We’ve organized a mostly informal Topics in Noncommutative Algebra seminar this semester, and I’m talking first in it. I’m eventually going to be talking about a paper of Ginzburg’s connecting Hochschild and cyclic cohomology to the equivariant cohomology of representation schemes. Unfortunately, the trouble about talking about fun results like that is that you need to cover alot of background material; as such, I’m doing what is turning out to be a two lecture series on equivariant cohomology and its deRham version. I figured I’d mirror these talks with a couple of posts, and maybe even talking about Ginzburg’s paper if I get enough prereqs covered.

Today I’m just going to be talking about topological equivariant cohomology. Let’s start with a nice space (say, a CW complex) and a Lie group which acts on . Unless this action is free and proper, the quotient space might be a poorly behaved space. Take, for example, acting on by some irrational rotation; the quotient isn’t even Hausdorff.

The motivating question of equivariant cohomology is: “Is there a good cohomology theory for the pair , which is if acts freely and properly?” The hope is that this will shine some more light on the hidden internal structure of the bad quotients.