Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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

2 Answers 2

up vote 12 down vote accepted

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!

share|improve this answer
    
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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.