Smith equivalence of representations for odd order cyclic groups

Abstract

THEOREM A. There exist odd order cyclic groups which have Smith equivalent but nonisomorphic representations. After a short background discussion, we shall give a list of conditions (1.1) which are important in the development of this paper and state more precise versions of our results. Theorem B provides nonisomorphic Smith equivalent representations if one has nonisomorphic representations of G which satisfy (1.1). Corollary C shows the existence of infinitely many groups which have nonisomorphic representations which satisfy (1.1). The orders of some cyclic groups of this type are described by Lemma 2.5 and Corollary 2.6. Taken together, Theorem B and Corollary C prove Theorem A. The concept of Smith equivalence is based upon a question posed by P. A. Smith in a 1960 survey article [32, p. 4061. In our terminology the question is whether Smith equivalent representations are linearly isomorphic. Atiyah-Bott [3] and Milnor [17] established an affirmative answer for cyclic groups of odd prime power order and for representations with semifree action. By definition a group acts semifreely if each point is left fixed by only the trivial group element or by each group element. An extension of this work by Sanchez [28] implied the answer also to be affirmative if G is cyclic of order pq where p and q are odd primes. Bredon [6] showed for 2-groups that Smith equivalent representations are isomorphic if their dimension is large in comparison to the order of the group. The first negative answers to Smith’s question were established by the second author for odd order abelian groups with at least four noncyclic Sylow subgroups. See [23], [24]. Subsequently a number of papers have established a negative answer for even order groups, but the case of odd order cyclic groups has remained open. For cyclic groups of even order see the papers [7], [25], [ 111, [3 13, For noncyclic abelian groups and nonabelian 2-groups, see [34] and [S]. There are many interesting results and ideas in these papers. Results and techniques used in answering Smith’s question varj markedly according to the parity of the order of G. In Remark 1.2(ii) we explain the difference of the geometric and number theoretic treatment of even and odd order groups as it arises from the Atiyah-Bott extension of the Lefschetz Fixed Point theorem. One interesting question, untouched here, is to relate the differentiable structure of a homotopy sphere Z and the isotropy representations of an action of G on Z with exactly two fixed points. Some general remarks on this appear in the problem section of the forthcoming proceedings of the 1983 conference on group actions at Boulder [393. It can be shown that the homotopy spheres with group actions that are constructed by our method are all standard spheres. The methods of Schultz (e.g. in [29] and other papers) are relevant to realizing these actions on exotic spheres.