Reading list for Equivariant Cohomology

Question

I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, in particular where basic examples are worked out?

I think that the following notes are quite good, but I would like something more detailed and containing more examples/exercises: Anderson - Introduction to equivariant cohomology in algebraic geometry.

Answer 1

I recommend Equivariant de Rham cohomology: theory and applications which in my opinion is a well-made introduction in equivariant cohomology. Moreover the article of Tymoczko "An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson" in Snowbird lectures in algebraic geometry contains a lot of computations and examples of equivariant cohomology of GKM spaces.

Answer 2

I would like to point out that the term "equivariant cohomology'' is ambiguous. To those unfamiliar with modern algebraic topology, it means Borel cohomology, the cohomology theory that is the subject of this question. That was introduced by Borel in the late 1950's or early 1960's. However, in 1966, Bredon introduced a more general version of equivariant cohomology (I know the year since I was at a talk he gave that year at IAS). Intuitively, it is reasonable to think of orbits $G/H$ as like equivariant points, since their orbit spaces are points. Bredon cohomology starts with a contravariant functor, $M$ say, from the homotopy category of orbits to abelian groups. Then, on $G$-spaces, an ordinary cohomology theory $H^*_G(X;M)$ with coefficients in $M$ is a homotopy functor to graded abelian groups that satisfies the obvious equivariant analogues of the Eilenberg-Steenrod axioms, but with dimension axiom stating that $$H^*(G/H;M) = H^0(G/H;M) =M(G/H).$$ These are the ordinary cohomology theories of modern algebraic topology. Let $A$ be an abelian group. One example of a coefficient system is the constant coefficient system at $A$, denoted $\underline{A}$. It has $\underline{A}(G/H) = A$ for all $H\subset G$. Let $EG$ be a free contractible $G$-space and define the homotopy orbit $X_{hG}$ of a $G$-space $X$ to be the orbit space $(EG\times X)/G$, where $G$ acts diagonally on the product. Then the Borel cohomology of $X$ is defined to be $H^*(X_{hG};A)$. However, an easy comparison of definitions or axioms shows that Borel cohomology is just the special case $H^*_G(EG\times X;\underline{A})$ of Bredon cohomology. For that reason, no algebraic topologist today would consider writing a book just about Borel cohomology. Unfortunately, there are no textbooks about Bredon cohomology either as far as I know. However, one already somewhat outdated book, not a textbook but a 1995 status report on a subject, ``Equivariant homotopy and cohomology theory''

http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf

starts out with an exposition of equivariant cohomology, including how Borel and Bredon cohomology each apply to the proof of classical P.A. Smith theory, which dates from the late 1930's.

Answer 3

For the desired symplectic emphasis, I’d warmly recommend these (of which Panagiotis’ ref. cites #2–5):

1. Atiyah, Michael F.; Bott, Raoul, The moment map and equivariant cohomology, Topology 23, 1-28 (1984). ZBL0521.58025.

2. Berline, Nicole; Getzler, Ezra; Vergne, Michèle, Heat kernels and Dirac operators. Berlin etc.: Springer-Verlag. vii, 369 p. (1992). ZBL0744.58001.

3. Guillemin, Victor W.; Sternberg, Shlomo, Supersymmetry and equivariant de Rham theory. With reprint of two seminal notes by Henri Cartan, Berlin: Springer. xxiii, 228 p. (1999). ZBL0934.55007.

4. Audin, Michèle, Torus actions on symplectic manifolds, 2nd edition. Basel: Birkhäuser. viii, 325 p. (2004). ZBL1062.57040.

5. Meinrenken, Eckhard, Equivariant Cohomology and the Cartan Model, Encyclopedia of mathematical physics, Vol. 2, pp. 242-250 (2006). ZBL1170.00001.

6. Vergne, Michèle, Applications of Equivariant Cohomology, Proc. Int. Cong. Math. Zürich, Vol. 1, pp. 635-664 (2007). MR2008m:53194.

Answer 4

Some additional resources, which are more on the algebraic side than the symplectic side:

1) Bill Fulton's Eilenberg lectures on Equivariant Cohomology in Algebraic Geometry, available at David Anderson's website. My understanding is that the plan is for these notes to be compiled into a book at some point.

2) Michel Brion's Equivariant Cohomology and Equivariant Intersection Theory.

Answer 5

As mentioned in the comments, Loring Tu has since written a book on equivariant cohomology which assumes little background and is quite clear. I think it prepares one quite well for reading The Moment Map and Equivariant Geometry by Atiyah and Bott.

Here is Tu's book: https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology.