Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
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Sign up to join this communityIs there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
Crossposted on StackExchange.
The answer to your question is yes, of course. The theory has been around at least since the late 60s! See Wasserman's paper
A Wasserman. Equivariant differential topology, Topology 1969; 8(2):127-150.
I think the first "big application" is due to Atiyah and Bott, who used it to study (what else?) Yang-Mills on surfaces. See the first section of this 90 page behemoth
MF Atiyah and R Bott. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences (1983): 523-615.
I personally found the exposition in Section 2 of Hingston's subsequent paper to be somewhat cleaner and more transparent:
N. Hingston. Equivariant Morse theory and closed geodesics. Journal of Differential Geometry 19 (1984), no. 1, 85--116.
I'm sure this equivariant Morse theory is in many textbooks by now, but the original papers cover much of what it is and how it is used. There's even a recent discrete version for $G$-simplicial complexes due to Freij!
If you like a Morse homological treatment, you could also look at the paper of Austin and Braam.
Austin, D. M.(1-IASP); Braam, P. J.(4-OX) Morse-Bott theory and equivariant cohomology. The Floer memorial volume, 123–183, Progr. Math., 133, Birkhäuser, Basel, 1995.