Compact Lie Groups Which Act on Euclidean Space without Fixed Points

It is shown that a compact Lie group G with identity component Gq acts without fixed points on euclidean space if and only if G„ is nonabelian or G/G0 is not of prime power order, which completes earlier work of P. E. Conner and E. E. Floyd, Conner and D. Montgomery, and W.C. Hsiang and W.-Y. Hsiang. In this note we describe completely the class of compact Lie groups which can act without fixed points on euclidean space, thus completing earlier work of P. E. Conner and E. E. Floyd [2], followed by that of Conner and D. Montgomery [3] and W.-C. Hsiang and W.-Y. Hsiang [4, 1.9] who extended Conner and Floyd's original idea. Theorem. Let G be a compact Lie group with identity component G0. Then G acts without fixed points on some euclidean space if and only if G0 is nonabelian or G/Gq does not have prime power order. If G0 is abelian and G/G0 has prime power order pk and G acts on R",then by P. A. Smith theory the fixed point set (R„)G° is nonempty and Z-acyclic. Then (R")G° is a G/G0-space and (R")G = [(R")C°]G/G<> is nonempty and Zpacyclic, again by Smith theory. We now begin to consider the converse. The following is a fairly well-known topological analogue of induced representations. Induction Lemma. If G is a compact Lie group, H C G is a subgroup of finite index k, and X is an H-space, then there is an action of G on X , the product of k copies of X, such that the fixed point set {Xk) is the diagonal copy of XH. Further, if'/: X -* X is an H-map, then the natural map fk: Xk —> Xk is a Gmap. We indicate a proof, following a suggestion of G. E. Bredon. See also [6, 1.2]. Notice that, by choosing coset representatives for H in G, X can be identified with Maps (G,X), the set of 77-maps from G to X, where 77 acts on G by left translation. Then G acts on Maps (G,X) by (g X is an 77-map, then fk: Xk -» Xk is Received by the editors January 10, 1975 and, in revised form, May 16, 1975. AMS (MOS) subject classifications (1970). Primary 57E25, 57E10. 1 Supported in part by NSF grants GP-33031X and GP-36418X1. 2 Supported in part by the National Science Foundation and the Alfred P. Sloan Foundation. © 1976, American Mathematical Society 416 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use COMPACT LIE GROUPS WHICH ACT ON EUCLIDEAN SPACE 417 identified with/* : Maps "(G, A) -> Maps"(G, A), where/*( S of degree 0. For one can then form the infinite mapping telescope T associated to /, a contractible simplicial complex on which G acts without fixed points. One then embeds T in a representation of G. An open invariant regular neighborhood of T finally provides the desired action. The proof of the Proposition proceeds by induction on the order of G. We may assume, using the Induction Lemma, that every proper subgroup of G has prime power order. Then by an elementary result of group theory [7, 6.5.7] G must be solvable. Therefore we may find a surjection G —> Zp and an injection Zq —> G where p and q are distinct primes and Z. and Z denote the cyclic groups of orders p and q, respectively. Let Dp and Dq be standard 2-disks with boundary circles Sp and Sq on which Tp and Zq act by rotations. The surjection G —> Zp makes Dp into a G-space with one fixed point and the Induction Lemma shows that the 2/c-disk (Dq) is a G-space with one fixed point, where k is the index of Z in G. Let S2k~x denote the boundary (2k l)-sphere, on which G acts without fixed points and consider the join S = Sp * S2k~x, a (2k + l)-sphere on which G acts without fixed points in the obvious way. We claim that S admits a simplicial G-map of degree 0. To construct the desired G-map, begin by noticing that there are equivariant maps a: Sp —> Sp and B: Sq —» Sq having degrees 1 + mp and 1 + nq, respectively, where m and n are arbitrary integers. (E.g., a(z) = zx+mp, where Z acts on Sp by multiplication by exp(2iri/p).) Let cB: Dq -» Dq be the obvious conical extension of B to all of D . Now a is also a G-map, as is the map (cB)k: (Dq)k -* (Dq)k. Let y: S2k~x -» S2k~x be the restriction of (cB) to the boundary. Notice that degree(y) = (1 + nq)k. In his discussion of the Conner and Floyd construction, Bredon [1, pp. 58-62] shows how to construct, given G-maps such as a and y, another simplicial G-map a □ y: Sp * S2k~x -> Sp * S2k~x such that degree (a □ y) = degree (a) + degree(y) 1. Thus degree (a □ y) = mp + (I + nq) . Since (p, q) = 1 we can choose integers n and / such that 1 + nq = lp. Then degree (a □ y) = mp + (lp) , where m is still an arbitrary integer. Let m = —lkpk~x. Then degree (a □ y) = 0 as desired. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 418 a. l. edmonds and ronnie lee