In a paper by Cohen, Jones, Segal "Morse theory and classfying spaces", they

constructed a flow category of a Morse function and

showed the classifying space of the flow category is homotopic to the

underlying manifold. Is there any such analogue theory for G-invariant

functions on a compact manifold M, where G is a compact Lie group acting on

M? e.g, a classifying space of such an G-invariant function is homotopic to

the homotopy quotient in the Borel construction?

Another question: In the paper by Cohen, Jones and Segal, where did they use

the information of critical points in the proof of the classifying space of

the flow category is homotopic to the underlying manifold?