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](https://math.stackexchange.com/q/829417/60713).



Equivariant version of Morse theory
The answer to your question is *yes, of course*. The theory has been around at least since the late 60s! See Wasserman's [paper](http://www.maths.ed.ac.uk/~aar/papers/wasser001.pdf) 



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> A Wasserman. *Equivariant differential topology*, Topology 1969; 8(2):127-150. 
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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](http://www.csee.wvu.edu/~xinl/library/papers/math/geometry/Atiyah1983.pdf)



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> 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.
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I personally found the exposition in Section 2 of Hingston's [subsequent paper](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.jdg/1214438424&page=record) to be somewhat cleaner and more transparent:



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> N. Hingston. *Equivariant Morse theory and closed geodesics*. Journal of Differential Geometry 19 (1984), no. 1, 85--116.
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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](http://www.sciencedirect.com/science/article/pii/S0012365X08006201) for $G$-simplicial complexes due to Freij! 


