Semifree Actions of Finite Groups on Homotopy Spheres John Ewing Transactions of the American Mathematical Society, Volume 245 (Nov., 1978), 431-442. Stable URL: http ://links.j stor. org/sici ?sici=0002-9947 %28197811%29245 % 3C431% 3ASAOFGO% 3E2.0.CO% 3B2-5 Your use of the JSTOR archive indicates your acceptance of JSTOR' s Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Transactions of the American Mathematical Society is published by American Mathematical Society. Please contact the publisher for further permissions regarding the use of this work. Publisher contact information may be obtained at http://www.jstor.org/joumals/ams.html. Transactions of the American Mathematical Society ¸ 1978 American Mathematical Society JSTOR and the JSTOR logo are trademarks of JSTOR, and are Registered in the U.S. Patent and Trademark Office. For more information on JSTOR contact j stor-info@umich.edu. ¸2002 JSTOR http://www.j stor. org/ Wed Jul 24 14:41:53 2002 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 245, November 1978 SEMIFREE ACTIONS OF FINITE GROUPS ON HOMOTOPY SPHERES BY JOHN EWING 1 ABSTRACT. We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial. 0. Introduction. A group G is said to act semifreely on a space X if every point is either fixed by every element of G or fixed only by the identity. The classification of smooth semifree actions of finite groups on homotopy spheres has been discussed by Browder and Petrie [3] and Rothenberg [6]. We briefly summarize the basic scheme. Given a finite group G we fix a representation p: G --> O (d) such that p restricted to the unit sphere S d-  is fixed point free. A (G, p)-manifold M is a smooth manifold together with a smooth, semifree action of G on M such that the fixed point set F is nonempty and locally the representation of G on the normal bundle of F in M is equivalent to p. In a natural way this defines a reduction of the structure group of the normal bundle to Z(p), the centralizer of p(G) ih O (d). A (G, p)-orientation is a specific reduction of the structure group of the normal bundle to Z(p). We then define C Jr(p) to be the set of (G, p)-oriented h-cobordism classes of (G, p)-oriented manifolds which are homotopy N-spheres with fixed point set a homotopy (N- d)-sphere. The set C Jr(p) has the structure of an abelian group under the connected sum operation. The object of the present paper is to prove the following qualitative result about the groups CN(p). THEOREM A. Let p: G --> O (2d) be a fixed point free representation of a finite group G and suppose that 2N - 2d > 2. Then c2V(p) is a finite group. We point out that the condition that p has even dimension is a restriction only when I GI = 2. This case is already well understood [3], [6]. The essential ingredient for proving this theorem is well known. According Received by the editors September 16, 1977. AMS (MOS) subject classifications (1970). Primary 57E25; Secondary 57D85. I Partially supported by NSF Grant # MCS76-05973. ¸ American Mathematical Society 1979 431 JOHN EWING to [7] there is a long exact sequence -)nS(D2n+l x L,S 2n x L).)C2N(p)  F2n where 2n = 2N - 2d, HS denotes the group of homotopy smoothings, L is the orbit space of S 2a- 1 under p, F:n is the group of homotopy spheres and A (p) is the monoid of G-equivariant self-maps of S :a- 1. From the surgery exact sequence and the finiteness of the odd dimensional Wall groups it follows that HS(D 2n+l X L, S 2n X L) is a finite group. It is well known that F2n and r2n(A(p)) are finite groups also. Therefore we must consider the composite: c (0) L r:n ß (0), z (0)) $proj '2n(A(p), Z(p)) - Yl'2n_l(Z(p)) which assigns to an element of c2N(p) the element of r_l(Z(p)) which classifies the equivariant normal bundle to the fixed point set. In order to prove Theorem A we must prove the following. THEOREM B. Let p: G--> O (2d) be a fixedpointfree representation of afinite group G and suppose 2N - 2d  2. Then the image of  o proj o q: c2N (p)  r2N-2a-l(Z(P)) contains no elements of infinite order. A number of special cases of this result have previously been shown. In [8] Schultz obtained the result for G cyclic of prime order p and either p small (3, 5 or 7) or N - d large (N - d  2 log 2 p - 1). (There is a sign error in [8] which does not affect the conclusions except forp = 7; see [9].) Schultz' work easily extends to cyclic groups of nonprime power order, but again for N - d sufficiently large. In [12] Wang has extended the calculations using a different approach; but again there is a sign error which invalidates some of the results, especially Corollary 3.8 and Theorem 4.7 for even order groups. The work of [4] is essentially a proof of Theorem B in the case where G is cyclic of prime order. Our proof here is similar, but both the algebraic and topological arguments require more careful analysis. The proof of Theorem B will proceed in several stages. In õ 1 we make some easy observations about the centralizers of representations which show that it is sufficient to prove the theorem for G cyclic. In õ2 we prove the result for cyclic G using the Atiyah-Singer G-signature Theorem together with an algebraic lemma. The proof of the algebraic lemma is then given in õ3. It should be remarked that the techniques of the present paper yield considerable information about the groups cN(p) when N is odd. In particu- lar, these groups, even when they are finite, tend to be of rather large order, and the orders are related to both the homotopy groups of spheres (which is FINITE GROUPS ON HOMOTOPY SPHERES 433 expected) and to the order of certain ideal class groups (which is unexpected). The author would like to thank Reinhard Schultz, Bob Stong and Leonard Scott for valuable conversations concerning various aspects of this work. 1. Reduction to the cyclic case. In this section we will show that it is sufficient to prove Theorem B in the case where G is cyclic. From our. previous remarks it is evident we must consider the commutative square C2y(p) > 7f2n - i(Z(P)) ( c2N(p lH) > ( lr2n_l(Z(P lH)) HCG HCG H cy-lic H cyclic PROPOSITION 1.1. Let p: G --> O (2d) be a fixedpoint free representation of a finite group G. For any integer n > 1 the restriction map res.' gf2n_l(Z(p)) ( Q ' is a monomorphism. HC_G H cyclic ':n-l(Z(Plh,))  Q We prepare for the proof of the proposition by making some easy obser- vations about the centralizers of representations. Given an irreducible representation O: G--> 0 (d) let V denote the repre- sentation space. By Schur's Lemma we know that HomG(V, V) -- R, C or H. We call such a representation type I, II, or III accordingly. Now it is easy to see that the transpose operation defines an involution on HomG(V, V) which is either trivial, complex conjugation or quaternionic conjugation in each case. It follows that Z(p)c O(d) is isomorphic to 0(1), U(1) or $p(1) respectively. Similarly we may consider any real representation p: G --> 0 (d) and write P = E aiPi + E bj. + E Ck'k, i j k where the Pi, ' and 'k are irreducible representations of type I, II or III respectively. Then the centralizer Z (p) c O (d) is isomorphic to the product II O(a,) x II U(bj) x II Sp(ck). i j k The situation for complex representations is, as usual, easier. If p: G--> U(d) is a complex representation we can write p = i aiPi where the p are irreducible. Exactly as before we see that Z(p)c_ U(d) is isomorphic to II U(a). There is then an obvious homomorphism 434 JOHN EWING rl: r2n-l(Z(p))---> R[ G ] given by ,/([f])= Y'i f*(o*c(ø)Pi, where o*c 0 is the cohomology suspension of the nth Chern class of BU(ai). It is clear that ,/© Q is a monomorphism. Finally we consider a real representation p: G--> O (d) without any irre- ducible constituents of type I; that is, using our previous notation, j k Let Pc denote the complexification of p. By the usual arguments each (oi) c is the sum of an irreducible and its (distinct) conjugate while each (*k)c is twice a self conjugate irreducible. It follows that Z(p) U(b.) x II Sp(ck) and j k Z(oc) x x II J k Letting i: Z(p)--> Z(pc) denote the inclusion, it is now easy to see that i# © Q is a monomorphism on homotopy groups. PROOF OF PROPOSITION 1.1. Clearly the proposition is true for G cyclic and hence we may assume that I G] > 2. From the classification of fixed point free representations of finite groups given by Wolf [13], we see that for [G] > 2 there are only irreducible fixed point free representations of type II or III. Consider the commutative diagram: 7r2n- 1 (Z(p)) i#  7r2n- 1 (/(PC))  ) R [O] res res res  lr2n-- I(Z(P IH)) > D JT2n-- I(Z(Pc IH)) >  R [H] HC_G HC_ G HC_ G H cyclic H cyclic H cyclic We have observed that i# © Q and ,/© Q are monomorphisms. It is well known that the restriction map for the representation rings is a monomorphism. The proposition is now immediate. [] 2. The proof when G is cyclic. We now give the proof of Theorem B for G a cyclic group. Since the result is well known for G - Z 2, we can assume that G is cyclic of order q > 2. First, we establish some notation and recall some facts. Let T denote a generator of G and let X = e 2*ri/q. The irreducible unitary representations of G are given by Ok, 0 < k < q, where ok(T) = X k. Clearly FINITE GROUPS ON HOMOTOPY SPHERES 435 Pk is fixed point free if and only if (k, q)= 1. We note that pk is real equivalent to k = Pq-k. Given a fixed point free representation p: G - O (2d), it is a standard fact that p is the realization of a unitary representation. Hence p is a real equivalent to YeA dkPk, where the d are nonnegative integers whose sum is d and A = { k E Z[ 1 < k < q/2, (k, q) = 1 }. As in the previous section it follows that Z(p) c O (2d) is isomorphic to IIA U(d). Now given an element of c2N(p) we choose a representative homotopy sphere x2N with (G, p)-action, and let x2n denote the fixed point set, where 2n - 2N - 2d. Let , denote the normal bundle of the fixed point set. The reduction of the structure group of , to Z(p) gives , the structure of a complex G-vector bundle, and as in [11] we may write kA where , is a complex vector bundle Theorem B we need only show that H:n (E:n, Q) are zero for k E A. of dimension d. In order to prove the rational Chern classes Cn(k) TO accomplish this we compute the G-signature (see [1, p. 578]) in two ways. First, since the middle dimensional cohomology of y:N is trivial, it is clear that sign(T, 27 v) - 0. On the other hand we can compute sign(T, x2N) from the G-signature Theorem [1, p. 582]. We see that where Xk+ 1 ) c=2n 1 kA -- and where (X a, a) is the characteristic class defined by the power series associated to xk+ 1 Xke z -- 1 ' Since the only possible nonzero, rational Chern classes are Co()= 1 and cn(), we can write: = 1 + where ( k) is some number in Q(X). We can "determine" the numbers () by using the defining power series for (, ) and a standard trick. This gives a generating function for n( k) as follows. 436 JOHN EV/ING E (--1)nn(Xk)z n= 1--z z log  Xke z-- n----0 1 z [XeZ+l XeZ-1 ] XeZ- ' Elementa mapulation of the generating function now yields e follong two leas. LEMMA 2.1. (I),( -) = (- 1)'(I),(). [] LEMMA 2.2. If q is even and n > 0 then (I),,(q/2-,) _- (_ 1),,+l(I),,(,). To summarize, we have shown that 0 = sign( T, E 2v) = _+ C { kA (2.3) where, of course, (Cn(,), [y,2n])  Q and C-- 0. It is now clear that the following is the crucial result concerning the n(;'). LœM, 2.4. (i) If q  0mod4 then the numbers {n(;')[k E A} are linearly independent over Q for n > 1. (ii)/f q -= 0 mod 4 then the numbers {n(;'l k E ,'[)} are linearly indepen- dent over Q for n > 1, where/i= { k EZll < k < q/4 and (k, q) = 1). We shall defer the proof of this purely algebraic lemma until the next section. The proof of Theorem B is now almost immediate. If q : 0 mod 4 then from (2.3) and Lemma 2.4 we conclude that c n (,) = 0 for k  A. If q _= 0 mod 4 we must work a little harder. We prove the result by induction on q. If q = 4 then A contains only one element and a rational pontrjagin class argument shows that Cn(l) ---- 0. Suppose that q  0 mod 4 and q > 4. Consider the subgroup H of G generated by T 2 of order q/2. Clearly y.2,v is an (H, 0[n)-manifold and exactly as before the normal bundle of the fixed point set decomposes into: (]),e, 'b,. By considering the restriction of the representation 0, to H, it is evident that , = , ( q/2-,. From the induction hypothesis we conclude that Cn(k) = Cn(k ) '{- (-- 1)ncn(q/2_k ) = O. Therefore, using Lemma 2.2, we see that 0 = sign(T, Z 2v) = _+ C Y. 2qn(') 1 and real number 0 < t < 1 let ½p(n, t) = E v--O 1 (-1) n + (v+ 1- t) n (v+ t) n It is easy to see that qo(n, t) is convergent for n > 1 and absolutely convergent for n '> 1. PROPOSITION 3.3. For any real number 0 < t < 1 e2*rite z Jr 1 _--  2 e2'me z -- 1 n--o (2ri) n+ l ½p(n + 1, t)z n. PROOF. The proof follows easily from the identity e 2ix + 1 e 2ix -- 1 - ictn(,rx), and the well-known partial fraction decomposition ,rctn(rx) = 1 +  X v=l X "{-  1 We leave the manipulation to the reader. [] 1 COROLLARY 3.4. For any real number 0 < t <  , e2rite z-- 1 =  2 e2rite z Jr 1 n=O (2'i) n+l 1 ½p (n + 1, t Jr  )z n. ' in the previous proposition. [] PROOF. Replace t by t +  FINITE GROUPS ON HOMOTOPY SPHERES Using Proposition 3.3 and Corollary 3.4 together with the generating function for n(X'), we conclude: COROLLARY 3.5. For k  A, 'L(") = (-1) n (2i) n+l [c(n,k/q) - c(n,k/q + «)]. [] We can now quite easily evaluate the requisite character-sums. Our answer will involve the Dirichlet L-series defined by  x(,') L(s, x) = Z ,,s , where X is a character mod q. PROPOSITION 3.6. Let X be a character mod q such that X(-1) = (-1) n. Then (i) Z,4 x(k)(n, k/q) = (- 1)nqnL(n, X), (ii) Y,4 x(k)(n, k/q + «) - (- 1)nqn(2n(2) -- 1)L(n, YO, q odd, 1 (iii) Z,4 x(k)(n, k/q + 5) = (- 1)nqn( 1 + q/2)L(n, X), q = 0 mod 4. PROOF. The calculations are all similar. For the first we use the fact that X(v) -- 0 if (v, q) 4: 1. From the definition of qo(n, t) we see that Z x(k)qo n,  = Z kA kA xq,) Z v=O =qnEE kA v=O kE.,/ v=O 1 (--1) n + (t' + 1 -- k/q) n (t' + k/q) n x(k) (-1)nx(k) + (qv + q- k) n (qt' + k) n (-- 1)nx(q -- k) (qv + q- k) n (-1)nX(k) + (qv + oo x(,') : (--1)nq n E t,n = (-1)nqnL( n, X)' The second sum is similar. Using the definition of v(n, t) we see that 440 JOHN EWING + «) k(EA v=0 (v4-«-k/q) n (-1) n + 1 (v +. +k/q) n =2nqnE E kA v=O (2vq + q- 2k) n (-1)nx(k) ((2v + l)q + 2k) n (-- 1)n2nqn(2) E E kEA v=O  x(,') = (-- 1)n2nqn(2)  vn v odd ß = (-- l)n2nqn(2) {  X(---- v) v=l pn = (--1)nqn(2n(2) -- 1)L(n, X). x(q- 2k) x(2k) + (2v + q- 2k) n  x(2,,) } v=l (2V) n ((2v + 1)q + 2k) n Finally, for the third case we use the fact that for q = 0 mod 4 and k odd, k(1 + q/2) = k + q/2 = k - q/2 modq. Then Z x(k)cp(n, k/q + «) kA : E x(:) E kA v=O 1 (--1) n + l--k/q)n (V+ 1 (V +   + k/q) n kEA v=O X( k ) + (qv + q/2 + k) n (qv + q/2- k) n = (-- 1)nqn(1 + q/2) : (--1)nqn(1 + q/2) ,4  I x(q/2- k) n + x(q/2 + k) ,,=o (q/'  -/{ '-- ) (q; ' -{ .. )n ] ' x(,') Z vn = (--1)nqn(1 + q/2)L(n, X). [] t,:l We are now in a position to prove Lemma 2.4. Combining Corollary 3.5 and Proposition 3.6, we see that for any character X mod q such that X( 1) -- (- l) n, FINITE GROUPS ON HOMOTOPY SPHERES 441 = kA 2 (2.i) n +, 1 (2rti) ,+ ' qn( 1 -- 2n-'(2))L( n, X) qn(1 -- (1 + q/2))L(n, X) q odd, q --= 0 mod 4. By Lemma 3.2 it is enough to determine when these sums are nonzero. It is immediate from the Euler product formula that L(n, X) 4:0 for n > 1. For the remaining part we must consider the various possibilities for q separately. If q is odd then we simply note that 1(2)l = 1. Hence 1 - 2 n- 1(2) 4:0 for n > 1 and the lemma is proved. The case when q = 2 mod 4 is easily reduced to the preceding one by noting that n(--;X ') = --n(;X'). If q = 0 mod 4 we note that by Lemma 2.2 span{On(?,')l k  A} = span { On(?, ' ) l k  Now the character sum vanishes precisely for those characters X mod q for which X(1 + q/2) = 1. It is easy to show that this is true if and only if X is induced from a character rood q/2. Moreover, there are precisely (q)/4 even characters and (q)/4 odd characters with this property. Since the cardinality of A is (q)/2, we conclude that the dimension of the span of (n0?)lk A) is'(q)/4. However, the cardinality of  is also (q)/4. It follows that (n(?,')l k 6 ) are linearly independent over Q. [] REMRI<. Lemma 2.4 is in general false for n = 1, and it is not hard to show that Theorems A and B are consequently false in general in case 2N - 2d = 2. (See [10].) However Lemma 2.4 does hold for n = 1 if X(2)4:1 for every odd character mod q when q is odd. This is true if and only if - 1 is a power of 2 mod q. The following facts are all elementary except the last, which follows from a theorem of Tchebotarev [5, p. 169]. (1) Write q- p[,,...,pk, where Pi is prime, and suppose the order of 2 modpi is 2t,(odd). Then -1 is a power of 2 mod q if and only if l,=12=... = > 0. Let p be an odd prime and suppose the order of 2 mod p is 2 t (odd). Then (2) If p --= 7 mod 8 then l = O. (3) If p = 5 mod 8 then l = 2. (4) If p = 3 mod 8 then l = 1. (5) If p = 1 mod 8 then l can be any nonnegative integer. Moreover, for each fixed value of l there are an infinite number of primes p for which 2 has order 2 t (odd) mod p. JOHN EWING REFERENCES [1] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. 87 (1968), 546-604. [2] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. [3] W. Browder and T. Petrie, Diffeomorphisms of manifolds and semifree actions on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 160-163. [4] J. H. Ewing, Spheres as fixed point sets, Quart. J. Math. Oxford Ser. 27 (1976), 445-455. [5] L. J. Goldstein, Analytic number theory, Prentice-Hall, Englewood Cliffs, N.J., 1971. [6] M. Rothenberg, Differentiable group actions on spheres, Proc. Advanced Study Inst. Algebraic Topology, Aarhns, 1970, 455-475. [7] R. Schultz, Homotopy sphere pairs admitting semifree differentiable actions, Amer. J. Math. 96 (1974), 308-323. [8] , Rational h-corbordism invariants for lens space bundles, Quart. J. Math. Oxford Ser. 25 (1974), 497-511. [9] , Corrigenda, Quart. J. Math. Oxford Ser. 28 (1977), 128. [10] , Spherelike G-manifolds with exotic eclui.variant tangent bundles (preprint). [11] G. B. Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 129-151. [12] K. Wang, Semifree actions on homotopy spheres, Trans. Amer. Math. Soc. 211 (1975), 321-337. [13] J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. DEPARTmEnt OF MATHEMATICS, INDIANA UNIVERSITY, BLOOMINGTON, INDIANA 47401 DEPARTmEnt OF MTICS, UNIVERSITY OF VIRGINIA, CHARLOTrESVILLl% Vmon,IA 22903