h i s s e c t i o n , we s h a l l introduce v a r i o u s ideas from group cohomology theory which s h a l l be fr e q u e n t l y used i n subsequent s e c t i o n s . R e c a l l the f o l l o w i n g . D e f i n i t i o n 1.1. Let G be a f i n i t e group and l e t A be a G-module. For a l l n ^ 0, we def i n e H n(G;A), the n*"*1 cohomology group of G w i t h c o e f f i c i e n t s i n A, by H n(G;A) = Ext^(Z,A), where Z has t r i v i a l G-module s t r u c t u r e . t h S i m i l a r l y , we de f i n e H n(G;A), the n homology group of G Q w i t h c o e f f i c i e n t s i n A, by H (G;A) = Tor (Z,A) . n n In t h i s chapter and i n chapter 3 the concept of norm w i l l appear. D e f i n i t i o n 1.2. Let G be a f i n i t e group, and l e t Z(G) denote the i n t e g r a l group r i n g . We d e f i n e the norm element, N e Z(G), by N = I x . xeG I f A i s a G-module, we de f i n e the norm homomorphism N: A —>• A by Na = £ xa . xeG We a l s o d e f i n e the norm of a homomorphism. Let A and C be G-modules. Define N: Horn (A,C) »- Horn (A,C) by (Nf) (a) = £ x f ( x _ 1 a ) , xeG where f e Hom„(A,C) . Note that i f f e Hom„(A,C), then Nf = (order G)-f. L b Now, l e t I denote the augmentation i d e a l of Z(G) . Then we have - 7 -Q I'A 5_ ker{N: A —>• A} as well as in N c A . Consequently, the norm * G homomorphism N: A — y A induces N : A — y A , where A = A/I«A and A G = {a e A|ga = a V g e G } . Noting that H ° ( G ; A ) = A G and H ( G ; A ) = A_, we have defined N*: H Q ( G ; A ) y H ° ( G ; A ) . D e f i n i t i o n 1.3. We define the Tate cohomology H ( G ; A ) of G with c o e f f i c i e n t s i n the G-module A as follows: H N ( G ; A ) = H N ( G ; A ) -for n > 0 H ° ( G ; A ) = coker{N*: H Q ( G ; A ) — > H ° ( G ; A ) } = AG/N-A H _ 1 ( G ; A ) = ker{N*: H Q ( G ; A ) — y H ° ( G ; A ) } = ker{N: A — A } . / I H N ( G ; A ) = H , ( G ; A ) for n < - 1 . —n-± "-2 "-1 "0 "1 "2 I t follows that . . . . H , H , H , H , H ,... forms an exact connected sequence of functors, often referred to as the complete derived sequence of G . As implied above, we s h a l l denote by H ( G ; A ) the graded module which includes a l l elements of the complete derived sequence. Tate cohomology may also ,be defined d i r e c t l y , by means of complete re s o l u t i o n s . D e f i n i t i o n 1.4. A complete r e s o l u t i o n X for a f i n i t e group G i s an d n d 0 exact sequence .... >• X >- X n — > . . . > X„ y X ^ y . . . y X n n-1 0 -1 -n of f i n i t e l y - g e n e r a t e d p r o j e c t i v e G-modules together with an element Q e e C^_^) s.t. e generates im d^ . - 8 -Remark 1.5. This l a t t e r condition implies that the map d^ admits a f a c t o r i z a t i o n X^ y Z y X _ l ' w n e r e e i s a G-epimorphism and u i s a G-monomorphism. Consider the exact sequences (1.6) — y X — X 1 y ... • Xn Z 5-n n-1 0 (1.7) 0 — y Z X , — • ... — y X ,. — y X — y -1 -n+1 -n (1.6) i s a G-projective r e s o l u t i o n of Z, and the "dual" of (1.7) — • X° — y X° ,. — • ... — • x°. — y Z — y 0 -n -n+1 -1 i s as w e l l . Here, X^\ = Hom(X_^,Z) . Conversely, given (1.6) and (1.7), one may s p l i c e these sequences as follows i n order to obtain a complete r e s o l u t i o n of G . — y X — y ... — y X„ ^ X , — y X „ — y ... — > X — y n 0 - 1 - 2 - i i z Remark 1.8. Let X be a complete r e s o l u t i o n of the f i n i t e group G, and l e t A be a G-module. Then the Tate cohomology Hn(G;A) may be computed as H n(Hom„(X,A)) . Let us conclude t h i s section by sta t i n g an elementary lemma and giving a d e f i n i t i o n , both of which we s h a l l soon use. - 9 -Lemma 1.9. If G i s a f i n i t e group of order r , and A i s a G-module then rH(G;A) = 0 . D e f i n i t i o n 1.10. Let TT be a subgroup of the f i n i t e group G; l e t X be a complete r e s o l u t i o n of G . Then the i n c l u s i o n i : Hom_(X,A) — • Horn (X,A) induces a homomorphism, c a l l e d r e s t r i c t i o n , ft " i : H(G;A) > H(TT;A) . f Define a homomorphism t: Horn (C,A) > Horn (C,A) by ( t f ) ( c ) = £x.f(x.^c), where A and C are G-modules, f e Horn (C,A), i and where X^TT, . . . >x IT are the l e f t cosets of TT i n G . Replacing C by a complete r e s o l u t i o n X of G and passing to cohomology, we have the induced map, c a l l e d the transfer homomorphism, ft * t : H(TT;A) >- H(G;A) . I I . PERIODICITY The most important concept, for our purposes, i n t h i s chapter, i s the concept of p e r i o d i c i t y . Our main theorem (1.18) r e l a t e s the notion of p e r i o d i c i t y to free actions of groups on spheres. Remark 1.11. In the proof of theorem 1.13, which follows, we assume f a m i l i a r i t y with the cup product for Tate cohomology. The cup product i s "a homomorphism ty: H P(TT;A) 0H^(TT;B) —> H P + <^(TT;A®B) , where TT i s a f i n i t e group and A and B are Tr-modules. For the d e f i n i t i o n and properties of ty, see [2, ch. 12]. We s h a l l denote ty(aGb) e H P + q(rr;A©B) by a'b . - 10 -D e f i n i t i o n 1.12. An element g e H (G;Z) i s c a l l e d a maximal generator i f i t i s a generator and has order |G|, where |G| denotes the order of G . Theorem 1.13. Let g e H Q(G;Z) . The f o l l o w i n g are e q u i v a l e n t : (a) g i s a maximal generator (b) the order of g i s |G| (c) there e x i s t s h e H _ Q(G;Z) s . t . h-g = 1 ^ XI ^ XI I Q (d) the map a 1—> a*g i s an isomorphism H (G;A) >• H (G;A) f o r a l l n and A . Proof: (a) => (b) Obvious (b) =>. (c) Let g e H Q(G;Z) have order r = |G| . By lemma 1.9, the order of any element of H (G;Z) d i v i d e s r . Thus, there e x i s t s a map c(> : H Q(G;Z) • Z__ s . t . (g) = 1 • Then the d u a l i t y theorem H _ Q(G;Z) = Hom(H Q(G;Z),Z ) [2, ch. X I I , thm. 6.6] i m p l i e s that there e x i s t s h e H Q(G;Z) w i t h h«g = (j)(g) = 1 . (c) => (d) Consider H N(G;A) H n + q(G;A) H N(G;A) defined by a(a) = a«g, B(a) = a«g 1 . Then ag = 1 and 3a = ( - l ) q , and i t f o l l o w s that a and 3 are isomorphisms. (d) => (a) Consider the isomorphism H^(G;Z) > H Q(G;Z) given by a y a*g . Since H^(G;Z) i s c y c l i c of order |G|, so i s H Q(G;Z) . Since H°(G;Z) i s generated by 1, H Q(G;Z) i s generated by g • Remark 1.14. I f g e H Q(G;Z) i s a maximal generator, so i s g" 1 e H _ Q(G;Z) . - 11 -"S QH~S If h e H ( G ; Z ) i s another maximal generator, then g-heH ( G ; Z ) i s also a maximal generator. D e f i n i t i o n 1.15. An integer q i s c a l l e d a period for the group G i f H ^ ( G ; Z ) contains a maximal generator i . e . i f H ^ ( G ; Z ) i s c y c l i c of order |G| . Remark 1.16. Anti-commutativity of the cup product implies that periods are even. Proposition 1.17. If G has period q, so does every subgroup TT of G . In f a c t , i f g e H ^ ( G ; Z ) i s a maximal generator, then so i s ft ft i (g) £ H ^ ( T T ; Z ) , i being the r e s t r i c t i o n homomorphism. Proof: F i r s t , i t i s a straightforward consequence of d e f i n i t i o n 1.10 * * that t i (g) = [ G : - r r ] g , where [ G : T T ] denotes the index of TT i n G . i i * Now [ G : T r ] g has order | TT | , and therefore i (g) has order at le a s t | T T| . But, by lemma 1.9, each element of H ^ ( T r ; Z ) has order d i v i d i n g I TT | . Thus, i (g) has order p r e c i s e l y | TT | , so i (g) i s a maximal generator for H ^ ( 7 r ; Z ) by theorem 1.13. We s h a l l now conclude chapter 1 by st a t i n g and proving our main r e s u l t . Theorem 1.18. Let TT be a f i n i t e group acting fixed point f r e e l y on a space £ n having the homotopy type of a sphere S n . Then TT has peri o d i c cohomology of period (n+1) . Lemma 1.19. Let rr be a f i n i t e group acting f i x e d point f r e e l y on a space £ n having the homotopy type of the sphere S n, and assume n - 12 -Is even. Then, either T T = { 1 } , or TT = Z^ . Proof: Any map f: E n —>- £ n has Lefschetz number A(f) = l + ( ~ l ) n deg f. Then the Lefschetz fixed-point theorem says that i f f has no f i x e d points, then 1 + ( - l ) n deg f = 0 . So i f n i s even, deg f = -1, and f i s o r i e n t a t i o n reversing. So every s e T T , S f 1, reverses o r i e n t a t i o n of S n . So i f s ^ S j e u, s^ ^ 1, i1 1, then both s^ and reverse o r i e n t a t i o n and thus S-^S2 preserves o r i e n t a t i o n . Hence S-^S2 = Since t h i s r e l a t i o n holds for a l l elements of T T , i t follows that TT = Z^. Proof of theorem 1.18. By lemma 1.19, we need only consider the case i n which n i s odd. In t h i s case, each element of rr i s o r i e n t a t i o n -preserving . Let be the t o t a l space of the universal p r i n c i p a l TT-bundle; E ^ may be obtained by taking the i n f i n i t e j o i n E ^ = T T * T T * s u i t ably topologized. There ex i s t s a free (right) action of rr on E ^ given by (t,x. + ... + t x ,y) = t.x^y + ... + t x y, where x, ,. . . ,x ,y e T T , 1 1 n n 1 1 n n I n Et. = 1 . The c l a s s i f y i n g space B = E . / 7 T i s a K ( T T , 1 ) . Now, consider £ n x E . There e x i s t s a free action of rr on TT E n x E ^ given by (u,v)«x = (u*x, v x ) , u e £ n , v e E ^ , x e TT . Denote the o r b i t space by E nx E . The projections E*1 x E —> E n ^ ^ IT Tr o 7 7 n I n and E x E — > E induce projections E x E > E / T T and - TT TT TT Tl 2 E x E >- B . Both 6., and 6~ are f i b r a t i o n s with f i b r e s E and n rr TT 1 I rr E n r e s p e c t i v e l y . To sum up, we have the following diagram: E n „n „ 1 „n, E • E x E > E / T T TT TT rr '2 K ( r r , l ) - 13 -Note that, since E i s c o n t r a c t i b l e , 0- induces isomorphisms TT 1 * n * n on cohomology H (E /T T) • H (£ ) . Using t h i s f a c t , together with the Gysin sequence of the f i b r a t i o n y i e l d s the following exact sequence: . . . . — H P ( T T ) -> H P ( E n / r r ) — H P _ n ( T r ) — H P + 1 ( T T ) — H P + 1 ( E n / r r ) * n But H (E / T T ) = 0 for k > n . Thus, for a l l i > 0, we have i . . _ i+n+1 H (TT) = H (rr) which i s enough to e s t a b l i s h p e r i o d i c i t y . - 14 -CHAPTER 2: THE SOLUTION OP THE CLIFFORD-KLEIN SPHERICAL SPACE FORM PROBLEM Recall that the c l a s s i c a l s p h e r i c a l space form problem requires the c l a s s i f i c a t i o n up to isometry of a l l complete, connected, Riemannian manifolds of constant curvature K > 0 . In the introduction, we stated that the f i r s t s o l u t i o n of t h i s problem was given by Wolf i n h i s book 'Spaces of Constant Curvature' [23]. In t h i s chapter, we give an account of Wolf's work. As a reference for d e t a i l s which at c e r t a i n points we s h a l l omit, we c i t e [23, chs. 5,6,7]. What we s h a l l do i s present a l l the main points of the argument, and attempt to make the basic procedure c l e a r . We s h a l l make constant use of the c l a s s i c a l representation theory of f i n i t e groups; i n p a r t i c u l a r , we s h a l l r e l y heavily on the notion of induced representation. That portion of group representation theory needed for the s o l u t i o n of the sphe r i c a l space form problem may be found, concisely stated, i n [23, ch. 4]. I. OUTLINE OF VINCENT'S PROGRAM We begin with the following reformulation of part of the Hopf-K i l l i n g theorem. Theorem 2.1. The isometry classes of complete, connected, Riemannian manifolds M of dimension n >_ 2 and constant curvature K > 0 are i n 1-1 correspondence with conjugacy classes of f i n i t e subgroups T of the orthogonal group 0(n+l) i n which only the i d e n t i t y element has +1 for an eigenvalue. Proof: The H o p f - K i l l i n g theorem implies that M i s isometric to S n / r , - 15 -where Y i s a proper group of isometries of S n . But the f u l l group of isometries of S n i s 0(n+l) ; thus T <=_ 0(n+l) . Since S n i s compact, r acts properly i f f T i s f i n i t e and acts f i x e d point f r e e l y . T acts f i x e d point f r e e l y means that f o r a l l g e T, g ^ 1, we have gx ^ x for a l l x e S n i . e . only l j , has +1 for an eigenvalue. If r = hAh ^ for some subgroup A c_ 0(n+l) and some h e 0(n+l), then h induces an isometry h: S n/T > S n/A by h(Tx) = A(hx) . ^ n n n n Conversely, i f h : S /V > S /A i s an isometry, and i f p: S >- S /T n n *^ n n and q: S >- S /A are the covering projections, then h p : S > S /A has a l i f t h: S n >• S n r e l a t i v e to the covering q: S n > S n/A . h i s an isometry of S n which conjugates T to A . From t h i s theorem we see that the C l i f f o r d - K l e i n s p h e r i c a l space-form problem has been reduced to the following two-step problem: (i) f i n d a l l f i n i t e subgroups V c_ 0(n+l) which can act fixed point f r e e l y on. S n ( i i ) solve the r e s u l t i n g conjugacy problem. D e f i n i t i o n 2.21 Let TT be a representation of a group G . If g e G, g ^ 1 implies that Tr(g) does not have +1 for an eigenvalue, then TT i s said to be fixed point f r e e . A f i x e d point free group i s an abstract f i n i t e group admitting a fix e d point free representation. We s h a l l generally abbreviate 'fixed point free' to ' f . p . f . ' Remark 2.3. ( i ) If TT i s equivalent to a f . p . f . representation, TT i s f . p . f . ( i i ) If TT i s f . p . f . , TT i s a d i r e c t sum of i r r e d u c i b l e f . p . f . representations. - 16 -( i i i ) If {TT.} i s a family of f.p . f . representations, then £ T K i i s f . p . f . (iv) If rr i s f . p . f . , then TT i s f a i t h f u l . G. Vincent's chief conceptual device was to regard V _c Q(n+1) as the image of a f a i t h f u l orthogonal representation p: G —> 0(n+l) of an abstract f i n i t e group G . We u t i l i z e t h i s idea i n the next theorem, which i s b a s i c a l l y a rephrasing of theorem 2.1 i n a more e x p l i c i t manner. Theorem 2.4. Given a f i n i t e group G and f . p . f . i r r e d u c i b l e orthogonal representations {a^,...,a^} of G, we associate the complete, connected, n-dimensional Riemannian manifold S n/(a^ $ . . . @ o^)(G) of constant r curvature K > 0, where (n+1) = £ (degree a.) . i = l 1 Every, complete, connected Riemannian manifold of constant curvature K > 0 i s isometric to a manifold obtained i n t h i s manner. (G,{a^,...,0^}) and (G,{x^,...,x }) give isometric manifolds i f f r = s and there e x i s t s a permutation k —* i ^ of {l,2,...,r} and an automorphism a: G —>• G s.t. T v " a ^ s equivalent to a. k X k Proof: The f i r s t statement i s true because of theorem 2.1 and the fa c t that every orthogonal representation i s equivalent to a d i r e c t sum of i r r e d u c i b l e orthogonal representations Theorem 2.1 says that (G, {a^,. . . , 0 " ^ } ) and ( G , { T ,. .. ,x s)) determine isometric manifolds i f f (a^ $ ... ® a )(G) and (x n €> ... © x )(G) are conjugate i n 0(n+l) . Therefore, i n order to prove the second statement, we note that, for orthogonal f a i t h f u l representations a and x, there e x i s t s f e 0(n+l) s.t. - 17 -T ( G ) = f > a ( G ) * f I f f there e x i s t s an automorphism a: G —»• G s.t. T * a i s equivalent to a . This can be seen as follows: F i r s t , the implication <= i s c l e a r . To prove =>, given such an f, define a: G —*• G by o(g) = T _ 1 ( f - a ( g ) - i f _ 1 ) . We are now i n a p o s i t i o n to describe the program undertaken by Vincent i n h i s approach to the sp h e r i c a l space form problem. We f i r s t f i n d necessary conditions on an abstract f i n i t e group G i n order that G be f . p . f . Then we c l a s s i f y the abstract f i n i t e groups s a t i s f y i n g these necessary conditions, thereby obtaining a family {G,} of groups. A Next, we determine the equivalent classes of i r r e d u c i b l e f . p . f . complex representations of each G^ . We use t h i s c l a s s i f i c a t i o n to f i n d the equivalence classes of i r r e d u c i b l e f . p . f . orthogonal representations. The f i n a l step consists i n determining the automorphism group of each G^, and deciding when two i r r e d u c i b l e f . p . f . orthogonal representations are equivalent modulo such an automorphism. Suppose we e x p l i c i t l y carry out each step of t h i s program. Then theorem 2.4 says that we have i n fact solved the C l i f f o r d - K l e i n s p h e r i c a l space form problem. I I . NECESSARY CONDITIONS ON FIXED POINT FREE GROUPS This section i s quite c e n t r a l to our presentation, providing the basis for much of our subsequent discussion. In t h i s section we determine necessary conditions, i n terms of the structure of c e r t a i n subgroups, that a f i n i t e group be f. p . f . We also determine a r e l a t i o n s h i p with the ideas of group cohomology introduced i n chapter 1. D e f i n i t i o n 2.5. Let G be a f i n i t e group and l e t p and q be primes, not n e c e s s a r i l y d i s t i n c t . If every subgroup of G of order pq i s - 18 -c y c l i c , we say that G s a t i s f i e s a l l pq-conditions. Theorem 2.6. Let F be a f i e l d of c h a r a c t e r i s t i c zero, and l e t G be a f i n i t e group which admits a f.p. f . representation over F . Then G s a t i s f i e s a l l pq-conditions. Proof: Let TT be a f . p . f . representation of G on a vector space V over the f i e l d F . Suppose H i s any n o n - t r i v i a l subgroup of G, and l e t v e V . Since Tr(h') £ Tr(h)v = ][ Tr(h)v and TT|H i s f . p . f . heH heH i t follows that I Tr(h)v = 0 . heH Now, suppose that H i s a non-cyclic subgroup of order pq, where p and q are primes with p <^ q . Let {S^,...,S^} be the proper subgroups of H . Note that k >_ 2, for i f p < q, then H has subgroups of order p and of order q, while i f p = q, then 2 H i s non-cyclic of order p , i . e . H = Z^ © Z , and so there e x i s t two subgroups of order p . Now, since each S^ i s c y c l i c of order p or q, i f h e H, h 4 1, then h belongs to p r e c i s e l y one of 'k k the S. . Therefore, the i d e n t i t y £ £ T T ( S ) V = £ 0 = 0 may be 1 i = l seS. i = l x rewritten as \ Tr(h)v + kv = 0 . Subtracting £' Tf(h)v = 0, we obtain heH heH h * l ( k - l ) v = 0 for a l l v e V . But F i s of c h a r a c t e r i s t i c zero, and, since k >_ 2, (k-1) 4- 0 . Hence we must have v = 0 for a l l v e V But t h i s contradicts the f a c t that G has a f . p . f . representation on Remark 2.7. Theorem 2.6 i s also v a l i d i n c h a r a c t e r i s t i c p, but we s h a l l not use t h i s f a c t . 2 By theorem 2.6, a f . p . f . group s a t i s f i e s a l l p -conditions. The - 19 -2 p -condition i s a p a r t i c u l a r l y u s eful c r i t e r i o n , as i s shown by the next theorem. Theorem 2.8. Let G be a f i n i t e group. The following are equivalent: (i ) G has periodic cohomology. 2 ( i i ) G s a t i s f i e s a l l p -conditions, ( i i i ) Every abelian subgroup of G i s c y c l i c , (iv) Every p-subgroup i s c y c l i c i f p i s an odd prime. Every 2-subgroup i s ei t h e r c y c l i c or generalized quaternionic. (v) Every Sylow p-subgroup i s c y c l i c i f p i s an odd prime. Every Sylow 2-subgroup i s ei t h e r c y c l i c or generalized quaternionic. 2 Proof: ( i ) => ( i i ) : I f G does not s a t i s f y a l l p -conditions, G has a subgroup of the form £B Z^ . If G has period q, so does each subgroup of G, by (1.17); thus i t s u f f i c e s to show that Z^ © Z^ has no period. Consider homomorphisms Z >• Z ® Z > Z s.t. P P P P f3a = l z • Passing to cohomology, we have P * ft (2.9) H q(Z ,Z) H q(Z ©Z ,Z) ^ - y H q(Z ,Z) s.t. a*g* = 1 . P P P P Now, as i s well known, H q(Z^,Z) = Z^ for q even. Thus, by (2.9), H q(Z ® Z ,Z) has a d i r e c t summand which i s c y c l i c of order p . So, for q even, H q(Z^©Z^,Z) cannot be c y c l i c of order p That i s , there does not ex i s t a maximal generator for H q(Z^®Z^,Z) and so Z © Z i s not p e r i o d i c . P P ( i i ) •*> ( i i i ) : Let A be an abelian subgroup of G . A non-c y c l i c implies that A contains a subgroup of the form Z © Z , 2 contradicting the f a c t that G s a t i s f i e s a l l p -conditions. Conversely, - 20 -we note that any group of order p i s abelian. ( i i ) => (iv) : Let rr be a p-subgroup of G . Then the center of rr i s n o n - t r i v i a l . Let a be a n o n - t r i v i a l c e n t r a l element; then a has order a power of p, say p , and a s u i t a b l e power of a, c a l l t h i s element f3, has order p and generates a c e n t r a l c y c l i c subgroup TT 1 of order p . We claim that rr' i s the only subgroup of order p . For, i f rr" i s another such subgroup of rr, we c l e a r l y have rr1 n rr" = {1}, and so, since TT' i s c e n t r a l , TT' © rr" i s a subgroup 2 of TT . But rr' © TT" i s a non-cyclic abelian group of order p , c o n t r a d i c t i o n ( i i ) . Thus rr has only one subgroup of order p . But any group s a t i s f y i n g t h i s condition i s e i t h e r c y c l i c or generalized quaternionic [24, p. 118]. (iv) => (v): Obvious. (v) => ( i ) : I t i s well known that c y c l i c groups and generalized quaternionic groups are p e r i o d i c , the former of period two, the l a t t e r of period four. Thus, we l e t IT^,. . . , b e Sylow subgroups corresponding to the primes p ',. . . ,p which d i v i d e |G|, and we assume that rr. has X S X period q^ and maximal generator g^ e H (rr^;Z) . One can prove the existence of an integer u, a common multiple of q^,...,q s.t. t * ( g i U / ' C 1 i ) e H U(G;Z;p i) has order | TT_J . Here H^GjZjp^) denotes the p_^-primary component of H U(G;Z) . Then the sum of these elements i s a maximal generator i n H U(G;Z), and so G has period u . D e f i n i t i o n 2.10. We s h a l l r e f e r to a group s a t i s f y i n g the equivalent conditions of theorem 2.8 as p e r i o d i c . - 21 -I I I . THREE CRUCIAL LEMMAS In t h i s section we state and prove three lemmas which are v a l i d for a l l f . p . f . groups. Although these lemmas are easy, they are the most important tools i n the determination of f . p . f . representations of f . p . f . groups, and t h i s i s why we devote a separate section to them. The f i r s t two lemmas r e l a t e f . p . f . and induced representations; the t h i r d r e l a t e s f.p . f . representations and external tensor products. We s h a l l see i n section V that these two operations, induced represent-at i o n and tensor product of representations, allow us to get a l l f.p . f . representations of a l l f . p . f . groups. Further comment i s deferred u n t i l section V . Lemma 2.11. Let H be a normal subgroup of index k i n a f i n i t e group k G ; G = u b^H, b^ e H . Let a be an i r r e d u c i b l e complex representation i = l of H and define o,..: h •—> a (b. "'"hb.) , 1 < i < k . Then the induced ( l ) x x - -Q representation a of G i s i r r e d u c i b l e i f f the a(-j_) a r e mutually inequivalent representations of H . Proof: Let TT = a° . Then rr |H = a . © . . . @ o . . Let V = V . , © . . . ® 1 (1) (k) 1 be the representation space of T T , where V i s the representation space of ' Suppose the °(±) a r e mutually inequivalent. If U i s a T T ( G ) - i n v a r i a n t subspace of V , U i s of the form V . @ . . . © V . , where i ^ < ... < i ^ , since rr(H)-U = U . But, for a l l i , £_ U, and i t follows that V = U . Therefore, TT i s i r r e d u c i b l e . Conversely, suppose TT i s i r r e d u c i b l e . If a n d a r e t n e characters of TT and ° f±~) r e s p e c t i v e l y , then one can ca l c u l a t e = 1 + 2/k £ X.> ( I f a>$: G —^ denotes ^ ^ i = 0 for 1 < j geG 1 3 and so the a a r e mutually Inequivalent. Lemma 2.12. Let TT be a representation of a f i n i t e group G; l e t H be a subgroup of G the order of which i s d i v i s i b l e by each prime d i v i s o r of |G| . Then (a) TT i s f . p . f . i f f rr r e s t r i c t s to a f. p . f . representation on each subgroup of prime order i n G . 2 (b) I f G s a t i s f i e s a l l p -conditions, the following are equivalent: ( i ) TT i s f.p.f . ( i i ) If p i s a prime s.t. p||G|, then rr i s f . p . f . on some subgroup of order p . ( i i i ) TT i s f . p . f . on H . Proof: Let g e G , g £ 1 . Note that g~ has prime order f o r some integer t . Then, to prove (a) , we need only recognize that i f rr(g) has +1 for an eigenvalue, so does rr(g t) . 2 Suppose G s a t i s f i e s a l l p -conditions. Then, two subgroups of the same prime order are conjugate, proving the equivalence of ( i ) and 2 ( i i ) of (b). Also, H s a t i s f i e s a l l p -conditions and contains a conjugate of every prime order subgroup of G by assumption. This proves the equivalence of ( i i ) and ( i i i ) of (b). Lemma 2.13. Let a and 3 be complex representations of the f i n i t e groups A and B res p e c t i v e l y . (a) a ® 3 i s a f. p . f . representation of A x B i f f - 23 -( i ) a i s f . p . f . representation of A ( i i ) 3 i s f . p . f . representation of B ( i i i ) |A| i s r e l a t i v e l y prime to | B | (b) a ® 3 i s i r r e d u c i b l e i f f a and 3 are i r r e d u c i b l e (c) every i r r e d u c i b l e representation of A x B i s an e x t e r i o r tensor product of the form a (8> 3 • Proof: Let a <8> 3 be f . p . f . d^ and d^ being the degrees of a and 3 r e s p e c t i v e l y . Then a ® 3 | A - r i 1 = d -a and a ® Bj f 1 , „ = d * J 1Ax{l} 6 M 1{1}XB a Let p be a prime s.t. p||A|,p||B| . Choose a e A and b e B to be of order p . We replace a and b by suitable powers s.t. a(a) i 2iTi/p , „„ x . , -2Tri/p _ has eigenvalue e and 3(b) has eigenvalue e . Then r « „ \ / i \ • i 2iTi/p -2Tri/p , , . . , (a®3)(a,b) has eigenvalue e • e = 1 , contradicting the f a c t that a ® 3 i s f . p . f . Thus the orders of A and B are r e l a t i v e l y prime. Assume ( i ) , ( i i ) , and ( i i i ) . If x e A x B has prime order p, then either x e A or x e B, by condition ( i i i ) , so (a®3)(x) cannot have 1 for an eigenvalue by ( i ) and ( i i ) . Now a ® 3 i s f . p . f . by lemma 2.12. This proves (a). Statements (b) and (c) are well-known. For a proof, see, for example [8, theoreme 10]. IV. VINCENT'S PARTIAL SOLUTION OF THE SPHERICAL SPACE FORM PROBLEM A. C l a s s i f i c a t i o n of Fixed Point Free Groups Having A l l Sylow Subgroups C y c l i c : Theorems 2.6 and 2.8 y i e l d two classes of f . p . f . groups: ( i ) those i n which a l l Sylow subgroups are c y c l i c and ( i i ) those i n which the - 24 -Sylow p-subgroups are c y c l i c for p an odd prime, while the Sylow 2-subgroups are generalized quaternionic. The case of the simplest type of f.p . f . group, class ( i ) , w i l l now be treated i n d e t a i . This was the case considered at length by Vincent. The general case w i l l be treated i n section V, but usually d e t a i l s w i l l not be included i n the discussion. However, t h i s s p e c i a l case i s i n d i c a t i v e of Wolf's general procedure, and, i n any case, r e s u l t s i n the c l a s s i f i c a t i o n of s p h e r i c a l space forms of dimension k t 3 (4) . Those f i n i t e groups having a l l Sylow subgroups c y c l i c are completely characterized by the following c l a s s i c a l theorem of Burnside. The statement concerning pq-conditions i s due to Zassenhaus. Theorem 2.14. Let G be a f i n i t e group of order N i n which every Sylow subgroup i s c y c l i c . Then G i s generated by two elements A and B subject to the r e l a t i o n s ,m „n , 1 .r A = B =1, BAB = A , N = mn (2.15) ((r-l)n,m) =1, r 1 1 = 1 (m) . Let d be the order of r i n K , the "group of residue classes modulo m . Then d|n . Furthermore, G s a t i s f i e s a l l pq-conditions i f f n' = n/d i s d i v i s i b l e by every prime d i v i s o r of d . Conversely, any group defined by the r e l a t i o n s (2.15) has order N and has every Sylow subgroup c y c l i c . B. Representation Theory of F i n i t e Groups Having A l l Sylow Subgroups C y c l i c Let us b r i e f l y r e c a p i t u l a t e . We have found that i n order that a f i n i t e group G be f . p . f . , i t i s necessary that G be p e r i o d i c . We - 25 -have r e s t r i c t e d ourselves to the simplest type of pe r i o d i c groups, those having a l l Sylow subgroups c y c l i c , and, i n theorem 2.14, we have given an e x p l i c i t d e s c r i p t i o n of these groups i n terms of generators and r e l a t i o n s . R e c all that the next step i n Vincent's program consists i n studying the representation theory of such groups i n an e f f o r t to c l a s s i f y the f.p.f . orthogonal representations. In the following theorem, we r e t a i n the notation of 2.14. Also, when we speak of a "representation" without a d d i t i o n a l q u a l i f i c a t i o n , we mean "finite-dimensional complex representation". The main point of the following c l a s s i f i c a t i o n theorem i s that a l l f . p . f . complex representations of G turn out to be induced from f . p . f . representations of c e r t a i n subgroups H, defined below. As we s h a l l see i n section V of t h i s chapter, such behaviour i s quite t y p i c a l . Theorem 2.16. Let G be a group of order N having a l l Sylow subgroups c y c l i c and s a t i s f y i n g a l l pq-conditions. Let H be the subgroup of G generated by A and B d . (a) H i s a c y c l i c normal subgroup of G of order mn' generated by AB d . (b) The i r r e d u c i b l e complex representations of H are of degree J • x. /-AT>CU 2Trik/mn' • . ., _ one and are given by a^(AB ) = e i . e . the vector space auto-/.r,ds . , i . . , . , 2Tfik/mn' morphism o^CAB ) i s scalar m u l t i p l i c a t i o n by e Let 71^ . be the representation of G induced by o^, nec e s s a r i l y of degree d . Then (c) TT, i s equivalent to TT i f f k = £(n') and k = r U£(m) , where 0 <_ u < d . (d) The following conditions are equivalent: - 26 -( i ) TT i s an i r r e d u c i b l e f a i t h f u l representation of G . ( i i ) TT i s an i r r e d u c i b l e f . p . f . representation of G . ( i i i ) TT i s equivalent to some where (k,N) = 1 . (e) The i r r e d u c i b l e f . p . f . representations of G a l l have degree 2 d; the number of such representations i s 0(N)/d , 0 Euler's 0-function. Proof: Induction shows that B d commutes with A, and i t follows that {Bd} i s the centre of G; consequently, H = {A,Bd} i s normal i n G . Since A has order m, B d has order n', and (m,n) = 1, i t follows that H has generator AB d . That i s H i s the c y c l i c group {AB d}, as claimed. It i s c l e a r that the are the i r r e d u c i b l e complex represent-ations of H . Note that o^ i s f a i t h f u l i f f (k,|H.|) = 1 i . e . (k,mn') = 1, and since every prime d i v i s o r of d also divides n', we have (k,mn') = 1 i f f (k,N) = 1 . So o i s f a i t h f u l i f f (k,N) = 1 . Define b. = B ~ 1 + 1 , 1 < i < d . Then G = b^H u ... u b„H, with l — — 1 2 b e H . Define CL ,.. (h) = a, ( b T ^ b O . Then a. ,..(ABd) = I 3 • _i k ( j ; a k ( B J AB JB ) = a k(A B ) = a 3 , where a = a (A), 3 = a±(B ). Now, a i s a p r i m i t i v e m*"*1 root of unity, while 3 i s a p r i m i t i v e n ' ^ root of unity and (m,n') = 1 . Thus i t i s easy to see that i f (k,m) = 1, then { ^ ( i ) » • • • »°k(d) ^ a r e m u t u a x - ' - y inequivalent, and d k k secondly, w r i t i n g ^ ( A B ) as a 3 , i t i s immedicate that i s equivalent to k = fc(n') and k = r^ "*"£(m) . Now, we have seen that o^ i s f a i t h f u l i f f (k,N) = 1 . This f a c t , combined with lemmas 2.11 and 2.12 imply that TT^ i s i r r e d u c i b l e and f . p . f . i f (k,N) = 1, and TT^ i s not f a i t h f u l , therefore not f . p . f . , i f (k,N) ^1 - l i -lt remains only to prove ( i ) => ( i i i ) of (d) and to prove (e). Let TT be a f a i t h f u l i r r e d u c i b l e representation of G with represent-ation space V . Let h e H ; Tr(h) e GL(V) i s diagonalizable. So there e x i s t s a d i r e c t sum decomposition V = 0 V ^ s.t. each rr(h) acts by scalars on but not on any © V with i ^ j . Now, 77(G) permutes the V_^ ; the r e s u l t i n g permutation group i s t r a n s i t i v e on the set {V.}, since 77 i s assumed to be i r r e d u c i b l e . Now, a 1 standard r e s u l t of group representation theory [23, p. 142] states that, Q under these circumstances, 77 = T , where x i s the representation of Thus L = {g e G|Tr(g)V1 = V } on v 1 . ' One shows that i n f a c t L = H Q 77 = a where a i s the representation of H on . Furthermore, G G i f a = a' © a", then 77 = a'! © a" , contradicting i r r e d u c i b i l i t y of 77 , So a i s i r r e d u c i b l e and therefore a = a, for some k . Let k K = ker • Since H i s c y c l i c , K i s stable under every auto-morphism of H . Thus K _c ker 77 . But i r i s f a i t h f u l ; hence K = 0, 0 " ^ i s f a i t h f u l and so (k,N) = 1 . Thus 77 i s equivalent to some 77 with (k,N) = 1, proving ( i i i ) . • To v e r i f y (e) , we count the number of inequivalent f a i t h f u l T T ^ . There are 0(mn') f a i t h f u l ; now, p r e c i s e l y d of these y i e l d the same 77, , and thus the number of inequivalent f a i t h f u l 77, i s k k 0(mn')/d . Now, an integer i s prime to mn' i f f i t i s prime to mn'd = N. 2 Accordingly, 0(mn')/d = 0(N)/d , proving the f i n a l claim. C. Action of Automorphisms of G on Representations We s h a l l now give a more e x p l i c i t formulation of theorem 2.16. In addition, we s h a l l determine the automorphism group of G, where G i s given by 2.15, and determine when two representations are equivalent - 28 -modulo such an automorphism. Theorem 2.17. Let G be a group of order N having a l l Sylow subgroups c y c l i c and s a t i s f y i n g a l l pq-conditions given i n terms of generators and r e l a t i o n s by 2.15. Whenever k and £ are integers s.t. (k,m) = (£,n) =1, we define V*(A) 2 i r i k/m 2iTikr/m 2 i r i k r d ^/m V*(B) = 2 T r i£'/n e 0 Whenever s,t,u are integers s.t. (s,m) = (t,n) = 1 and t = 1 (d), we define * S , t , U ( A ) = ^ * 8 , t , u W = B V (a) The i r r e d u c i b l e f a i t h f u l complex representations of G are ju s t the TT ; they have degree d and are f. p . f . (b) The automorphisms of G are j u s t the ty s, t ,u (c) TT, „ • ty i s equivalent to i , „, and TT , i s k,£ s,t,u sk,t£ a,b equivalent to TT , , i f f b = b'(n') and a' = ar (m) for some 3. > D integer c with 0 < c < d - 29 -(d) If G i s c y c l i c , then TT, i s equivalent to i t s conjugate i f f N < 2, and, i n that case, TT, i s equivalent to a r e a l representation. (e) If G i s not c y c l i c , then TT i s not equivalent to a r e a l K j Jo representation, and i s equivalent to i t s conjugate i f f n' = 2 . Proof: (a) i s a restatement of part of theorem 2.16. In f a c t , TT i s equivalent to T T^ of 2.16 where k = q (m) and £ = q (n') . (b) Suppose ty: G —> G; then ty(A) = A S B V and IJJ(B) = BtAu for some integers s,v,t,u . A generates the commutator subgroup G', so i f ty i s an automorphism, we must have (s,m) = 1 and v = 0 . Also BG' generates G/G', so i f ty i s an automorphism, (t,n) = 1 . Conversely, suppose (s,m) = (t,n) = 1 and v = 0 . Then ty i s an — 1 r I- c: —t" ^ r automorphism i f f KB) • *(A) • ip(B ) = TJI(A) i f f B A B = A i f f t A S r = A S r i f f t = 1 (d) . (c) The condition f o r equivalence of TT , and TT , , , has been a,b a ,b established i n 2.16. As for the other condition, i f G i s c y c l i c then A = 1, m = r = l , and the r e s u l t i s c l e a r . So, suppose G i s not i • J J r • 2iTik/m . 2Tfi£/n' , , , 2 j-1 c y c l i c , and define a = e , g = e , f ( j ) = l + r + r + . . . + r for 1 < j <_ d . Note that r-f(d) = f(d)(m), thus (r-l)«f(d) 5 0(m) . Since (r-l,m) =1, i t follows that m|f(d), a fa c t which we s h a l l use i n the c a l c u l a t i o n s i n the next paragraph. Looking at the matrix formulation of the representation TT, , we K . 5 A/ see that the representation space V has a basis {x ,...,x,} s.t. j - l r TT, 0(A)x. = a x. for a l l j , while TT (B)x. = x. - for j < d K . , J C 3 2 k,x, j j+I and TT. „(B)x, = gxn . Define y. = ^ V a U f ^ ~^x. , where k,£ d 1 j 3 t = vd + 1, and define TT = TT, „ • ty .We claim that TT i s k,£ s,t,u - 30 -equivalent to t ^ • F i r s t , {y^,...,y(j} i s a basis f o r V and i i ^ - u 4-1- _ / , i 0 ( j - l ) * v u ' f ( j - l ) sr"' sr^ c a l c u l a t i o n shows that Tr(A)y. = 3 a a x . = a y J 2 J £ i i j r . j / T ) s n ( j - l ) * v u * f ( i - l ) u«r^ v for a l l j and for j < d, Tr(B)y^. = g J a J a g Xj+1 = g j V a U ' f ( : i ) x j + 1 = y while for j = d, u(B)y d = D ( d - l ) v u - f ( d - l ) u - r d _ 1 v+1 .dv+1 u-f(d) D t 3 a a 3 x ^ ^ = 3 a x = 3 x 1 . (d) Again, i f G i s c y c l i c , the assertions follow e a s i l y . (e) If G i s non-cyclic, neither of m,n',d i s equal to 1; furthermore m i s odd, since m i s r e l a t i v e l y prime to both r and r - 1 . Let IT = T T^ ^ a n d l e t a be the representation of d G H = {A,B } s.t. TT = a . If TT i s r e a l , o i s r e a l , and since H i s c y c l i c , H must have order 1 or 2, y i e l d i n g a contradiction since m||H| . Now, l e t x = char T T , x = char TT . If n' > 2, we have , d. 2fri£/n' , ... -2Tri£/n' — ,„dN , . . X(.B ; = d*e ? d*e = X ( B ) , and so TT I S not equivalent to i t s conjugate TT . On the other hand, i f n' = 2, then since each prime d i v i s o r of d must divide n', we have d = 2d', where d' i s a power of 2; also r d = -1 (m). Now, t h i s implies that i f 0 z d'+c c ZTTik/m , r - r . . . . . „ a = e , then a = a and i t i s t r i v i a l to v e r i f y that X(A) = x(A) . In addition x(B WA y) = 0 = x(B WA y) for 1 i B W 4 B d and, f i n a l l y , x(B dA y) = ~x(Ay) = -x(A Y) = x(B dA Y) . Thus we have X = x which implies that TT i s equivalent to TT . D. Real Fixed Point Free Representations It remains f o r us to use the information obtained i n theorem 2.17 to determine the orthogonal f . p . f . representations of G . - 31 -Theorem 2.18. Let G be a group of order N having a l l Sylow sub-groups c y c l i c , given i n terms of generators and r e l a t i o n s by 2.15. Suppose G i s not c y c l i c of order one or two. Let R(0) denote the r o t a t i o n matrix cos 2TT0 - s i n 2TT0 s i n 2TT0 cos 2TT0 Suppose k and £ are integers s.t. (k,m) = (£,n) = 1 . Let TT be the r e a l representation of G of degree 2d defined i n 2 x 2 blocks by k,£ \ , £ ( A ) = R(k/m) R(kr/m) R(kr d \ , £ ( B ) = R(£/n') 0 (a) A r e a l representation of G i s f . p . f . i f f i t i s equivalent to a sum of representations TT . K . j Ay (b) TT^ ^ i s equivalent to TT^, ^, i f f there e x i s t numbers e = ±1, and c = 0,1,...,d - 1 s.t. k' = ekr C(m) and £' = e£(n') 2 (c) If n' = 2, there are 0(N)/d inequivalent TT 2 If n' ^ 2, there are 0(N)/2d inequivalent TT k,£ - 32 -Proof: Note that IT, = TT_, , and (TT ) = TT, . + T r , where (IT ) denotes the complexification of TT . Now theorem 2.18 follows from theorems 2.17 and 2.16. E. C l a s s i f i c a t i o n of Spherical Space Forms of Dimension Q t 3 (4) We s h a l l now present Vincent's p a r t i a l s o l u t i o n of the C l i f f o r d -K l e i n s p h e r i c a l space form problem. Let M be a complete, connected Riemannian manifold of constant p o s i t i v e curvature. Let us f i r s t note that i f the dimension of M,Q, i s even, the problem i s t r i v i a l ; the only s p h e r i c a l space forms are the sphere S^ and the p r o j e c t i v e space P^ . Indeed, we proved a more general statement i n lemma 1.19. The key theorem which we require i s the following; i t i s due to Vincent. Theorem 2.19. Let M be a complete, connected, Riemannian manifold of constant curvature K > 0, having dimension Q t 3 (4) . Then the fundamental group of M , TT^(M) , has every Sylow subgroup c y c l i c . Proof: Let H be a Sylow 2-subgroup> of TT^(M) . By theorems 2.6 and 2.8, we need only show that H cannot be generalized quaternionic. Now, H acts f i x e d point f r e e l y on S^ ; thus, by theorem 1.18, H has p e r i o d i c cohomology of period Q + 1 . But generalized quaternionic groups are known to have period 4 . Therefore, i f H were generalized quaternionic, we would have Q + 1 = 0 (4) i . e . Q = 3 (4) i n contra-d i c t i o n of our hypothesis. I t follows that every Sylow subgroup of TT^(M) must be c y c l i c . - 33 -According to Vincent's theorem, our c l a s s i f i c a t i o n theorems 2.4 and 2.18 y i e l d a complete c l a s s i f i c a t i o n of a l l C l i f f o r d - K l e i n s p h e r i c a l space forms of dimension Q £ 3 (4) . Let us give an e x p l i c i t d e s c r i p t i o n of these manifolds. So, l e t M be a complete, connected Riemannian manifold having dimension Q t 3 (4) . Let G = T T ^ ( M ) . F i r s t , i f G = {1}, then M = S Q . Second, i f G = Z^, then M = P^ . Otherwise, M i s isometric to the following manifold: (2.20) S 2 v d _ 1 / ( T r , )(G) . 1 1 v' v Here we have retained the notation of theorem 2.18. Now, M determines the set {(k^,£ ),...,(k v,£ v)} i n the following sense: M i s also isometric to a manifold S^ v d (TT , © . . . © TT , ) (G) i f f there a n , b a , b 1 1 v v exis t s a permutation j —*• i _ . and integers e^.,c^,s,t such that e.. = ±1, 0 <_ c < d, (s,m) = (t,n) =1, t = 1 (d) c. k. = e.sa.r 3 (m), I. = e.tb. (n') . i j 3 3 i j 3 3 Note that t h i s r e s u l t i s j u s t a d i r e c t a p p l i c a t i o n of theorems 2.4 and 2.18. Example 2.21. This example i s due to Vincent. Any number d > 1 can occur i n (2.20), for l e t m be any one of the i n f i n i t e number of primes of the form 1 + bd, and l e t G be defined by 2.15. This example shows that there e x i s t an i n f i n i t e number of sph e r i c a l space forms, i n every odd dimension Q > 1, whose fundamental groups are non-abelian and do not occur i n lower dimensions. - 34 -This completes our discussion of Vincent's p a r t i a l s o l u t i o n of the s p h e r i c a l space form problem. In section V, we s h a l l investigate the general case i n which we consider a l l p e r iodic groups. V. WOLF'S SOLUTION OF THE SPHERICAL SPACE FORM PROBLEM A. Some Motivation The basic procedure which we s h a l l adhere to and the techniques which we s h a l l use are v i r t u a l l y i d e n t i c a l to those of the s p e c i a l case handled i n section IV. However, the group theory involved i n the general case i s s u b s t a n t i a l l y more complicated. But before we begin to state theorems, we s h a l l give a short general discussion of the types of groups appearing i n the C l i f f o r d - K l e i n s p h e r i c a l space form problem. The e a r l i e s t examples of sp h e r i c a l space forms were, associated with regular polyhedra; i n f a c t , the corresponding space groups are the double coverings of r o t a t i o n groups of these polyhedra. To be more s p e c i f i c , l e t us define to be the dihedral group of order 2m i . e . the group of symmetries of the regular plane m-gon; l e t us define T to be the tetrahedral group, the group of symmetries of the regular tetrahedron; define 0 to be the group of symmetries of the regular octahedron, c a l l e d the octahedral group; f i n a l l y , define I to be the group of symmetries of the regular icosahedron c a l l e d the icosahedral group. Note that a l l these groups may be regarded as sub-groups of the s p e c i a l orthogonal group S0(3) . The following proposition i s fundamental. Proposition 2.22. Every f i n i t e subgroup of S0(3) i s c y c l i c , d i h e d r a l , tetrahedral, octahedral, or icosahedral. - 35 -Now, l e t Q denote the algebra of r e a l quaternions; regard R as embedded i n Q as the space of pure imaginary quaternions and 3 regard S as the group of unit quaternions. Then the map 3 -1 TT: S —> S0(3) defined by Tr(q)(q') = qq'q i s a two-fold 3 d i f f e r e n t i a b l e covering of S0(3) by S . We define the binary * -1 dihedral groups to be the groups = TT ( D m) > a n < ^ t n e binary tetrahedral, binary octahedral, and binary icosahedral groups to be * -1 ft _ i ft _1 T = TT (T) , 0 = TT (0) and I = TT (I) r e s p e c t i v e l y . These are the space groups associated with regular polyhedra which we mentioned above. Proposition 2.22 implies the following: 3 Proposition 2.23. Every f i n i t e subgroup of S i s c y c l i c , binary dih e d r a l , binary tetrahedral, binary octahedral, or binary icosahedral. 3 Now, i t i s these f i n i t e subgroups of S which play a c e n t r a l r o l e i n the C l i f f o r d - K l e i n s p h e r i c a l space form problem. In f a c t , i t turns out that a l l p e r i o d i c groups are b u i l t up by taking extensions of one basic type with another; these basic types are c l o s e l y r e l a t e d to the c y c l i c , binary d i h e d r a l , and binary polyhedral groups. If G i s a solvable periodic group, there are four classes of extensions. The f i r s t c onsists of a s p l i t extension of two c y c l i c groups, a metacyclic group. This i s the type considered i n section IV and given by 2.15 i n terms of generators and r e l a t i o n s . The second class consists of a s p l i t extension of a metacyclic group with a generalized quaternion group; the t h i r d c l a s s , a metacyclic group with a generalized binary tetrahedral group and the fourth c l a s s , a metacyclic group with a generalized binary octahedral group. If G i s not solvable, i t i s again constructed by taking extensions of basic types; here, the basic types are metacyclic - 36 -and binary icosahedral. Let us now begin the work leading to e x p l i c i t descriptions of a l l these groups. We divide our class of. periodic groups into two sub-classes as suggested above, according to whether the group i n question i s solvable or not solvable. The work on solvable groups, which we treat f i r s t , i s due to Zassenhaus. Now, i t i s possible 2 that a group s a t i s f i e s a l l p -conditions or pq-conditions, while 2 a homomorphic image of that group does not s a t i s f y the 2 -condition. As a r e s u l t , d i f f i c u l t y a r i s e s i n the formulation of induction arguments, a p r i n c i p a l t o o l i n the theory of solvable groups. To r e c t i f y t h i s s i t u a t i o n , Zassenhaus considers a l l those solvable groups whose Sylow p-subgroups are c y c l i c for p an odd prime, and whose Sylow 2-subgroups have c y c l i c subgroups of index two. For a complete l i s t of these groups see [23, p. 176]. However, not a l l such groups are p e r i o d i c . The following theorem gives a c l a s s i f i c a t i o n of a l l solvable p e r i o d i c groups. B. Solvable Fixed Point Free Groups Theorem 2.24. Let G be a f i n i t e solvable group. Then G i s periodic i f f G i s isomorphic to one of the groups l i s t e d i n the following table: - 37 -Type Generators Relations Conditions Order A,B II A,B,R I I I A,B,P,W A m = B n = 1 -1 BAB As i n I; also R 2 = B n^ 2, RAR = A ; RBR = B IV As i n I; also P =1, P 2 = Q 2 = (PQ) 2, AP = PA, AQ = QA, BPB BOB - 1 -1 PQ A,B,P,Q,R As i n I I I ; also R" = P RPR _ 1 = QP, RQR - 1 ~ - 1 2 „2 Q - I P - I k RAR = A , RBR = B m>_l, n>_l, ( n ( r - l ) ,m) = 1 rn = l(m) 2 As i n I; also, I = l(m) n = 2 U«v, u ^ 2 , k 2 E l ( n ) , k = -1(2 U) As i n I; also n = 1 (2) n = 0 (3) As i n I I I ; also k =1 (n) k = —1 (3), r k _ 1 = £.2 = 1 (m) mn 2mn 8mn 16mn Furthermore, the following conditions are equivalent: (a) G has a f . p . f . complex representation. (b) G s a t i s f i e s a l l pq-conditions. (c) G i s of type I, I I , I I I , or IV above with the a d d i t i o n a l requirement: If d i s the order of r i n K , the m u l t i p l i c a t i o n m group of residues modulo m, then n' = n/d i s d i v i s i b l e by every prime d i v i s o r of d . Sketch of proof: One obtains the four classes of solvable p e r i o d i c groups i n a case by case check of Zassenhaus' l i s t [23, p. 176]. Also note that - 38 -(a) => (b) by theorem 2.6. Apply theorem 2.14 to the subgroup {A,B} of G to see that n T i s d i v i s i b l e by every prime d i v i s o r of d; thus (b) => ( c ) . I t remains to prove (c) => ( a ) . For groups of type I , we have already seen t h i s i n theorem 2.16. For groups of types I I , I I I , and IV, the procedure i s analogous. F.p.f. rep r e s e n t a t i o n s of G are induced from those of some subgroup known to admit f . p . f . r e p r e s e n t a t i o n s . Again, we use lemma 2.12 to conclude that these induced r e p r e s e n t a t i o n s are f . p . f . C. Non-solvable Fixed P o i n t Free Groups Our main theorem determines a l l non-solvable f . p . f . groups. But before s t a t i n g t h i s theorem, we need a few p r e l i m i n a r y r e s u l t s . D e f i n i t i o n 2.25. SL(n,q) denotes the m u l t i p l i c a t i v e group of (nxn) matrices of determinant 1 over the f i e l d F of q elements. q H ft Remark 2.26. I t i s a standard f a c t that I = SL(2,5); t h i s isomorphism w i l l be u s e f u l i n the f o l l o w i n g theorem. We d e f i n e an automorphism 0 of SL(2,5) as f o l l o w s : 0 i s conjugation by 0 -1 2 0 2 Then 0 = 1 , and 0 generates the group of outer automorphisms of SL(2,5) . Theorem 2.27. Let G be a f i n i t e non-solvable group which admits a f . p . f . complex r e p r e s e n t a t i o n . Then G i s of one of the f o l l o w i n g two types. - 3 9 -TYPE V: G = K x S L ( 2 , 5 ) , where K i s a solvable f . p . f . group of type I and order prime to 3 0 . TYPE VI: G = {G ,S}, where G^ = K x S L ( 2 , 5 ) i s a normal subgroup of index 2 and type V, S 2 = -I e S L ( 2 , 5 ) , S L S - 1 = 6 ( L ) , where L e S L ( 2 , 5 ) , 0 i s the automorphism of S L ( 2 , 5 ) defined above, and where S normalizes K . Conversely, every group of type V or VI i s a f i n i t e non-solvable group admitting a f. p . f . complex representation. Sketch of proof: Let G have a f . p . f . complex representation. G non-solvable implies that some higher derived group H c_ G i s perfect i . e . H = H * , the commutator subgroup of H . The c r u c i a l f a c t needed here i s that the only n o n - t r i v i a l perfect group admitting a f. p . f . ft complex representation i s the binary icosahedral group I . This i s a n o n - t r i v i a l theorem; for a group-theoretic proof, see [ 2 3 , pp. 1 8 1 -1 9 5 ] ; f o r a somewhat more general r e s u l t and a d i f f e r e n t approach to * the proof, see [ 9 ] . I t now follows that H = I . Z, the centre of H , i s of order two, and i f we set M equal to the c e n t r a l i z e r of H i n G, i t turns out that Z i s the Sylow 2-subgroup of M . Now i t follows by a theorem of Burnside [ 2 3 , p. 1 5 9 ] that the kernel K of the transfer homomorphism 'V £ i s a normal subgroup of index 2 i n M . K i s solvable of type I because i t has odd order. Define G^ = {geG|conjugation by g i s an inner automorphism of H } . 2 Then G^ = M « H = K x H . G s a t i s f i e s a l l p -conditions implies that ( | K | , | H | ) = 1 , which implies ( | K | , 3 0 ) = 1 . So i f G = G , then G i s of type V . Suppose G ^ G^ ; i t can be shown that G i s then of type VI. - 40 -Conversely, i f G i s of type V and a and g are f .p.f. representations of K and SL(2,5) r e s p e c t i v e l y , then i t follows from lemma 2.13 that a ® B i s a f . p . f . representation of G . If G i s of type VI and i s a f.p . f . representation of G^, i t turns out Q that if i s a f . p . f . representation of G . Remark 2.28. Suzuki [14] has shown that the f i n i t e non-solvable periodic groups are of the form (a) K x SL(2,r), r prime, r >^ 5, K a group of type I with | K | r e l a t i v e l y prime to r ( r -1) . (b) { K x SL(2,r),S} where K x SL(2,r) i s a normal subgroup of index 2 and type (a), S 2 = -I e SL(2,r), S L S _ 1 = 8(L) for a l l L e SL(2,r) and S normalizes K . Here 6 i s conjugation by ^ ^ w u where w generates the m u l t i p l i c a t i v e group of non-zero elements of F Furthermore, Suzuki has'shown that SL(2,r) s a t i s f i e s a l l pq-conditions i f f r i s of the form 2 + 1 for some k . Thus, unlike the solvable case, (2.24), i f G i s a f i n i t e , non-solvable periodic group, i n order that G have a f . p . f . complex representation i t i s not s u f f i c i e n t that G s a t i s f y a l l pq-conditions. D. Representation Theory of Fixed Point Free Groups (2.24) and (2.27) give a complete c l a s s i f i c a t i o n of f . p . f . groups. Now our task i s to study the representation theory of each of these types of groups, c l a s s i f y i n g up to automorphism the equivalence classes of f . p . f . orthogonal representations. The representation theory of metacyclic groups described i n section IV of t h i s chapter serves as a reasonable model f or the general - 41 -case, although each type of periodic group seemingly must be handled i n d i v i d u a l l y and complications do a r i s e . However, i t does not seem i n s t r u c t i v e to methodically go through each case (and subcases) , l i s t i n g the r e s u l t s . Some remarks of a general, explanatory nature may be more worthwhile. In any case, an e x p l i c i t l i s t i n g of a l l f . p . f . complex representations of the s i x types of f . p . f . groups, and related i n f o r -mation, may be found i n [23, pp. 198-211]. Generally speaking, one determines the f . p . f . complex representations of a f . p . f . group G as follows: F i r s t , one determines the f . p . f . complex representations of subgroups of c e r t a i n types. In f a c t , groups of type I I I (see 2.24) are studied v i a a close examination of t h e i r * generalized binary tetrahedral subgroups with d e f i n i n g r e l a t i o n s X = P = 1, P = Q , XPX = Q, XQX _ 1 = PQ, POP"1 = Q - 1, v >_ 1 . The representation theory of groups of type IV i s studied by means of the representation theory of generalized binary octahedral subgroups ft 0 with defining r e l a t i o n s v X 3 = P 4 = 1, P 2 = Q 2 = R 2, POP - 1 = Q - 1, XPX" 1 = Q, XQX _ 1 = PQ, RXR _ 1 = X _ 1, RPR - 1 = QP, RQR _ 1 = Q _ 1, v _> 1 . ft ft Note that T^ i s the binary tetrahedral group, while 0^ i s the binary octahedral group. The representation theory of groups of type V and ft VI i s studied v i a subgroups of binary icosahedral type, I Assuming that one has determined the i r r e d u c i b l e f . p . f . complex 42 -representations of these designated subgroups of G , one obtains the f . p . f . complex representations of G from these using the operations of induced representation or tensor product of representations. The key tools i n t h i s analysis are lemmas 2.11, 2.12, and 2.13. Let us elaborate. We have pointed out i n part A of t h i s section that a l l p e r i o d i c groups are either metacyclic or are obtained as extensions of metacyclic groups with other types. S t i l l assuming G i s a f . p . f . group, we consider two cases: (i) G i s a d i r e c t product of a metacyclic sub-group K and a subgroup L which i s generalized binary tetrahedral, generalized binary octahedral or binary icosahedral, as the case may be. In t h i s case, f.p . f . complex representations of G are obtained as tensor products of f.p . f . complex representations of K and L . ( i i ) G i s an extension which i s not a d i r e c t product. In t h i s case, one can always f i n d a subgroup, say G Q , of G s.t. the i r r e d u c i b l e f . p . f . complex representations of G are induced from those of G ^ . Let us i l l u s t r a t e these comments by looking at one type of f . p . f . group at greater length. Consider those groups of type I I I . As mentioned above, t h e i r generalized binary tetrahedral subgroups are the key to t h e i r representation theory. In what follows, our notation i s that of (2.24) and (2.27). Also, we denote by F ( G ) the set of a l l equivalence classes of i r r e d u c i b l e f . p . f . complex representations of the group G . There are three subcases to consider. Subcase A. n £ 0 (9) and therefore d i 0 (3) . Then i t i s easy to check that G i s a d i r e c t product G = {A,B"^ } x {B n^,P,Q} ; 3 n / 3 here, {A,B } i s a group of type I and {B ,P,Q) i s isomorphic to * T . The elements of r ( G ) are given by v, = TT ( _ T , where - 43 -\ z€ f c { a ' b 3 } a n d T e f c ( T * J ' b o t h f c ^ a ' b 3 } a n d F C ( T * ) b e i n g previously determined, of course. To obtain t h i s l a s t statement we use lemma 2.13. Subcase B. n = 0 (9), but d f 0 (3) . Then n = 3 Vn" with (3,n") = 1, and v > 1 . Again, G s p l i t s into a d i r e c t product, G = {A,B "} x {B n ,P,Q} where {A,B } i s of type I and {B n ,P,Q} = T . Again, by lemma 2.13 F„(G) consists of a l l v, „ . = TT, „ <£> x., v C k,£,3 k,£ 2 \,i £ F C { A ' B 3 } ' T 3 £ F C ( V * Subcase C. d = 0 (3) . Here we f i n d that G does not s p l i t as a d i r e c t product. The i r r e d u c i b l e f . p . f . complex representations of G are obtained not as tensor products, but are induced from represent-ations of the normal subgroup K = {A,Bd,P,Q} . K does s p l i t as a d i r e c t product; K = {A} x {Bd} x {P,Q} with {P,Q} = Q8, the quaternion group of order 8 . Thus, F C(K) consists of a l l a ka £®a, a f c e F^A}, a £ e F c ( B d } , a e F C(P,Q} . Now, F C(G) consists of a l l V k = ( C T k ® °£<8> °0 , by lemmas 2.11 and 2.12. For the sake of completeness, l e t us b r i e f l y mention the other f. p . f . groups. For a group G of type IV, F C(G) i s obtained i n a manner ft b a s i c a l l y s i m i l a r to that of type I I I , except that 0^ assumes the ft r o l e of T . v If G i s of type I I , the elements of F C(G) are e s s e n t i a l l y Q obtained as induced representations a = TT , where ir e ¥ {A,B} and {A,B} i s of type I. ft If G i s of type V, G = K x I with K of type I . F (G) consists of a l l i ^ ^ . = \ > z ® ± . , ^ e F ^ K ) i . e VQ(I) . - 44 -If G i s . o f type VI, elements of F ( G ) are induced from sub-groups of type V. Let t h i s somewhat b r i e f i n d i c a t i o n conclude our discussion of how the i r r e d u c i b l e f . p . f . complex representations of f . p . f . groups are obtained, and l e t us now turn to the next stage of our program, which consists i n describing the action of automorphisms of G on ^ Q ( G ) . E . Action of Automorphisms of G on Representations If i s an automorphism of G and TT i s an i r r e d u c i b l e f . p . f . complex representation of G , then there exists an action F C ( G ) y } R ( - , ( G ) given by TT y TT«^ . We s h a l l denote by U ( G ) the group of transformations of F ( G ) L» Li induced by automorphisms of G . Now, we have seen that the elements of F ( G ) were constructed u using two basic operations: tensor product of representations and induced representations. The following two lemmas give information about \]^,(G) i n both cases. Lemma 2.29. Let H be a c h a r a c t e r i s t i c subgroup of the f i n i t e group G , and l e t a be a representation of H . If \> i s an automorphism G G of G , then ( c i M ) i s equivalent to a -vp . r i Q Proof: If TT = a i s induced from a v i a a coset decomposition G = ub.H, b = 1, then TT* 1 be a prime s.t. p|d ; then p|nT and so p divides the order of TT1 ( M ) , contrary to hypothesis. Thus d = 1 and TT ( M ) i s cyclic. - 48 -CHAPTER 3: SWAN'S WORK ON THE HOMOTOPY SPHERICAL SPACE FORM PROBLEM I. INTRODUCTION In the introduction to t h i s t h e s i s , we discussed how the C l i f f o r d -K l e i n s p h e r i c a l space form problem could be generalized from the Riemannian case to the t o p o l o g i c a l case, asking the following: "Find a l l f i n i t e groups which can act f i x e d point f r e e l y on S n by homeo-morphisms, and c l a s s i f y the r e s u l t i n g " t o p o l o g i c a l space forms" up to homeomorphism. More generally, we may ask which f i n i t e groups can act f i x e d point f r e e l y on a space having the homotopy type of a sphere. We have seen i n chapter 1 that a f i n i t e group which acts f i x e d point f r e e l y on a space having the homotopy type of a sphere must have periodic cohomology. The following converse r e s u l t was established by Swan [16]. Theorem 3.1. Let TT be a f i n i t e group of order n . Let d be the greatest common d i v i s o r of n and (n), where cf> i s Euler's -function. Suppose TT has periodic cohomology of period q . Then there exists a f i n i t e s i m p l i c i a l complex X of dimension dq - 1 which has the homotopy type of a (dq-1)-sphere and on which TT acts f r e e l y and s i m p l i c i a l l y . It follows from a construction of Milnor, which we s h a l l describe i n section II of t h i s chapter, that theorem 3.1 i s a consequence of the following purely algebraic theorem concerning the existence of free p e r i o d i c resolutions of TT . Theorem 3.2. Let TT be a f i n i t e group of order n, and l e t d = (n,(n)). Suppose TT has p e r i o d i c cohomology of period q . Then Tr has a periodic free r e s o l u t i o n over Z of period dq . - 49 -Right now, some explanation of terminology seems to be i n order. D e f i n i t i o n 3.3. Let R be a commutative r i n g with i d e n t i t y . A f i n i t e group IT i s said to have a p e r i o d i c free r e s o l u t i o n of period k over R i f there e x i s t s an exact sequence of f i n i t e l y generated modules (3.4) 0 • R W. , • ... • W„ R — > 0 k-1 0 with a l l W RTT-free, where RTT i s the group r i n g of TT over R . Of course, i t i s assumed that a l l maps i n t h i s sequence are Re-maps, and that i r acts t r i v i a l l y on the two terms R . I f , i n the above s i t u a t i o n , the are assumed merely to be p r o j e c t i v e , i t i s said that TT has a periodic p r o j e c t i v e r e s o l u t i o n of period k . In f a c t , the term "periodic r e s o l u t i o n " i s quite d e s c r i p t i v e . For suppose we are given a sequence of the form (3.4); then we can define a complete r e s o l u t i o n , i n the sense of d e f i n i t i o n 1.4, by s p l i c i n g together copies of (3.4), as follows: -> w —>• w —>• ... —>• w —> w k-1 0 k-1 / \ / (3.5) R R /\ / \ 0 0 0 0 This complex obviously has an automorphism of degree k . Conversely, given a complete r e s o l u t i o n together with an auto-morphism of degree k as i n (3.5), i t i s clear that we can break t h i s sequence apart into resolutions of the form (3.4). - 50 -Remark 3.6. Throughout our discussion we s h a l l work over Z, the r i n g of integers. However, we should l i k e to point out that a l l the r e s u l t s we obtain hold i n greater generality. For example, our p r i n c i p a l r e s u l t , theorem 3.1, has the following generalization: Let P be a set of primes. Let R be the l o c a l i z a t i o n ^(p) . Let CP be the Serre c l a s s of f i n i t e abelian groups having no p-torsion for any prime p e P . Then the "mod C " counterpart of theorem 3.1 i s Theorem 3.7. Let TT be a f i n i t e group of order n, and l e t d = (n, 2 . Then there e x i s t s a. f i n i t e s i m p l i c i a l complex of dimension dq - 1 which i s simply connected, whose i n t e g r a l homology groups are isomorphic mod CP to those of a (dq-1)-sphere, and on which TT acts f r e e l y and s i m p l i c i a l l y . Note that the condition dq > 2 i n theorem 3.7 merely excludes t r i v i a l cases ( i f d = 1, TT i s c y c l i c and therefore acts on S^ " by r o t a t i o n s ) ; the reasons for i t s appearance are purely geometric. No such r e s t r i c t i o n occurs i n the following generalization of theorem 3.2. Theorem 3.8. Let fr be a f i n i t e group of order n, and l e t d = (n,cj>(n)). Suppose that IT has a f i n i t e P-period q . Then TT has a periodic free r e s o l u t i o n over R of period dq . We s h a l l say no more about these possible generalizations; i n the sequel, we s h a l l confine ourselves to Z . Remark 3.9. Let us state once and for a l l that a l l modules treated i n t h i s chapter w i l l be assumed to be f i n i t e l y - g e n e r a t e d unless we make an e x p l i c i t statement to the contrary. In f a c t , i t i s only i n section IIIB that we encounter modules which are not f i n i t e l y generated. T - 51 T - . I I . MILNOR'S CONSTRUCTION A. Algebraic Preliminaries Milnor's construction y i e l d s a CW-complex from a re s o l u t i o n of type (3.4); however, i n order to obtain a simply-connected complex, which we need, i t i s e s s e n t i a l that we have a periodic r e s o l u t i o n having low-dimensional terms of a c e r t a i n s p e c i f i e d form. In lemma 3.13 we w i l l show that i t i s always possible to construct such a per i o d i c r e s o l u t i o n . The main t o o l i n the proof of 3.13 i s the following w e l l -known lemma of Schanuel. Lemma 3.10. Let 0 — > B — > P — * - A — > 0 and 0—> B' —>- P* —> A 1 —> 0 be exact with P and P' pr o j e c t i v e . If A = A', then P + B' = P' + B. D e f i n i t i o n 3.11. We define two modules A and A 1 to be equivalent, and we write A ^ A' i f there e x i s t p r o j e c t i v e modules P and P' s.t. P + A = P' + A' . If P and P' can be chosen to be free, we s h a l l write A ^ A' . Coroll a r y 3.12. Suppose we have two exact sequences 0 —> B —> P — y ... —> P„ —* A —> 0 m 0 0 —> B' —> P* —> ... —y P' —y A' —> 0 . m 0 If a l l P and P^ are pr o j e c t i v e , and A ^ A', then B ^ B' . If a l l P. and P'. are free, and A 'V A', then B ^ £ B' . i i f f Proof: I f m = 0, replace given sequences by the exact sequences 0 — > B — > P Q + P — * A + P —^ 0 0 B' — y P i + P' ~y A' + P* 0 - 52 -where A + P = A 1 + P' . Applying Schanuel's lemma, we f i n d that PQ + P + B' = PQ + P' + B i . e . B ^ B' . Now, i f m > 0, break the given long exact sequences into a series of short exact sequences i n the usual manner, and proceed i n d u c t i v e l y . We wish to construct a p e r i o d i c r e s o l u t i o n over the i n t e g r a l group r i n g ZTT . As a f i r s t step, we merely choose any exact sequence of ZTT-modules 0 —>- A —>- W, > W.. WA — * Z —> 0 k-1 1 0 where the W_^ 's are a l l ZTr-free. Suppose there e x i s t s a periodic free r e s o l u t i o n of period k, as follows: 0 — > Z — y w; —>• W' —> W' Z —* - 0 . k-1 1 0 Then c o r o l l a r y 3.12 implies that A ^ Z . The next lemma shows that t h i s condition i s also s u f f i c i e n t . Lemma 3.13. Suppose we have an exact sequence over ZTT of the form (3.14) 0 —> A X k_ x X k_ 2 ^ where u s p l i t s as a Z-map. Suppose there e x i s t s a module A' and pr o j e c t i v e modules P and P' s.t. P + A = P' + A' . Then there ex i s t s an exact sequence of the form (3.15) 0 — y A' —>- X . + P X, . + P' ( 8 , 0 ) > X. ~ ~~y~ k-1 k-2 - k-3 where a l l modules and maps from X^ on are the same as i n (3.14). Proof: F i r s t , we add P to both A and X ^ - l a n c * obtain from (3.14) - 53 -0 A + P (' y ? 1- ) > X k _ x + P —> X k_ 2 -> Then we replace A + P by the isomorphic module A' + P' . Since A i s a Z-direct summand of x k _ ^ » A + P i s a Z-direct summand of X + P and hence so i s P' . Since P' i s Zn-projective, there e x i s t s a Z-endomorphism P' —*- P' having 1 , as norm [2, ch. X I I , prop. 1.1]. Let p: P' —>• P' be such a map, and l e t f: X, , + P —> P' be a Z - r e t r a c t i o n . Then N(pf) i s a Zn-retraction. k-1 Let g: X^ ^ + P —> P' be any ZiT-retraction. Then the following sequence i s exact: 0 — A' -*X. . + P ^ \ , + P' ^ X v ^ .... k-1 k-2 k-3 Corol l a r y 3.16. Consider the exact sequence of ZTr-maps with a l l free (3.17) 0 —• A W W WQ -> Z —> 0 Then A ^ Z i f f TT has a periodic p r o j e c t i v e r e s o l u t i o n over Z . Also A ^ Z i f f TT has a periodic free r e s o l u t i o n over Z . Proof: Suppose A ^ Z . By lemma 3.13, we need only show that (3.17) s p l i t s over Z . But t h i s i s c l e a r , because Z and WQ,...,Wk_^ are a l l Z-free. The converse follows immediately from c o r o l l a r y 3.12. B. Milnor's Construction The following lemma contains the c r i t i c a l step i n Milnor's construction. Lemma 3.18. Suppose the f i n i t e group TT acts f r e e l y on a simply-connected CW-complex X; assume that R\(X;Z) = 0 for 1 <^ i <_ m - 2 - 54 -Let F be a f r e e ZTT-module and f: F —> Z . (X) a ZTT-map. Then m-1 we can a t t a c h m-cells to X to get aXW-complex Y s . t . (a) TT acts f r e e l y on Y (b) C (Y) = C (X) + F . m m Proof: F i r s t , i f m >_ 3, the Hurewicz homomorphism h: ^..j^™ ^) —*" ^ ^ H ..(X ) i s an isomorphism and i s an epimorphism i f m = 2 . Here m-1 X m _ 1 denotes the (m-1)-skeleton of X . A l s o , H .(X™ - 1) = Z 1 ( X ) , m-1 m-1 where Z ,, as u s u a l , denotes the (m-1)-cycles. Consider the m-1 f o l l o w i n g diagram: m-1 m-l m-l (3.19) Now, si n c e F i s f r e e , there e x i s t s a g: F —> J completing (3.19) to a commutative diagram, as f o l l o w s : (3.20) TT . ( X m X ) • H . ( X m 1 ) = Z . (X) m-1 m-1 m-1 Now, l e t {e } be a Zir-base f o r F ; l e t g(e ) e TT . (X ) a a m-I be represented by a map . s»-l _ X™-1 . a Now, we wish to extend the a c t i o n of TT to these spheres; furthermore, we wish to do i t i n such a way that when we a t t a c h c e l l s to X v i a - 55 -these maps w e have a w e l l - d e f i n e d a c t i o n of IT on the a d j u n c t i o n space. For each t e TT and each index a, choose an m - c e l l tE™ ; l e t ts™ ^ denote the boundary of tE™ . Define E to be the d i s j o i n t a a union of a l l the tE™ . Now TT acts on E by t . (t„Em) = t , t„E m . a 1 2 a 1 2 a „ ,-. „m—1 „m—1 , / v , _ Define g t: tS >• X by g f t x ) = tg (x) . The c o l l e c t i o n of a l l the g 's d e f i n e s a map g: u'tS m ^ >• X, and a,t a i t i s a simple matter to check that g i s T r-equivariant. As a r e s u l t , we have a w e l l - d e f i n e d a c t i o n of IT on Y = Xu E . g I t i s c l e a r that the a c t i o n of IT on Y i s f r e e i . e . c o n d i t i o n th (a) holds. As f o r c o n d i t i o n ( b ) , we note that the m chain group C (Y) i s the f r e e a b e l i a n group on the m - c e l l s of Y, and since we have attached one c e l l to X f o r each Zir-generator of F and each element of TT, we c e r t a i n l y have C (Y) = ^ ( X ) + F as a b e l i a n groups. Furthermore, C (Y) = C (X) + F as Tr-modules because we have extended m m the a c t i o n of TT e q u i v a r i a n t l y to Y . Theorem 3.21. Suppose IT has a p e r i o d i c f r e e r e s o l u t i o n over Z of period k >_ 4 . Then TT can act f r e e l y and s i m p l i c i a l l y on a simply-connected f i n i t e s i m p l i c i a l homotopy (k-1)-sphere of dimension (k-1) . Proof: We s h a l l f i r s t c o n struct X as a CW-complex; we use an i n d u c t i v e procedure. We s t a r t by choosing a f i n i t e two-dimensional complex K having TT as fundamental group. Since TT i s f i n i t e , and t h e r e f o r e f i n i t e l y p resentable, there i s no problem i n choosing K to be f i n i t e . Now, we note that TT acts f r e e l y on the chains of the u n i v e r s a l covering complex K . Since K i s simply connected, the augmented chain complex - 56 -3 3 ( 3 . 2 2 ) C 2(K) — ^ C X ( K ) —X> C Q ( K ) > Z — > 0 forms an exact sequence w i t h ( K ) , C ^ ( K ) , C Q ( K ) f r e e ir-modules. By choosing a r e s o l u t i o n f o r Z^(K) = ker 9 ^ , we extend ( 3 . 2 2 ) to the f o l l o w i n g exact sequence ( 3 . 2 3 ) 0 —y A —• Wk_x —> ... —>W X—> WQ Z —> 0 —*- where ( 3 . 2 4 ) W2 -> W 1 —*• WQ — • Z —*- 0 i s isomorphic to ( 3 . 2 2 ) , and where a l l W^ , 0 <_ i <_ k - 1 , are f r e e TT-modules. But TT has a p e r i o d i c f r e e r e s o l u t i o n over Z of p e r i o d k . Thus, by c o r o l l a r y 3 . 1 2 , A Z . I t i s at t h i s p o i n t that we need to c o n s t r u c t a p e r i o d i c r e s o l u t i o n w i t h p r e s c r i b e d low-dimensional terms; we can do t h i s by lemma 3 . 1 3 . Consequently, we get a p e r i o d i c r e s o l u t i o n ( 3 . 2 5 ) 0 -*• Z -> V . . - > V . 0 - * • W , .->'...-> W. -»• Z -y 0 . k - 1 k - 2 k - 3 0 Note t h a t , beginning at the term W k - 3 ' t h i s sequence c o i n c i d e s w i t h ( 3 . 2 3 ) . For k >_ 5 , the terms W Q , W ^ , W 2 are not a f f e c t e d by the a p p l i c a t i o n of lemma 3 . 1 3 . For k = 4 , the term W2 would be a l t e r e d ; however, we can circumvent t h i s problem by an a p p l i c a t i o n of lemma 3 . 1 8 . We s h a l l t a c i t l y assume t h i s has been done; t h e r e f o r e , we may assume, without l o s s of g e n e r a l i t y , that the terms W Q , W ^ , W 2 are unaffected by our c o n s t r u c t i o n . The importance of t h i s i s that i f provides a s t a r t i n g p o i n t f o r our i n d u c t i o n . For convenience, l e t us r e w r i t e ( 3 . 2 5 ) as - 57 -(3.26) 0 - > Z — > F F C _ 1 — > . . . — > F 1 — > F Q - ^ - Z - > 0 . We s h a l l now prove, that for a l l m, there e x i s t s a simply-connected m-dimensional CW^complex X ™ on which TT acts f r e e l y , and whose chain complex over ZTT has the form F —>• F —>- ... —>• F — y F . m m-1 1 0 For m = 2, the complex K s u f f i c e s since —y F^ —y F n — y Z—y 0 i s isomorphic to (3.22) and ir acts f r e e l y on K . Now, suppose X has been constructed. So TT acts f r e e l y on rn*-1 X and X has chain complex F n — y F „ — y . . . —>- F.. — y F_ . m-1 m-2 1 0 Observe that a l l conditions for a p p l i c a b i l i t y of lemma 3.18 are s a t i s f i e d . Applying lemma 3.18 with X = X ™ \ F = F , f = 9: F — y Z , ( X ) y i e l d s a complex Y s.t. TT acts f r e e l y on m m-1 Y and s.t. C ( Y ) = F . Moreover, the chain complex of Y over ZTT m i s F — y F . —>• . . . — y F_ — y F„ . We complete our induction by m m-1 1 (J se t t i n g X M = Y . k-1 Consider the complex X = X constructed i n t h i s manner. TT acts f r e e l y on X by construction; furthermore, X has chain complex • F k - l ^ F k - 2 ^ ••• - ^ - 0 and i t follows that X i s a homology (k-1)-sphere. But X i s simply-connected, and i s therefore a homotopy (k-1)-sphere. So X s a t i s f i e s a l l the requirements of theorem 3.21 except that i t i s not s i m p l i c i a l . But i f some X M ^ i s s i m p l i c i a l , we may choose the attaching maps g^ ^ of lemma 3.18 to be s i m p l i c i a l by applying the s i m p l i c i a l approximation theorem to the g^ and then - 5 8 -defining g a fc = t g ^ as before. Thus X i s obtained from a s i m p l i c i a l complex by making s i m p l i c i a l i d e n t i f i c a t i o n s and therefore some bary-c e n t r i c subdivision of X i s s i m p l i c i a l . Furthermore, such a sub-d i v i s i o n i s stable with respect to the action of TT . Remark 3.27. Our reduction from theorem 3.1 to theorem 3.2 i s now complete. The remainder of the chapter s h a l l be devoted to the proof of theorem 3.2. I I I . PERIODIC PROJECTIVE RESOLUTIONS A. Existence of Periodic Projective Resolutions In theorem 3.29, we s h a l l prove that "shortest p o s s i b l e " periodic p r o j e c t i v e resolutions always e x i s t f o r per i o d i c groups. In part B of th i s section, t h i s w i l l y i e l d a r e s u l t concerning actions of f i n i t e groups on CW-homotopy.spheres. In addition, theorem 3.29 i s the f i r s t step i n the proof of theorem 3.2, which i s the ultimate objective of our e n t i r e discussion. But before proving theorem 3.29, we need a lemma [16, lemma 4.2]. Lemma 3.28. Let A be a ZTT-module, tors i o n — f r e e over Z . Then A ^ Z i f f both of the following conditions are s a t i s f i e d : (a) A ^ Z over a l l Sylow p-subgroups of TT (b) H^(TT;A) contains an element of order n, n = |TT| . We s h a l l not give a proof of t h i s lemma; note, however, that one impl i c a t i o n i s f a i r l y obvious. For, suppose A ^ Z over TT ; then condition (a) i s c e r t a i n l y s a t i s f i e d . As for condition (b), we have "0 "0 H (TT;A) = H (TT;Z) = Z . n - 59 -Theorem 3.29. Let TT be a f i n i t e group having p e r i o d i c cohomology of period k . Then TT has a pe r i o d i c p r o j e c t i v e r e s o l u t i o n of period k over Z . Proof: Choose an exact sequence over ZTT of the form (3.30) 0 —>• A —> MS —* ... -> WQ —*• Z —*• 0 with a l l p r o j e c t i v e . By c o r o l l a r y 3.16, i n order to show the existence of a periodic p r o j e c t i v e r e s o l u t i o n for ir, we need only show that A 'v Z . Our plan i s to show that c r i t e r i a (a) and (b) of lemma 3.28 are s a t i s f i e d ; we may then conclude A ^ Z . So, l e t T\_ be a Sylow p-subgroup of TT . TT i s p e r i o d i c ; thus, by theorem 2.8, ir i s c y c l i c i f p i s an odd prime and i s ei t h e r c y c l i c or generalized quaternionic. Also, by proposition 1.17, the period of TT divides that of TT . Therefore, 2 Ik, and i f TT„ p 1 2 i s generalized quaternionic, 4|k . Here we use the well-known facts that c y c l i c groups have period 2, while generalized quaternionic groups have period 4 . Now, for c y c l i c and generalized quaternionic groups, we can construct e x p l i c i t p e riodic free resolutions of lengths 2 and 4 r e s p e c t i v e l y [2, pp. 250-254]. It follows that T\_ has a pe r i o d i c free r e s o l u t i o n of period k . Applying c o r o l l a r y 3.12, we immediately i n f e r that A "y Z over Hp for a l l Sylow p-subgroups Tt_ of IT, v e r i f y i n g (a) . Consider again the sequence (3.30). We break t h i s sequence up into short exact sequences, and we consider the long exact cohomology sequence of each r e s u l t i n g short exact sequence. It i s clear that, - 60 -since a l l are p r o j e c t i v e and consequently H~'(Tr;W^) = 0 for a l l ~0 ~k j , we obtain the isomorphism H (TT;A) = H (TT;Z) . Since TT has ~0 period k, we have H (TT;Z) = H (TT;Z) . Thus, we have "0 "0 I I "0 H (IT;A) = H (TT;Z) = Z^, n = [ TT | and consequently, H (TT;A) has an element of order n, v e r i f y i n g (b). B. A p p l i c a t i o n to Group Actions on Homotopy Spheres Remark 3.31. If P i s a pro j e c t i v e module, there e x i s t s an i n f i n i t e l y generated free module F s.t. P + F i s free. One sees t h i s as follows: Since P i s p r o j e c t i v e , we may choose Q s.t. P + Q i s free. Then define F = Q + P + Q + P + Q + I t i s clear that F i s free and that P + F i s free. In addition, i f 0 Z P . . — > ... —* P n —* Z —*- 0 i s a k-1 0 periodic p r o j e c t i v e r e s o l u t i o n , we may convert i t into a free r e s o l u t i o n by taking the d i r e c t sum of t h i s r e s o l u t i o n with exact sequences of the form 0 0 — > F. —* F. 0 —*- 0, 0 < i < k - 1, i x — — where the F. are chosen to be free and s.t. P. + F. i s free, as x x x above. Theorem 3.32. Suppose TT has periodic cohomology of period q . Then TT can act f r e e l y and s i m p l i c i a l l y on a finite-dimensional homotopy (q-1)-sphere of dimension (q-1) . Proof: By theorem 3.29, TT has a periodic p r o j e c t i v e r e s o l u t i o n of period q . By remark 3.31, we may construct from t h i s an i n f i n i t e l y generated periodic free r e s o l u t i o n of period q . Now, Milnor's construction applies to i n f i n i t e l y generated r e s o l u t i o n s ; the r e s u l t follows. - 61 -IV. PERIODIC FREE RESOLUTIONS A. The P r o j e c t i v e Class Group We proved i n theorem 3.32 that i f a f i n i t e group has cohomology of period q over Z, then TT can act f r e e l y and s i m p l i c i a l l y on a finite-dimensional s i m p l i c i a l homotopy (q-1)-sphere of dimension q - 1 . But t h i s homotopy sphere w i l l not, i n general, be a f i n i t e complex. By theorem 3.21, i n order to obtain f i n i t e n e s s , we must show that TT has a f i n i t e l y - g e n e r a t e d , periodic free r e s o l u t i o n . The problem can be u s e f u l l y phrased i n terms of the p r o j e c t i v e class group. D e f i n i t i o n 3.33. Let P and P' be p r o j e c t i v e ZTT-modules. We say that P and P' are equivalent i f there e x i s t f i n i t e l y - g e n e r a t e d free ZTT-modules F and F' s.t. P + F = P ' + F ' . The equivalence class of P w i l l be denoted [P] . The p r o j e c t i v e class group C(ZTT) has as elements the equivalence classes of p r o j e c t i v e ZiT-modules. C(ZTT) c a r r i e s an abelian group structure, given by [P] + [P'] = [P + P'] . D e f i n i t i o n 3.34. Suppose that E i s the following periodic p r o j e c t i v e r e s o l u t i o n of TT: 0 —> Z — P , P. —»- z —*- 0 . We define k-1 0 the class of E, C^(E), to be the following element of C ( Z T T ) k-1 C (E) = I (-D^P ] . i=0 Remark 3.35. Because of i t s importance, the r o l e of the " f i n i t e l y -generated" condition has been emphasized above. Now we s h a l l revert to our custom of omitting the term " f i n i t e l y - g e n e r a t e d " , although a l l modules are t a c i t l y assumed to be so. - 62 -We now a r r i v e at the c r u c i a l theorem used i n the proof of theorem 3.2; i t i s the c r i t e r i o n used to decide whether a given periodic pro-j e c t i v e r e s o l u t i o n i s i n f a c t free. Generally speaking, our procedure i s as follows: We construct a p e r i o d i c p r o j e c t i v e r e s o l u t i o n (3.36) 0 —• Z —»• Q —*• ... —> Q Q Z 0 with a l l free except for Q^_i» which determines an "obstruction" i n C(Zir) to r e a l i z i n g (3.36) as a f i n i t e complex; we then show that t h i s obstruction may be k i l l e d by s p l i c i n g r e s o l u t i o n s . Theorem 3.37. There ex i s t s a p e r i o d i c free r e s o l u t i o n over ZTT of period k i f f there e x i s t s a p e r i o d i c p r o j e c t i v e r e s o l u t i o n E over ZTT of period k having ^ ( E ) = 0 • Proof: (=>) Clear. (<=) Suppose E i s a periodic p r o j e c t i v e r e s o l u t i o n for TT of period k with C (E) = 0 . TT We i n d u c t i v e l y construct a sequence of periodic p r o j e c t i v e resolutions E ^ : 0 — > • Z —*- P,^ P 0 l ) — > Z —> 0, k-1 0 -1 <_ i <_k - 2, as follows: Define E ^ - 1 ^ = E . Suppose E ^ has been constructed. We choose a p r o j e c t i v e P! s.t. P^"^ + P! i s f r e e , and we define E^"1""^ to be o -+ z -> p^1:? -> ... -> ?[XX -> pfti + P . + P-1^ + p- ^ P-1^ ••• P ! 1 ^ Z -> 0" k-1 i+2 I + I i i l l - l 0 We see that, i n t u i t i v e l y , the procedure i s to begin at the r i g h t and b u i l d up a free r e s o l u t i o n term by term. Note that our construction preserves exactness. Furthermore, we have C ( E ^ ) = C (E) = 0 for a l l i . TT TT - 63 -Now, consider E ; c l e a r l y , P • ,...,P ^ a r e a 1 1 f r e e , thus [ P ^ k _ 2 ) ] = 0 for 0 <_ j <_k - 2, and since c ( E ( k ~ 2 ) ) = 0, (k-2) i t follows that t ^ - l J = 0 • By d e f i n i t i o n , there e x i s t s a free F (k-2) s.t. P + F i s fr e e . Then the following i s a per i o d i c free r e s o l u t i o n of TT: o + z . P< K: 2 ) + F - p f : 2 ) + F + p f : 2 > + ... + P + z -> o . k-1 k-2 k-3 0 D e f i n i t i o n 3.38. Let E and E' be per i o d i c p r o j e c t i v e resolutions of TT . E: 0 + Z -> P L , -> ... -* Prt Z + 0 h-1 U E' : 0 - > Z ^ P ' - * . . . - * P ! - > Z - > 0 k-1 0 We define EE' to be the per i o d i c r e s o l u t i o n EE' : 0 ->- Z P,_ ., + . . . •+ Fn P' P' -> Z 0 h-1 0 k-1 0 We s h a l l r e f e r to th i s construction as " s t r i n g i n g together" or " s p l i c i n g " E and E' . Remark 3.39. If k i s even, then c (EE') = c (E) + c (E') . TT TT TT C o r o l l a r y 3.40. Let E be a per i o d i c p r o j e c t i v e r e s o l u t i o n over ZTT of period k, k even. Suppose dc^(E) = 0 for some integer d . Then there e x i s t s a per i o d i c free r e s o l u t i o n over ZTT of period dk . Proof: c (E d) = dc (E) = 0 and E d has period dk . TT TT - 64 -B. Existence of Periodic Free Resolutions We s h a l l now show how c o r o l l a r y 3.40 enters into the proof of our p r i n c i p a l r e s u l t that a pe r i o d i c group admits a periodic free r e s o l u t i o n . D e f i n i t i o n 3.41. A group G i s c a l l e d elementary i f i t i s the d i r e c t product of a p-group and a c y c l i c group. The basic procedure i n the proof of theorem 3.2 consists i n constructing a pe r i o d i c p r o j e c t i v e r e s o l u t i o n X of TT s.t. c^,(X) = 0 for a l l elementary subgroups I T ' of TT . Suppose for the moment that we have constructed such a r e s o l u t i o n X of TT, say of period q . Now, i f TT' i s any subgroup of T T , we define the r e s t r i c t i o n map i * : C ( Z T T ) • C ( Z T T ' ) by i"[P] = [P] for [P] e C ( Z T T ) . The c r u c i a l observation to be made here i s that c ,(X) i s j u s t the image of c (X) under i 77 77 Noting t h i s , we now apply a theorem of Swan [15, cor. 9.4] . The proof of t h i s theorem involves a deeper analysis of the proje c t i v e c l a s s group and s h a l l be omitted. However, that part of the theorem relevant to our needs may be stated as follows: i Theorem 3.42. If i (x) = 0 for a l l i : T T ' J = 77 with T 7 1 elementary, then dx = 0, where d i s as i n theorem 3.2. From theorem 3.42 and the f a c t that c .(X) = i (c (X)), we 77 77 immediately conclude that dc^(X) = 0 . At t h i s point we u t i l i z e c o r o l l a r y 3.40; i t says that there e x i s t s a periodic free r e s o l u t i o n - 65 -of TT of period dq, and thus our proof of theorem 3.2 i s complete. So we have reduced our task to the following: Construct a per i o d i c p r o j e c t i v e r e s o l u t i o n X of TT s.t. c^,(X) = 0 for a l l elementary subgroups TT' TT . We s h a l l devote the remainder of chapter 3 to t h i s construction. D e t a i l s omitted from t h i s account may be found i n [16]. Our f i r s t objective i s to obtain considerably more information about ^ ( E ) • We proceed to develop techniques for determining when c^(E) i s zero. One basic idea which appears i s the construction of a standard r e s o l u t i o n and the development of techniques for comparing t h i s "known" r e s o l u t i o n with an a r b i t r a r y r e s o l u t i o n i n order to obtain information about the l a t t e r . We s h a l l now show how to "compare" two a r b i t r a r y p e r i o d i c p r o j e c t i v e r e s o l u t i o n s . Suppose we have two per i o d i c p r o j e c t i v e resolutions of TT: E: 0 —> Z —> P, 1 — > P A —> Z —>- 0 k-1 0 E' : 0 — > Z — > P ' — > P ' — > Z — * 0 . k-1 0 Consider the copies of Z at the right-hand end of these sequences. There e x i s t s a chain map E —>- E' extending 1 . So, l e t f: E —>• E' be a map s.t. f ( l ) = 1 for 1 e Z . We now consider f r e s t r i c t e d to the copy of Z at the left-hand end of the sequence E . We have f ( l ) = r e Z but r £ 1 i n general. If f i s another such map, then we have f ' ( l ) = r ' at the left-hand end. Furthermore, f and f 1 must be chain-homotopic; l e t f - f = s9 + 9s, where s i s a chain-homotopy connecting f and f . I t follows that f ' ( l ) - f ( l ) =s9(l) . TT Note that 9(1) e P, ,, the subset of P .. invariant under the action - 66 -of TT . Since P ^ - l P r o J e c t l v e > ^(7I'»^>jc_i_^ = ^' a n c* th e r e f o r e P, , = N*P, ., . So we have 9(1) = N*x f o r some x e P, .. . Therefore, k-1 k-1 k-1 we have r ' - r = f ' ( l ) - f ( l ) = s 3 ( l ) = sN«x = Ns(x) = n«s(x), n = | i r | . Thus r i s determined uniquely modulo n . D e f i n i t i o n 3.43. d (E,E') w i l l denote the residue c l a s s of r i n Z Tr n Remark 3.44. Since d^(E,E) = 1, d^(E,E') gives some s o r t of measure of the d i f f e r e n c e between E and E' . Lemma 3.45. I f E,E',E" are three p e r i o d i c p r o j e c t i v e r e s o l u t i o n s of peri o d k, then d (E,E") = d (E,E')d (E',E") . Now, sin c e d (E,E')d (E',E) = 1 , d ( E E ' ) i s a u n i t i n Z . We ' TT TT TT n have the f o l l o w i n g converse r e s u l t . Lemma 3.46. Let E be a p e r i o d i c p r o j e c t i v e r e s o l u t i o n of per i o d k . Let r be a u n i t i n Z^ . Then there e x i s t s a p e r i o d i c p r o j e c t i v e r e s o l u t i o n E' of per i o d k w i t h d (E,E') = r . The usefulness of d (E,E') w i l l be apparent once we s t a t e lemma 3.51. This lemma f u r n i s h e s the c r i t e r i o n used to conclude that a c e r t a i n p e r i o d i c p r o j e c t i v e r e s o l u t i o n of an elementary group TT' i s f r e e , i n case TT' i s c y c l i c . But f i r s t we need some p r e l i m i n a r i e s . D e f i n i t i o n 3.47. Let r e Z, and assume (r,n) = 1, where n = | TT | . Then (r,N) s h a l l denote the l e f t i d e a l of ZTT generated by r and the norm element N . Remark 3.48. (r,N) i s ZiT-projective [15, prop. 7.1]. - 67 -Lemma 3.49. If TT i s c y c l i c , (r,N) i s Z i T - f r e e . D e f i n i t i o n 3.50. Let Z denote the group of units of Z . Define a n o r - n * map v: Z_ > C ( Z T T ) by v(r) = [(r,N)] . Lemma 3.51. Let E and E' be per i o d i c p r o j e c t i v e resolutions of period k . Then c (E) - c (E') = ( - l ) k v(d (E,E')) . TT TT ..TT We mentioned that a basic technique would be the comparison of a r b i t r a r y resolutions with standard re s o l u t i o n s . F i r s t , we s h a l l compare resolutions of a per i o d i c group TT with resolutions of a chosen Sylow subgroup T T ^ of TT . This procedure i s very useful since we know exactly what the Sylow subgroups of a pe r i o d i c group look l i k e . In f a c t , T T ^ must be either c y c l i c or generalized quaternionic. So, we s h a l l need to e x p l i c i t l y construct standard pe r i o d i c free resolutions for c y c l i c and generalized quaternionic groups. This i s a well-known procedure, and we have already remarked that e x p l i c i t constructions may be found i n [2, pp. 250-254]. We s h a l l assume we have these standard resolutions at our disp o s a l . Let us note that the standard p e r i o d i c free r e s o l u t i o n for c y c l i c groups has period two, while the standard p e r i o d i c free r e s o l u t i o n for generalized quaternionic groups has period four. We s h a l l also need to compare resolutions of the periodic group TT with resolutions of a c e r t a i n elementary subgroup T T ' of TT ; t h i s elementary subgroup s h a l l be of the form IT '. = I T " x 0, where T T " i s c y c l i c and 8 i s generalized quaternionic. In [16, p. 287], - 68 -Swan constructs a free , p e r i o d i c r e s o l u t i o n of period four for such groups TT' . We s h a l l also assume we have t h i s r e s o l u t i o n at our di s p o s a l . We must s t i l l derive several preliminary r e s u l t s p r i o r to commencing the proof of theorem 3.2. Throughout the remainder of t h i s chapter, we assume that k i s a fixed even integer d i v i s i b l e by the period of TT . We use the symbol [n;p] to denote the highest power of p which I I * divides n, n = TT . As usual, Z R , denotes the group of units 1 1 [n;p] of Z R , . [n;p] ft D e f i n i t i o n 3.52. The group A i s defined to be the quotient of Z R , P [n;p] th * by the (k/2) powers of elements of Z . , . Ln; p J Now, suppose E i s a periodic r e s o l u t i o n of period k over ZTT . For p d i v i d i n g n, choose a Sylow p-subgroup IT of TT . Since T ip i s necess a r i l y e i t h e r c y c l i c or generalized quaternionic, we have a standard p e r i o d i c free r e s o l u t i o n E of TT , as discussed above. P P These resolutions have length 2 or 4 ; however, we may " s p l i c e " them to form periodic resolutions of length k, which we s h a l l also denote by E . I t now makes sense to consider d ( E , E ) . P TTP P D e f i n i t i o n 3.53. OJ ( E ) w i l l denote the image of d ( E , E ) i n A P f p P P Lemma 3.54. Wp(E) i s well-defined. More p r e c i s e l y , Wp(E) 1 S independent of the choice of T r^ and the standard set of generators O f TT p The map w y i e l d s information about c , as the next lemma shows. p TT - 69 -Lemma 3.55. Assume TT i s ni l p o t e n t . Let E and E' be per i o d i c p r o j e c t i v e resolutions of TT of period k . Suppose that f o r a l l primes p d i v i d i n g n, ^ ( E ) = w (E') . Then ^ ( E ) = 0 implies that c (E 1) = 0 . TT We now look at the e f f e c t of r e s t r i c t i n g m_ to a subgroup of TT . Let TT' _C IT be a subgroup; l e t TT^ be a Sylow p-subgroup of TT' and l e t TT ^ TT1 be a Sylow p-subgroup of TT . The analogues of p — p A and co s h a l l be denoted A' and to' . P P P P Let n' = | IT'| ; c l e a r l y , [n' ;p] divides [n;p] . Thus there e x i s t s a map j : A^ —> A^ induced by the canonical p r o j e c t i o n Z[n;p] Z [ n ' ;p] Lemma 3.56. Let E be a per i o d i c r e s o l u t i o n for TT, thus also f o r the subgroup TT' of TT . Then w1 (E) = j u (E), where j : A —>• A' P P P P i s the map defined above: We have now completed a l l p r e l i m i n a r i e s . I t remains only to achieve a synthesis of these r e s u l t s which y i e l d s a proof of theorem 3.2. For convenience, l e t us once again state theorem 3.2. Theorem 3.2. Let TT be a f i n i t e group of order n, and l e t d= (n,(j>(n)). Suppose TT has per i o d i c cohomology of period q . Then TT has a periodic' free r e s o l u t i o n over Z of period dq . Proof: R e c a l l that we have shown that i t i s s u f f i c i e n t to prove that there e x i s t s a periodic p r o j e c t i v e r e s o l u t i o n X of TT s.t. c , (X) = 0 • for a l l elementary subgroups TT' of TT . So, l e t TT' be any elementary subgroup of TT . Then TT' = IT" X 0, - 70 -where TT" i s c y c l i c and 6 i s a p-group. Select any pe r i o d i c p r o j e c t i v e r e s o l u t i o n E of TT of period q, where q i s the period of TT . We consider the two possible cases: Case I: 8 i s c y c l i c , so that TT' i t s e l f i s c y c l i c . This case occurs whenever p i s an odd prime; i f p = 2, i t again occurs i f TT^ i s c y c l i c . Lemma 3.57. Let k be an even integer. Let G be a c y c l i c group and l e t X be a periodic p r o j e c t i v e r e s o l u t i o n of G of period k . Then c G(X) = 0 . Proof: G has a periodic free r e s o l u t i o n Y of period k obtained by s t r i n g i n g together k/2 copies of the standard r e s o l u t i o n of length 2 . Since G i s c y c l i c , (r,N) i s free f o r a l l r; t h i s i s lemma 3.49. Applying lemma 3.51, we have c G(X) - c G(Y) = ( - l ) k v(d G(X,Y)) = (-l) k[(d G(X,Y)),N] = 0 . But Y i s free and so c Q ( Y ) = 0 . Therefore, c Q ( X ) = 0, completing the proof of lemma 3.57. So lemma 3.57 implies that c^,(E) = 0 for a l l c y c l i c elementary subgroups TT' of TT, completing the proof of theorem 3.2 i n case I. Case I I : 9 i s generalized quaternionic. Of course, t h i s case can occur only when i s generalized quaternionic. Then TT' = TT" X 0, a d i r e c t product of a c y c l i c group of odd order and a generalized quaternionic group. In t h i s case, TT' has a standard free r e s o l u t i o n - 71 -of period 4 constructed by Swan i n [16] and discussed previously. Also notice that i n t h i s case 4|q, since generalized quaternionic groups have period 4 . Of course, we can construct a per i o d i c free r e s o l u t i o n of TT' of period q by s p l i c i n g q/4 copies of E , where E i s Swan's r e s o l u t i o n . The following lemma may be proved by computation. q/4 q/4 Lemma 3.58. For p an odd prime, U p ( E ) = > while, for p = 2, u 2 ( E Q / 4 ) = 1 . * q/4 ~ Now, l e t r e Z be a unit whose image i n A i s (-1) co ( E ) n P P for p odd and whose image i n i s c o ^ C E ) . Lemma 3.46 states that there e x i s t s a per i o d i c p r o j e c t i v e r e s o l u t i o n E ' of period q , s.t. d ( E , E ' ) = r " 1 . By lemma 3.45, d ( E , E ' ) = d ( E , E ) d ( E , E ' ) \ TT T T p TT P TT P P P where E i s the standard free r e s o l u t i o n of period q of the Sylow p p-subgroup 71^ . Thus c o ^ C E ' ) = t O p ( E ) r = ( - l ) q ^ 4 for p odd, and u ) 2 ( E ' ) = 1 . By lemma 3.56, c ^ ( E ' ) = j c o P ( E ' ) and i t follows that ( E ' ) = ( - l ) q ^ 4 for p odd and C J ^ ( E ' ) = 1 . But, by lemma 3.58, c j P ( E Q / ' 4 ) = ( - l ) ^ 4 and ( ^ ( E * ^ 4 ) = 1 . Furthermore, both E ' and E Q / ^ 4 have period q . q/4 Also, since E i s free, C 1 T ^ ) = 0 . Now we have shown that a l l the hypotheses of lemma 3.55 hold. So, applying 3.55, we conclude c , ( E ' ) = 0 , i . e . c ( E T ) = 0 for a l l elementary subgroups rr' <= TT, completing the TT proof of theorem 3.2 i n case I I . - 72 -CHAPTER 4: SOME RESULTS ON FREE TOPOLOGICAL ACTIONS ON SPHERES I. A RESULT OF MILNOR As a r e s u l t of Swan's work, we conclude that a group G has per i o d i c cohomology i f f i t acts f r e e l y on a complex having the homotopy type of a sphere. An obvious question i s the following: Can we extend t h i s r e s u l t to conclude that i f G has per i o d i c cohomology, then G acts f r e e l y on a to p o l o g i c a l sphere? The answer to t h i s question i s , i n general, no; t h i s was f i r s t proved by Milnor [5]. This r e s u l t on the non-existence of free actions on spheres i s obtained as a c o r o l l a r y of the following theorem. Theorem 4.1. Let M n be an n-dimensional manifold having the mod 2 homology of the n-sphere. Let T: M n —> M n be a map of period two having no fixed points. Then, for every map f: M n —>• M n having odd degree, there e x i s t s some x e M n s.t. Tf(x) = fT(x) . The proof of theorem 4.1 s h a l l be postponed u n t i l a f t e r we derive some simple but important c o r o l l a r i e s . C o r o l l a r y 4.2. Suppose that the group G acts fixed point f r e e l y on a manifold M n having the mod 2 homology of the n-sphere. Then any element of order two i n G belongs to the centre of G . Proof: Let T e G be of order 2, and l e t U be any other element of G . By theorem 4.1, there e x i s t s x e M n s.t. TU(x) = UT(x) . But G acts fixed point f r e e l y , and therefore we must have TU = UT . That i s , T belongs to the centre of G . - 73 -Coroll a r y 4.3. Suppose that the group G acts fixed point f r e e l y on a manifold M n having the mod 2 homology of the n-sphere. Then G s a t i s f i e s a l l 2p-conditions. Proof: The only groups of order 2p are c y c l i c and dihe d r a l . Now the r e s u l t follows immediately from c o r o l l a r y 4.2. Remark 4.4. Coro l l a r y 4.3 gives another important necessary condition for free actions of f i n i t e groups on spheres. Co r o l l a r y 4.5. S^j the symmetric group on three l e t t e r s , cannot act fixed point f r e e l y on the n-sphere. Proof: S^ has three involutions of order 2, but the centre of S^ i s t r i v i a l . Remark 4.6. It i s well-known that S^ has periodic cohomology of period 4 . Thus c o r o l l a r y 4.5 furnishes an example of a periodic group which does not admit a free action on a sphere. Remark 4.7. Consider the complex X ~. S n constructed i n chapter 3 on which a given group TT acts f r e e l y . C o r o l l a r y 4.2 implies that X cannot, i n general, be a manifold. The remainder of t h i s section s h a l l be devoted to the proof of theorem 4.1. Proof of theorem 4.1 D e f i n i t i o n 4.8. Z^ acts on M n x M n by (x,y) — • (y,x) . Dividing out by the action of Z we obtain a space M n x M n/Z„, which s h a l l - 74 -be denoted M * M , and c a l l e d the symmetric product. Let A denote the set of a l l points (x,Tx) i n M n x M*1 and l e t A' denote the set of a l l {x,Tx} i n M n * Mn, where {x,Tx} i s the image of (x,Tx) under the canonical map M n x M*1 —> M n * M1 Suppose that for a l l x e Mn, we have Tf(x) ^ fT(x) . Then there e x i s t s a commutative diagram (4.9) where i ^ and 1^ are the canonical i d e n t i f i c a t i o n maps, p^(x,y) =x, f x ( x ) = ( f ( x ) , f ( T x ) ) , f 2{x,Tx} = .{f(x), f(Tx)} . The c r u c i a l f a c t needed for the proof of theorem 4.1 i s contained i n the following lemma. Lemma 4.10. The homomorphism i : H k(M n x M*1 - A) —> H k(M n * M n - A') i s an isomorphism for a l l k, where H denotes singular homology with Z 2 _ c o e f f i c i e n t s . Proof: The pro j e c t i o n p t : M n x M n - A —> M n i s a l o c a l l y t r i v i a l f i b r e map with f i b r e M n - Tx . Since we are using Z 2 ~ c o e f f i c i e n t s , a l l our manifolds are orientable; i t follows that we may apply the Lefschetz d u a l i t y theorem to the pair (Mn,Tx) . Doing so, we obtain an isomorphism H q(M n - Tx) = H n _ q(M n;Tx) and i t follows that M n - Tx has the homology of a point. - 75 -Using the f i b r a t i o n M n - Tx >• M n x M n - A P l n o t i n g that M n - Tx i s hom o l o g i c a l l y a point and M n i s homologically a sphere and applying the Serre exact homology sequence, we f i n d that the induced map (4.11) p- : H A(M n x M n - A) • R*(M n) i s an isomorphism. Since T i s f i x e d p o i n t f r e e , the diagonal subset A c_M n x M n a c t u a l l y l i e s i n M x M - A . Let A' d enote tne image of A i n M n * M n - A' . I t i s c l e a r that the diagonal map M n — A <: M n x M n - A i s a s e c t i o n of the f i b r e map p., . I t f o l l o w s that p,. d,. = 1 * m , 1 1* * H (M n) and s i n c e p ^ i s an isomorphism by (4.11), d ? v i s a l s o an isomorphism. Using t h i s f a c t and the long exact homology sequence of the p a i r (M n x M n - A, A), we conclude that (4.12) H.(M n x M n - A, A) = 0 . The next step i n the argument c o n s i s t s i n showing that the " r e l a t i v e 2 - f o l d c o v e r i n g " (M n x M n - A, A) (M n * M n - A', A') has a Gysin cohomology sequence. A problem a r i s e s because c i s a covering only o f f the diagonal A . Let U be any symmetric neighbourhood of A, and l e t U' denote the image of U i n M n * M n - A' . Consider the 2 - f o l d covering - 76 -(s°,s°) = ( z 2 , z 2 ) -> (M n x M n - A - A, U - A) (M n * M n - A' - A' , U' - A') Cl e a r l y , c^ i s a well-defined covering; also, c^ has associated a Gysin cohomology sequence. Consider the diagram (M n x M n - A - A, U - A) — • (M n x M*1 - A, U) (M n * M n - A' - A' , U' - A') — 6 — * (M n *M n - A' , U 1) (M n x M n - A, A) (M n * M n - A', A') As above, c^ has associated a Gysin sequence. Since e and e' are exci s i o n maps, c 2 also has associated a Gypsin sequence. Now, both A and M n x M n - A are absolute neighbourhood r e t r a c t s ; therefore, H*(Mn x M n - A, A) = lira H*(Mn x M n - A, U) where U ranges over a l l symmetric neighbourhoods of A . Furthermore, * n n an analogous statement holds for H (M * M - A', A') . Now, since the d i r e c t l i m i t of exact sequences i s i t s e l f exact, we obtain a Gysin sequence for c . (4.13) -+ Hk(MnxMn-A,A) + Hk(Mn*Mn-A* , A ' ) -»• H k + 1 (Mn*Mn-A' , A ' ) + H k + l(M nxM n-A,A) It follows from (4.12) that the left-hand and right-hand groups i n (4.13) are zero. By induction, i t follows that the middle groups are zero as w e l l . Thus, the corresponding homology groups H, (M n * M n - A', A') are also zero. So i t i s clear that the homomorphism j ^ : H (A') -> H^(Mn * M n - A') induced by the i n c l u s i o n - 77 -A' — y M M A', i s an isomorphism. Since, by (4.12), H^(Mn x M n - A, A) = 0 the homomorphism j ^ : H^(A) —> H^M11 x M n - A) induced by j : A —> M n x M n - A i s an isomorphism as w e l l . Notice also that ±2|A: A — y A' i s a homeomorphism. Consequently, the commutative diagram H*(A) ( i 2 A ) # H (A') -> H A(M n x M n - A) H^(Mn * M n - A') shows that ( i 2 ) * : H^(Mn x M n - A) — y H j V(M n * M n - A') i s an isomorphism, complet ing the proof of lemma 4.10. Now consider the following diagram, obtained by applying the functor H ( ) to (4.9) n H (Mn/T) n ( f 2 ) , -y H (M n * M n - A') n F i r s t , we observe that, since we are using mod 2 c o e f f i c i e n t s , that ( i j _ ) * = 0 . By lemma 4.10, C ^ ) * i s a n isomorphism. Hence (f-^)* = 0, and so f^ = 0, contrary to our assumption that f i s of odd degree. This completes the proof of theorem 4.1. - 78 -I I . A RESULT OF PETRIE AND LEE In keeping with the sketchy nature of t h i s • s e c t i o n , we s h a l l not attempt to define any terms. In p a r t i c u l a r , we s h a l l assume f a m i l i a r i t y with 'surgery' and associated notions. Some discussion of these ideas i s contained i n chapter 5. A long-standing conjecture had been the following: If the f i n i t e group G acts f r e e l y on a sphere, then G neces s a r i l y has an orthogonal action. A l l examples of groups which did act f r e e l y on a sphere could i n fac t act orthogonally; also the r e s u l t of Milnor (4.3) which showed that a p e r i o d i c group such as S^ could not act f r e e l y on a sphere, tended to support t h i s view. On the other hand, the r e s u l t of Swan (3.1) showed that any periodic group could act f r e e l y on a CW homotopy sphere. Thus i t was d i f f i c u l t to guess whether or not there existed a free, non-orthogonal action of some per i o d i c group on a sphere. F i n a l l y , P e t r i e , and, independently, Lee, showed that such actions did e x i s t , thereby proving that the above conjecture i s f a l s e . We s h a l l give a very b r i e f resume of the work of P e t r i e and of Lee. In order to construct a free action of a periodic group ir on a sphere, one may proceed along the following l i n e s : (a) I t i s known that TT acts f r e e l y on a CW-complex X having the homotopy type of a sphere (3.1). (b) One shows that the o r b i t space X / T T s a t i s f i e s Poincare d u a l i t y . (c) Using surgery, one r e a l i z e s the homotopy type of X/TT by a manifold M . - 79 -(d) The u n i v e r s a l covering manifold M of M i s a sphere; the covering transformations of M y i e l d a free action of TT . B a s i c a l l y , what Lee does i s show that, for the metacyclic group Z , p,q odd primes, the obstruction to carrying out step (c) of the P > q. above program l i e s i n a zero group. Now, l e t us ind i c a t e P e t r i e ' s approach. The f i r s t step consists i n e x h i b i t i n g a free action of the metacyclic group Z_ on a c e r t a i n Brieskorn v a r i e t y K of dimension 2q - 1 . Homologically, K i s almost a sphere; i n f a c t R\(K;Z) = 0 for i 4- 0, q - 1, 2q - 1 and H^_^(K;Z) i s a tor s i o n group a n n i h i l i a t e d by a power of q . We focus attention on H (K;Z) . q-1 The c r i t i c a l step involves a study of the l i n k i n g number form (TT)/Z(TT) ) q-1 ZTT q-1 where TT = Z [7, p. 109]. One needs to know that d> i s of a p»q c e r t a i n form, i n order to apply the following theorem of Wall [21, p. 255] Theorem 4.14. A necessary and s u f f i c i e n t condition that surgery i s possible on K/TT y i e l d i n g a manifold N with TT^(N) = TT and such that the u n i v e r s a l covering manifold E of N i s a homotopy sphere i s that there e x i s t a free module F of f i n i t e rank k over ZTT and a (kxk)-matrix B such that B = -B and an exact sequence 0 —>• F — y F — y K K ) —>- 0 such that the l i n k i n g number form (y) (x) = Ex.B.^y. mod ZTT, where i i j J x = ( x ^ , . . . , x k ) , y = (y^,...,y k) e F . Here B_Jr denotes the inverse - 80 -transpose of B . P e t r i e undertakes an analysis of Hermitian forms on Z , using p,q the algebra thus developed to conclude that cf> i s indeed of the form s p e c i f i e d by (4.14). I t i s thus possible to perform equivariant free surgery on K, transforming i t into a homotopy sphere and y i e l d i n g the sought-after free action of Z Remark 4.15. Lee's proof requires that both p and q be odd primes; al s o , the minimum dimension of a homotopy sphere on which Z acts p,q 2 i s 4q - 1 . Petri e ' s r e s u l t i s the more general i n that he shows that every metacyclic group admits such free actions. Furthermore, i t i s shown that the group may act on a homotopy sphere of dimension (2q-l) ; since the period of Z i s 2q, t h i s r e s u l t i s best pos s i b l e . - 81 -CHAPTER 5: SUFFICIENT CONDITIONS FOR FREE ACTIONS I. INTRODUCTION So far we have derived various necessary conditions for the existence of a free action of a f i n i t e group G on a sphere. For example, G must 2 s a t i s f y a l l p -conditions ((2.6) and (2.8)) and a l l 2p-conditions (4.3). As for s u f f i c i e n c y conditions, i f G s a t i s f i e s a l l pq-conditions and i s solvable, t h i s i s s u f f i c i e n t to guarantee the existence of a free, i n f a c t , free orthogonal action of G on a sphere (2.24). But, by the r e s u l t of P e t r i e and Lee (ch. 4, sect. I I ) , that G s a t i s f y a l l pq-conditions i s not necessary for the existence of free t o p o l o g i c a l actions on spheres. Obviously, i t would be desirable to extract from our various group-theoretic c r i t e r i a necessary and s u f f i c i e n t conditions for the existence of free t o p o l o g i c a l actions on spheres. This has recently been done i n a paper by Madsen, Thomas, and Wall [4]. In t h i s chapter, we discuss t h e i r work. The main theorem we wish to prove i s t h i s : Theorem 5.1. Let TT be a f i n i t e group. There e x i s t s a free t o p o l o g i c a l 2 action of TT on a sphere i f f TT s a t i s f i e s a l l p - and 2p-conditions. The following r e s u l t holds for^smooth actions: Theorem 5.2. For each free action of TT on S n constructed i n the proof of (5.1), there e x i s t s a d i f f e r e n t i a b l e structure a on S n such that TT acts f r e e l y and smoothly on S n . - 82 -I n * p r i n c i p a l , our procedure i s that outlined i n chapter 4, section I I . A l i t t l e more e x p l i c i t l y , our pattern of proof i s as follows: We begin by constructing a c e r t a i n f i n i t e s i m p l i c i a l complex X . Then we construct a normal invariant for X, which determines a normal cobordism class of normal maps M —>• X, M some manifold. Next, we show that the r e s u l t i n g surgery obstruction vanishes. This y i e l d s a manifold homotopy equivalent to X whose un i v e r s a l cover i s homotopy equivalent to a sphere, thus homeomorphic to a sphere. Two c r i t i c a l points i n the procedure involve the choice of X and of normal inva r i a n t for X; i t i s e s s e n t i a l that we make these choices i n a very s p e c i f i c manner. This allows us to prove the vanishing of the surgery obstruction. We s h a l l have more to say on t h i s matter l a t e r . Most d e t a i l s omitted from the following account may be found i n [4], except where noted. I I . THE CHOICE OF HOMOTOPY TYPE Our s t a r t i n g point i s the work of Swan discussed i n chapter 3; we s h a l l presently generalize h i s main theorem i . e . (3.1). Let us introduce the class of spaces which we s h a l l work with. D e f i n i t i o n 5.3. Let Y be a CW-complex dominated by a f i n i t e complex and suppose n >^ 3 . Y i s said to be (ir,n)-polarized i f there e x i s t s an isomorphism ir^(Y,yQ) —>• TT and a homotopy equivalence Y —>• S n \ where Y i s the u n i v e r s a l cover of Y . Remark 5.4. The following observations are easy consequences of (5.3): (i ) If Y i s (ir,n)-polarized, TT i s p e r i o d i c of period n . ( i i ) Let X,Tr,d,q be as i n theorem 3.1, and l e t dq >^ 3 . Then - 83 -the coset space X/TT i s (rr , d q ) - p o l a r i z e d . D e f i n i t i o n 5.5. Two p o l a r i z e d spaces and are s a i d to be equivalent i f there e x i s t s a homotopy equivalence f : Y^ —^ ^ 2 w n i c n preserves the p o l a r i z a t i o n s . We s h a l l need the f o l l o w i n g p r o p o s i t i o n which i s proved i n [ 1 8 ] . P r o p o s i t i o n 5.6. Equivalence c l a s s e s of (TT,n)-polarized complexes correspond b i j e c t i v e l y ( v i a the f i r s t k - i n v a r i a n t ) to generators g 1 H n ( T r ; Z ) . Remark 5.7. R e c a l l that the o b s t r u c t i o n to r e p l a c i n g the complex Y p r o p o s i t i o n 5.6 by a f i n i t e complex i s the 'Euler c h a r a c t e r i s t i c ' 6 the a s s o c i a t e d r e s o l u t i o n i n C(ZTT) . Note that t h i s depends only on the homotopy type of Y, hence by p r o p o s i t i o n 5.6, 6 = 9(g) depend only on the generator g of H n ( T r;Z) . The important p o i n t from chapter 3 to bear i n mind i s t h i s : g may be chosen s . t . 6 vanishe and thus Y may be chosen to be f i n i t e ; furthermore, the complex Y so chosen i s (TT ,n) - p o l a r i z e d . More to the p o i n t , the f o l l o w i n g g e n e r a l i z a t i o n of theorem 3.1 w i l l be c r u c i a l i n the proof of our p r i n c i p a l r e s u l t , theorem 5.1. Theorem 5.8. There e x i s t s a (TT,n)-polarized complex Y = Y(TT) s . t . f o r each subgroup p c_ TT admitting a f i x e d p o i n t f r e e orthogonal r e p r e s e n t a t i o n , the covering space Y ( p ) of Y corresponding to P i s homotopy equivalent to a manifold. - 84 -I I I . EXISTENCE OF NORMAL INVARIANTS A Poincare complex i s e s s e n t i a l l y a CW complex which s a t i s f i e s an appropriate form of Poincare d u a l i t y . For a precise d e f i n i t i o n , r e f e r to [22]. Now, a ( T r,n)-polarized complex Y i s n e c e s s a r i l y a Poincare complex, hence has a 'Spivak normal f i b r a t i o n ' [12]. This i s a s p h e r i c a l f i b r a t i o n and thus i s c l a s s i f i e d by a map Y —> BG . Here, BG i s the c l a s s i f y i n g space for stable s p h e r i c a l f i b r a t i o n s [13]. Let us add a few explanatory remarks i n order to j u s t i f y i n t r o -duction of the above notions. We s h a l l assume f a m i l i a r i t y with the basic notions of surgery on manifolds [6], [1], [22]; however, we s h a l l o f f e r some explanation of how surgery enters into our p a r t i c u l a r problem. Let X be a CW complex, M a closed manifold, and Jj>: M — y X a map. Suppose we wish to perform surgery on the manifold M and the map i n order to make § into a homotopy equivalence. C l e a r l y , i f such a homotopy equivalence i s to e x i s t , c e r t a i n requirements must be imposed on X . For instance, X must s a t i s f y Poincare d u a l i t y . For t h i s reason, we r e s t r i c t ourselves to the class of Poincare - complexes. Secondly, there must e x i s t a bundle over X with s p h e r i c a l Thorn c l a s s . (To see t h i s , consider a manifold N; embed i t i n a high-dimensional Euclidean space, and l e t y be the normal bundle of the embedding. Then y has s p h e r i c a l Thorn class.) This i s the consideration which leads to our introduction of the Spivak normal f i b r e space v . v i s a s p h e r i c a l f i b r a t i o n having sph e r i c a l Thorn class (and i s unique up to f i b r e homotopy equivalence). It follows that i n order to prove the existence of a bundle over X with s p h e r i c a l Thorn c l a s s , i t s u f f i c e s to prove that we can reduce the structure monoid of v from G to TOP - 85 -(in the to p o l o g i c a l case) or to 0 ( i n the smooth case). Equivalently, we s h a l l prove the existence of normal in v a r i a n t s . D e f i n i t i o n 5.9. Let the c l a s s i f y i n g map Y — B G be as above. A top o l o g i c a l (resp. smooth) normal invariant for Y i s a homotopy class of l i f t i n g s of f to BTOP (resp. BO) . Remark 5.10. Normal invariants for a Poincare complex Y are i n 1-1 correspondence with normal cobordism classes of normal maps M — y Y, M a manifold (topological or smooth, as the case may be) [22]. Suppose we know that a normal invariant e x i s t s for Y . This normal invariant determines a normal cobordism class of normal maps M —> Y, as above. Given such a cobordism c l a s s , there i s a w e l l -developed procedure for se t t i n g up a surgery obstruction problem for the construction of manifolds homotopy equivalent to Y [22] . The following general existence theorem shows that we always have j u s t t h i s s i t u a t i o n . Theorem 5.11. Any f i n i t e (TT,n)-polarized complex Y admits a smooth no rma1 invar ian t. Proof: The canonical map BO — y BG i s a map of i n f i n i t e loop spaces. Noting that G/0 i s an i n f i n i t e loop space and thus also B(G/0), we consider the diagram y G/0 y BO y BG y B(G/0) y \ f \ ? \ v \ \ Y - 86 -where the h o r i z o n t a l sequence i s a sequence of i n f i n i t e loop maps and where v i s the c l a s s i f y i n g map f o r the Spivak normal bundle. Since BO —> BG —> B(G/0) i s a f i b r a t i o n , i t f o l l o w s that the o b s t r u c t i o n to the existence of a smooth normal i n v a r i a n t f o r Y i s an element ft ft of k ( Y ) , where k i s the cohomology theory represented by B(G/0) . Since t h i s group i s f i n i t e l y generated, i t s u f f i c e s to prove that t h i s o b s t r u c t i o n vanishes when l o c a l i z e d at any a r b i t r a r y prime p . We compare the o b s t r u c t i o n f o r Y = Y ( T T ) w i t h that f o r the covering space Y ( T T ) of Y corresponding to the Sylow p-subgroup c: TT ; we use a ge n e r a l i z e d t r a n s f e r argument. Now, i f f: X —>• X i s a f i n i t e ft covering and h i s any cohomology theory, there e x i s t s a t r a n s f e r ft ~ ft + ~+ homomorphism f ^ : h (X) —> h (X) induced by an S-map X —*• X Here S-map means a map of some high dimensional suspensions and X + i n d i c a t e s the d i s j o i n t union of X and a base-point. Suppose that f: X —>• X i s a j - f o l d cover and j e h (pt.) i s i n v e r t i b l e ; then f o f : h (X) —>• h (X) i s an isomorphism. To see t h i s , note that there i s induced an endomorphism of the Atiyah-Hirzebruch s p e c t r a l ft ft ft 2 sequence H (X;h (pt. ) ) => h (X), which, on the E -term, i s m u l t i -ft p l i c a t i o n by j . I t now f o l l o w s that f i s i n j e c t i v e . The i n c l u s i o n map i : TT C TT induces a covering Y ( T T ) — ^ Y ( T T ) P — P which i s compatible w i t h Spivak normal bundles, and, t h e r e f o r e , i : k ( Y ( T T ) ) —> k ( Y ( T T )) maps the o b s t r u c t i o n to existence of a smooth normal i n v a r i a n t f o r Y ( T T ) to that of Y ( T T ) . P B(G/0) • - 87 -Now the degree of the covering n i s [TT: TT ] and so i s prime to p . o u p ft Consequently, the above discussion implies that i i s i n j e c t i v e when we l o c a l i z e at p . Hence, i n order to complete the proof, i t s u f f i c e s to show that a l l Y ( T T ) admit smooth normal inv a r i a n t s . This involves P a case by case study. Of course, r r ^ i s either c y c l i c or generalized quaternionic. If the former, then Y(TT ) i s homotopy equivalent to a lens space. If the l a t t e r , then p = 2; assume i s of order 2 , with r >_ 3 . For Y C T ^ ) of f i x e d dimension, the polarized homotopy types correspond to odd integers £(mod 2 ) . For SL = ±1 (8), there e x i s t smooth normal invariants defined by fixed point free orthogonal representations, while for 'SL = ±3 (8), Y C T ^ ) i s not homotopy equivalent to a f i n i t e complex, therefore neither i s Y(TT) [4, p. 378]. This completes the proof. We now have existence of smooth normal: i n v a r i a n t s ; however, for our purposes, t h i s w i l l not quite s u f f i c e . In the proofs of theorems 5.1 and 5.2, we must be able to choose a normal invariant for Y(TT) i n such a way that i t behaves n i c e l y on r e s t r i c t i o n to Y(TT') , where TT' i s a c e r t a i n subgroup of TT . We consider two cases: ( i ) TT solvable ( i i ) TT not solvable. The following proposition i s proved i n [18]. Proposition 5.12. If TT i s solvable, then TT contains a subgroup x s.t. (a) T contains a Sylow 2-subgroup of TT . (b) The only prime d i v i s o r s of |x| are 2 and 3 . 2 2 (c) The r e s t r i c t i o n homomorphism H (TT;Z^) h H (x;Z^) i s an isomorphism. - 88 -(d) There ex i s t f i x e d point free orthogonal actions of x on spheres. The importance of the existence of such a x derives from the following lemma, which i s proved i n [18]. Lemma 5.13. Any (topological) normal inva r i a n t f o r Y(x) extends to one for Y(TT) . For TT not solvable, we have the following r e s u l t . Lemma 5.14. A (topological) normal invariant c for Y(TT2) extends to one for Y(TT) i f f for each subgroup Q of contained i n a ft . ft subgroup T of TT, the r e s t r i c t i o n c|Y(Q) extends to Y(T ) . ft Here Q i s the quaternion group of order 8 and T i s the binary tetrahedral group. A proof of lemma 5.14 may be found i n [4]. IV. SURGERY Theorems 5.1 and 5.2 are derived from two general r e s u l t s which we now state. Theorem 5.15. Let : M —>• Y be a normal map of degree 1, M a closed manifold, Y a f i n i t e Poincare complex of formal dimension >^ 5 having f i n i t e fundamental group TT . Then surgery on to obtain a homotopy equivalence i s possible i f f (a) For each 2-hyperelementary subgroup p e n , .the covering space Y ( p ) i s homotopy equivalent to a manifold. (b) Surgery i s possible f o r the covering normal map : M(TT ) —• Y(TT - 89 -where TJ i s the Sylow 2-suhgroup of TT and, i f the dimension of M i s . even (c) The e q u i v a r i a n t signature of M i s a m u l t i p l e of the r e g u l a r r e p r e s e n t a t i o n of TT . Theorem 5.16. Let X be a closed t o p o l o g i c a l manifold of odd dimension w i t h f i n i t e fundamental group IT . Suppose that ( i ) X has a smooth normal i n v a r i a n t ( i i ) the covering space X C T ^ ) i s smoothable. Then X i s homotopy equivalent to a smooth manifold. The proof of theorem 5.15 r e s t s on the n o t i o n of r e d u c t i o n to subgroups, i n p r i n c i p l e i n much the same way as the proof of theorem 3.2 does. S p e c i f i c a l l y , one uses theorems of Dress to show that i t i s s u f f i c i e n t to prove 5.15 f o r hyperelementary subgroups. The key r e s u l t here i s that the n a t u r a l r e s t r i c t i o n map L^(TT) — y Z{L_^(p)|p _c TT hyperelementary} i s i n f e c t i v e . Here L_^ denotes the i surgery o b s t r u c t i o n group. The proof i s completed by applying r e s u l t s of Wall on the c a l c u l a t i o n of L-groups. The proof of theorem 5.16 u t i l i z e s these r e s u l t s of W a l l , along w i t h another a p p l i c a t i o n of the g e n e r a l i z e d t r a n s f e r and r e s u l t s of S u l l i v a n on the homotopy type of G/TOP. Assuming the r e s u l t s of (5.15) and (5.16), we s h a l l now prove (5.1) and (5.2). Theorem 5.1. Let TT be a f i n i t e group. There e x i s t s a f r e e t o p o l o g i c a l 2 a c t i o n of IT on a sphere i f f rr s a t i s f i e s a l l p - and 2p-conditions. Proof: The stated c o n d i t i o n s are necessary by (2.6), (2.7) and (4.3). To prove s u f f i c i e n c y , we consider two cases, according to whether or not - 90 -TT i s solvable. Case I : TT solvable: By theorem 5.8, there e x i s t s a f i n i t e ( i r,n)-polarized complex Y = Y (TT) s.t., for a l l subgroups p £ TT admitting f i x e d point free orthogonal representations, the covering space Y ( p ) of Y corresponding to p i s homotopy equivalent to a manifold. Consider the subgroup T of proposition 5.12; by .5.12 (d), x has a fi x e d point free orthogonal action on some sphere S . Then (5.8) implies Y ( x ) - Z ( T ) , where Z ( x ) i s the smooth manifold S /x . We now apply (5.13); the normal invariant defined by Z ( x ) extends to a normal inva r i a n t for Y . Having t h i s , we seek to apply theorem 5.15; we must show that conditions (a) and (b) of that theorem are s a t i s f i e d . F i r s t , i f p _£ TT i s 2-hyperelementary, then i t i s solvable. Also, every subgroup of p of odd order i s c y c l i c , and by assumption, every subgroup of p of order 2p i s c y c l i c . So p s a t i s f i e s a l l pq-conditions and i s solvable. Hence, by theorem 2.24, p has a fi x e d point free orthogonal action on some sphere S n and thus Y ( p ) i s homotopy equivalent to a manifold by (5.8). This v e r i f i e s (a). To v e r i f y (b), note that, by construction, surgery on the covering map corresponding to x y i e l d s a homotopy equivalence Z ( x ) - Y ( x ) . Since, by 5.12 (a), x contains a Sylow 2-subgroup n , the same applies to the covering corresponding to TT^ . Case I I : TT not solvable. Again, choose Y as i n (5.8). Then Y(TT^) - Z(v^), where we have obtained t h i s homotopy equivalence v i a a fixed point free orthogonal representation X of n • Of course, Y(T\^) has corresponding a normal invariant; we claim that t h i s normal invariant extends to Y ( i ) , - 91 -Applying (5.14), we see that i t i s s u f f i c i e n t to check that we can extend i n the case i n which we have a subgroup Q of T T ^ contained i n a subgroup T of TT . Now, Q has a unique i r r e d u c i b l e fixed point free representation. Hence the r e s t r i c t i o n of X to Q must be a multiple of t h i s representation, and so i t extends to a fixed point * ft free representation X of T But Y was chosen as i n (5.8); thus, ft the covering space Y ( T ) i s homotopy equivalent to the manifold n * S /im X . Thus, i f n i s a normal invariant for Y(TT^) , we have defined an extension of n|Y(Q) to Y ( T ) . The proof i s completed as i n the solvable case, replacing T by TT^ • We have now shown that (a) and (b) of (5.15) are s a t i s f i e d . As a r e s u l t , we conclude that Y has the homotopy type of a manifold. Consequently, so does the u n i v e r s a l cover Y . But Y i s ( i T , n ) -p o l a r i z e d ; thus Y - S n and i t follows that Y i s homeomorphic to S n (we have assumed n >_ 5 ) . Now, again by d e f i n i t i o n of a ( T r , n ) -polarized complex, TT = TT^(Y), and thus the action of TT^(Y) on Y (by covering transformations) furnishes a free action of TT on S n . This completes the proof of theorem 5.1. Theorem 5.2. For each free action of TT on S n constructed i n the proof of (5.1), there e x i s t s a d i f f e r e n t i a b l e structure a on S n such that TT acts f r e e l y and smoothly on S n . a Proof: We have constructed a free action of TT on S n i n theorem 5.1; l e t the o r b i t space S n / f r be denoted X . I t i s c e r t a i n l y s u f f i c i e n t to prove that X i s homotopy equivalent to a smooth manifold. We do t h i s by an a p p l i c a t i o n of theorem 5.16, so we must show that conditions - 92 -( i ) and ( i i ) of that theorem h o l d . As f o r c o n d i t i o n ( i ) , the existence of a smooth normal i n v a r i a n t f o r X i s guaranteed by theorem 5.11. A l s o , the normal i n v a r i a n t n f o r X was obtained by extension of a c e r t a i n normal i n v a r i a n t , e i t h e r one f o r Z ( T ) , using (5.13), or one f o r Z(v^), using (5.14). In both cases, the normal i n v a r i a n t was determined by a f i x e d p o i n t f r e e orthogonal r e p r e s e n t a t i o n of the appropriate subgroup (T or n^) of TT . Thus the r e s t r i c t i o n of n to X(T^2) y i e l d s a normal i n v a r i a n t which i s smooth. Thus the covering space X(TI^) i s smoothable, v e r i f y i n g ( i i ) . 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