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 $X$ with submanifold $F$, I would like to construct nontrivial actions of cyclic groups on $X$ with $F$ 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 $T = {S^1}$ on $X$ with fixed point set $F$ was established in [Rau92]. In this paper, I assume that the rational homotopy types of $F$ and $X$ 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 $T$-action on the complement $X\backslash F$ that fits together with a given $T$-bundle action on the normal bundle of $F$ in $X$. In a subsequent paper [Rau94], I plan to show how to realize this $T$-action on an actual manifold $Y$ rationally homotopy equivalent to $X$ with fixed point set $F$ and how to "propagate" all but finitely many of the restricted cyclic group actions to $X$ itself.