Is there some variant on Morse or MorseBott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
Crossposted on StackExchange.
Is there some variant on Morse or MorseBott 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
I think the first "big application" is due to Atiyah and Bott, who used it to study (what else?) YangMills on surfaces. See the first section of this 90 page behemoth
I personally found the exposition in Section 2 of Hingston's subsequent paper to be somewhat cleaner and more transparent:
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.


