Circle actions on rational homology manifolds and deformations of rational homotopy types

Abstract

The aim of this paper is to follow up the program set in [LR85, Rau92], i.e., to show the existence of nontrivial group actions ("symmetries") on certain classes of manifolds. More specifically, given a manifold X X with submanifold F F , I would like to construct nontrivial actions of cyclic groups on X X with F F as fixed point set. Of course, this is not always possible, and a list of necessary conditions for the existence of an action of the circle group T = S 1 T = {S^1} on X X with fixed point set F F was established in [Rau92]. In this paper, I assume that the rational homotopy types of F F and X X are related by a deformation in the sense of [A1178] between their (Sullivan) models as graded differential algebras (cf. [Sul77, Hal83]). Under certain additional assumptions, it is then possible to construct a rational homotopy description of a T T -action on the complement X ∖ F X\backslash F that fits together with a given T T -bundle action on the normal bundle of F F in X X . In a subsequent paper [Rau94], I plan to show how to realize this T T -action on an actual manifold Y Y rationally homotopy equivalent to X X with fixed point set F F and how to "propagate" all but finitely many of the restricted cyclic group actions to X X itself.