Cyclic group actions on manifolds from deformations of rational homotopy types

Abstract

This paper finishes the series [12,16,17] of papers that aim to show the existence of non-trivial group actions (“symmetries”) on certain classes of manifolds. More specifically, we ask whether there is a semifree smooth action of the circle group T = S1 – resp., a non-trivial action of a cyclic group Z/p, p a prime – on a given manifold X with a fixed point set of a given rational homotopy type F . We assume that the rational homotopy types of X and F are related by a deformation in the sense of [1] between their (Sullivan) graded differential algebra models (cf. [22,9]): Roughly speaking, we assume the conclusion of the Borel localization theorem[10, 2] on the rational homotopy level. Under certain additional assumptions, we prove a converse of that theorem: we show that there is a semifree smooth T -action on a manifold Y rationally homotopy equivalent to X with fixed point set Y T rationally homotopy equivalent to F . Moreover, for all but finitely many primes p, we find non-trivial smooth actions of Z/p on X itself with fixed point set rationally homotopy equivalent to F .