# Reference request: Equivariant Topology

I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students have all taken a basic course in algebraic topology (they know homology/cohomology and fundamental groups), but some may not know much more topology than that. Topics will likely include equivariant cohomology, (equivariant) bundles and characteristic classes, equivariant K-theory, and important classes of examples interspersed, including toric varieties, homogeneous spaces, and the Hilbert scheme of points in $\mathbb{C}^2$. My personal goal is to learn a bit about Bredon cohomology for compact, connected Lie groups (I'm happy to restrict that a bit, but probably not to finite groups).

Some references that I already have in mind include those listed in David Speyer's question and answer about equivariant K-theory. For Bredon cohomology, there are two books: Equivariant Cohomology Theories by G. Bredon, and Equivariant Homotopy and Cohomology Theory by J.P. May (with many other contributors).

Reference request: What are classic papers in equivariant topology that every student should read?

• Should this not be CW? It is basically a big list question. – Benjamin Steinberg Jan 13 '12 at 0:04

If I may be so bold, I would actually strongly suggest you start with finite groups, rather than compact Lie. While many of the results in equivariant homotopy are true in both cases, the formulations for finite groups are often easier to understand. Additionally, there are twists that show up in the compact Lie case which just make exposition (and I find comprehension) a good bit trickier.

For a finite group, it is very easy to carry out computations with Bredon homology and cohomology. In fact, it's easy to write down chain complexes of Mackey functors which do everything for you. For compact Lie, you can of course, do the same thing; I personally find it substantially harder and less intuitive.

• Mike, you write it is very easy to carry out computations ...''. Of course, Mackey functors only apply when you are in an RO(G)-graded context. But that is a detail. As you know, you lucked out in the Kervaire invariant one problem and only needed "easy'' computations. To avoid giving a wrong impression, people should be warned that in fact there are relatively few fully worked computations, and even things like rational equivariant characteristic classes in Bredon cohomology are still unknown. This is an area in desperate need of hard work, even for cyclic groups of prime order. – Peter May Apr 14 '12 at 17:55
\bib{MR1413302}{book}{
author={May, J. P.},
title={Equivariant homotopy and cohomology theory},
series={CBMS Regional Conference Series in Mathematics},
volume={91},
note={With contributions by M. Cole, G. Comeza\tilde na, S. Costenoble,
A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G.
Triantafillou, and S. Waner},
publisher={Published for the Conference Board of the Mathematical
Sciences, Washington, DC},
date={1996},
pages={xiv+366},
isbn={0-8218-0319-0},
review={\MR{1413302 (97k:55016)}},
}

• I find parts of this book formidable. Any advice (other than chin up, push on!) or alternatives? – Tara Holm Jan 13 '12 at 16:42
• Chin up, push on, Tara. Ignore the most formidable parts: the parts are written so as to be independently readable. The more formidable parts are usually there to fill gaps in the literature, so precisely the parts that are hardest to find elsewhere. And thanks, Neil. – Peter May Jan 14 '12 at 15:54
• Thanks for your comment, Peter. It's good to know the parts are independently readable. That should make my goals of learning some parts of it, and attracting as many grad student participants as possible, both attainable! – Tara Holm Jan 16 '12 at 18:18

This answer is biased towards the relation to Group Cohomology, as that's where I first learned about equivariant cohomology while studying under Ken Brown:
1) Quillen's famous The Spectrum of an Equivariant Cohomology Ring I+II (which proves that the Krull dimension of $H^*(G,\mathbb{Z}_p)$ is the p-rank, i.e. maximum rank of an elementary abelian p-subgroup of $G$, and that the minimal prime ideals of the ring are in 1-1 correspondence with the conjugacy classes of maximal elementary abelian p-subgroups).
2) Atiyah & Bott's The Moment Map and Equivariant Cohomology (on localization, but also just a nice expository paper).
3) Duflot's famous Depth and Equivariant Cohomology (which proves that the depth of the equivariant cohomology ring $H^*_G(X;\mathbb{Z}_p)$ is at least the maximum rank of a central p-torus acting trivially on the space $X$).
These are the three main references I'd give, but others include:
4) Duflot's Localizations of Equivariant Cohomology Rings (which computes the localization of the equivariant cohomology ring localized at one of its minimal prime ideals).
5) Duflot's The Associated Primes of $H^*_G(X)$ (which proves that the associated primes of $H^*_G(X;\mathbb{Z}_p)$ are invariant under Steenrod operations, and in fact can be obtained by restricting the ring to that of a p-torus).
6) Adem's Torsion in Equivariant Cohomology (self-explanatory).

And perhaps just as a meal-time reading: Loring Tu's expository article What is Equivariant Cohomology? in AMS Notices.

The Borel seminar, which is the classic reference for equivariant (Borel) cohomology, containins a wealth of information and is quite readable.

Adams has notes on Equivariant stable Homotopy. They are called Prerequisite (On Equivariant Stable Homotopy Theory) for Carlsson's Lecture.

Also, what about tom Dieck's Transformaiton Groups?

PS let me know if you can't find Adams.

This is perhaps a bit of an obvious one, but The Moment Map and Equivariant Cohomology by Atiyah and Bott is a safe bet.

I would strongly suggest Matschke's diploma

which talks about many applications your students will enjoy, and has good references.

here are some notes of fulton.

http://www.math.washington.edu/~dandersn/eilenberg/

papers by william graham and dan edidin are also suggested.