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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[Math] Equivariant version of Morse theory
equivariantmorse-theory
Related Solutions
Massey products are discussed in Section 1.3 of
Fukaya, Kenji. Morse homotopy, $A_{\infty}$-category, and Floer homologies. Proceedings of GARC Workshop on Geometry and Topology '93 (Seoul, 1993), 1--102,
available here (pdf). The Massey products are obtained by counting gradient flow graphs with four external edges and one (finite-, possibly zero-, length) internal edge. Fukaya sketches a construction of an $A_{\infty}$ category whose objects are Morse functions $f$ and with morphisms from $f$ to $g$ given by the Morse chain complex of $f-g$. In particular the Massey products can then be seen as arising from the $A_{\infty}$ structure in a standard formal way.
There is a relation to Lagrangian Floer theory: a Morse function $f:M\to \mathbb{R}$ corresponds to a Lagrangian submanifold $graph(df)$ of $T^*M$ and intersections between $graph(df)$ and $graph(dg)$ are in obvious bijection with critical points of $f-g$. There are results, pioneered by
Fukaya, Kenji; Oh, Yong-Geun. Zero-loop open strings in the cotangent bundle and Morse homotopy. Asian J. Math. 1 (1997), no. 1, 96--180,
which relate the gradient flow graphs appearing in the Morse $A_{\infty}$ operations to the holomorphic curves appearing in the Lagrangian Floer $A_{\infty}$ operations.
A Morse function is a map of a manifold to the real line locally equivalent to:
$$f(x_1,\ldots, x_n)=-x_1^2\ldots -x_k^2+ x_{k+1}^2+\ldots+x_n^2$$
for some $k$. In other words,
for which the singularities are as simple as possible.
While a Lefschetz pencil is a map of a smooth projective variety to the projective line local analytically given by $f=x_1^2+\ldots+x_n^2$.
So in this sense, they are very analogous. There are differences, however. Given
a Morse function $f:X\to \mathbb{R}$, the collection of the above numbers $k$, called indices,
determine the homotopy type.[the number of cells in a complex homotopic to $X$]. For "Artin's" vanishing, it is enough to choose an $f$, where these indices are bounded by dimension. (Details can be found in the first few pages
of Milnor's Morse theory.)
I'm not aware that there is any complete substitute on the other side, given say a Lefschetz pencil $f:X\to \mathbb{P}^1$. However, the pencil does give a way to calculate the (etale) cohomology of $X$ in terms of the critical points of $f$ and the monodromy, or more formally in terms of the direct images $R^if_*\mathbb{Q}_\ell$. And this is quite powerful.
Perhaps it would be more instructive give a Morse-like pseudo-proof of Artin's theorem.
By "pseudo" I mean that there is step which I can't justify without a lot more effort than this is worth. Let $X\subset \mathbb{A}^n$ be an irreducible affine variety of dimension $n$ over an algebraically closed field. By generic projection, we get a nonconstant morphism
$f:X\to \mathbb{A}^1$ which will play the role of our Morse function. Suppose that (*) the direct images $R^jf_*\mathbb{Q}_\ell$ were constructible and commuted with base change.
Then by induction, $R^jf_*\mathbb{Q}_\ell=0$ for $j>n-1$. From the Leray spectral
sequence, we need only prove that $H^i(\mathbb{A}^1,F)=0$ for $i>1$ and any constructible sheaf $F$,
but this is easy. Note that if $f$ were proper, (*) would be automatic. In general,
one could get around it by generic base change [SGA 41/2, p 236] and devissage.
Best Answer
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?) Yang-Mills 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!