Free actions on products of spheres: The rational case

Abstract

Let X = S " ' x . . . x S"" be a product of spheres of total dimension n=nl+n2 +... +nk. A fundamental unanswered question is the determination of which finite groups can act freely on X and what actions on the cohomology so arise. In particular it is conjectured that if an elementary abelian group acts freely, then its rank is less than or equal to k. Great progress has been made recently on this question by Carlsson EC], Adem-Browder [A-B], and Hoffman [HI in the case where all the spheres have the same dimension. In this paper we completely solve the rational analogue of the above question. Given an action of a finite group G on the rational cohomology ring H*(X; •), we give necessary and sufficient conditions for G to act freely on a closed manifold Y having the rational homotopy type of X, so that the G-action induces the specified action on H* (Y; Q)= H* (X; Q). In particular the necessary conditions give new obstructions for G to act freely on X with a specified representation in H*(X; Q). Our method includes a general discussion as to when a space with finite fundamental group has the rational homotopy type of a closed manifold. We now give our necessary and sufficient conditions: (A) For all gEG, for all ni even, g*ESn']=~[S nJ] for some nonzero rational number ~. (B) For all g ~ G { e } , ~ ( 1 ) i ( t r ( g . : H,(X; Q ) ~ Hi(X; Q)) =0. (C) (i) For n even, some n i odd, the equivariant intersection form on H "/2 (X; Q) is hyperbolic, (ii) for all ni even, no further condition, (iii) for n odd, z 89 Q)ea*(O.(G, w) )cL , (~G, w).