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?