Resolution of smooth group actions

TL;DR

This paper proves a canonical resolution process for smooth actions of compact Lie groups on manifolds with corners, resolving all isotropy types simultaneously through equivariant blow-ups, with applications to representation compactifications.

Contribution

It introduces a canonical resolution method for smooth group actions that captures all isotropy types via equivariant fibrations, extending the Folk Theorem.

Findings

Resolution of all isotropy types via equivariant blow-ups

Application to compactification of group representations

Canonical resolution structure for quotient spaces

Abstract

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution structure' consisting of equivariant iterated fibrations of the boundary faces. This structure projects to give a similar resolution structure for the quotient. In particular these results apply to give a canonical resolution of the radial compactification, to a ball, of any finite dimensional representation of a compact Lie group; such resolutions of the normal action of the isotropy groups appear in the boundary fibers in the general case.