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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.

Crossposted on StackExchange.

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There is a version of such a theory due mostly to M. Braverman and M. Farber. For a brief summary of it and references see Section 5.1 of the paper arxiv.org/pdf/math/9904148.pdf –  semyon alesker Jun 10 at 17:18

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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!

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Thanks for the answer. I've begun looking at Atiyah and Bott's paper. However, from what I can see from the first pages, they don't define some kind of "equivariant Morse complex", or do they do it later? Or should I start from one of the other papers? –  Daniel Robert-Nicoud Jun 11 at 15:55
@DanielRobert-Nicoud I'd start with Hingston and then chase references, particularly Borel's construction in [10]. The problem with "equivariant Morse complex" is that the obvious quotient might develop singularities, so you have to resort to classifying spaces and blow-ups, with everything working only up-to-homotopy. Section 1.3 of Hingston outlines the basic program, and [10] fills in the gaps. –  Vidit Nanda Jun 11 at 16:04
Thanks for the really fast answer. I will go look there, then. –  Daniel Robert-Nicoud Jun 11 at 16:09

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.

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