Question

Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ acts by isometries. I can't necessarily pick a $G$-invariant morse function $f:M\to\mathbb R$, but nevertheless, I can certainly pick a smooth function $f:M\to\mathbb R$ which, though perhaps not Morse, still has only isolated "nice" critical points in some precise sense. We therefore conclude:

There is a "handle" decomposition of $M$ (where I haven't said what I mean by "handle") which is preserved by $G$. Thus $G$ just permutes (and/or acts on individually) the handles.

I am interested in knowing to what extent this can be generalized to the case of an "action up to homotopy". More specifically, suppose we have $G\to\operatorname{Homeo}(M)/\text{homotopy}$. To what extent can we "decompose" M into simple pieces in a $G$-invariant way? If it helps, then it is OK to assume that the action of $G$ is "close to the identity" in a vague coarse sense.

(I am essentially just interested on the case of high-dimensional $M$, but of course the question makes sense in any dimension).

Answer 1

This foundational paper: Arthur G. Wasserman, Equivariant Differential Topology, Topology, 8(1967), 127-150, has section 4 dealing with equivariant Morse theory for manifolds with a smooth action of a compact Lie group, including equivariant handle attaching, equivariant Morse Lemma, etc. See MathSciNet Review. The last result of the paper concludes that a compact G-manifold is a union of "handle bundles over orbits".