Replacement of fixed sets for compact group actions: The 2\rho theorem

TL;DR

This paper investigates the conditions under which fixed sets of compact group actions on manifolds can be replaced by homotopy equivalent sets, establishing a key theorem and exploring various examples.

Contribution

It proves a new theorem providing conditions for the replacement of fixed sets in equivariant topology, extending understanding of group actions on manifolds.

Findings

Replacement is always possible under certain bundle conditions.

The normal bundle condition involves doubling a complex bundle over a 1-skeleton.

Examples show the range from always possible to rigid fixed set replacements.

Abstract

If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally linear actions on topological or PL manifolds by compact Lie groups, the replacement is always possible if the normal bundle of the fixed set is twice of a complex bundle over a 1-skeleton of the fixed set. Moreover, we also study some specific examples, where the answer to the replacement problem ranges from always possible to the rigidity.