Replacement of fixed sets for compact group actions: The 2\rho theorem
Geometric Topology
2013-04-16 v2 K-Theory and Homology
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.
Cite
@article{arxiv.0909.5002,
title = {Replacement of fixed sets for compact group actions: The 2\rho theorem},
author = {Sylvain Cappell and Shmuel Weinberger and Min Yan},
journal= {arXiv preprint arXiv:0909.5002},
year = {2013}
}
Comments
17 pages