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?

  • $\begingroup$ Well, an Invariant Morse Theory is sort-of what Bott used to prove his Periodicity Theorems (which are about various classifying spaces </pun>), but that is somewhat removed from your present question. Hm! $\endgroup$ – Jesse C. McKeown Dec 15 '15 at 17:00
  • $\begingroup$ The quick answer to your first question is "no", at least if the question is interpreted in its most naïve form: CJS build a top-enriched category F whose objects are the critical cells and the hom-spaces F(x,y) are derived from the moduli space of reparametrized broken flow lines. If you follow their construction, there is no reason (at least, none I can see) why each F(x,y) should inherit the group action. I think you can make a very compelling case when the hom spaces are also equivariant: eg when rotating a sphere about the north-south diameter... $\endgroup$ – Vidit Nanda Dec 17 '15 at 21:09
  • $\begingroup$ In the paper you mention the authors claim a homeomorphism between the classifying space of the category and the manifold in the Morse-Smale case. For that, the papers by Lizhen Qin mentioned in this question are relevant. $\endgroup$ – Gustavo Granja Dec 18 '15 at 11:55

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