Posts Tagged ‘math.GT’

Equivariant DeRham Cohomology

February 10, 2008

    My lectures on equivariant cohomology are spinning a bit out of control.  The questions and lively discussion, while always welcome, have stretched what was meant to be a hand-waving tour through the basics into a three week mini-course (at least, I hope its only three weeks).  I’m starting to feel a bit sheepish, since the result I’m trying to get to might not really merit a full month-long preamble.

    Last time, I talked about how to define the equivariant cohomology of a space M in terms of the cohomology of some big infinite-dimensional space M\times E/G.  This is good on a conceptual level, but unless E is particularly nice, we will have a bitch of a time computing the cohomology of anything.  What we need is a more effective model for the cohomology of M\times E/G.

    The idea is to start by pining for the existance of a nice de Rham complex \Omega(E) on E.  We’ll say “Oh, if only it \Omega(E) existed, it would look like this, and this…”.  Since E was only defined by its properties (contractibilty and a free action), this amounts to listing what properties a differential graded algebra should have to correspond to those of E.  Such DGAs will be called ‘locally-free, acyclic G^*-algebras’.

    From there, its a three step process.  First, show that every such DGA computes the same cohomology.  Second, show that there is an (almost) universal locally-free, acyclic G^*-algebra called the ‘Weil algebra’, which is simple enough in structure to make computations effective.  Third, show that there exists any such DGA which correctly computes the equivariant cohomology (this last step should probably be first, but it isn’t very exciting).

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The Cohomology of Quotients

February 2, 2008

    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 complexM and a Lie group G which acts on M.  Unless this action is free and proper, the quotient space M/G might be a poorly behaved space.  Take, for example, \mathbb{Z} acting on S^1 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 (M,G), which is H^\bullet_{CW}(M/G) if G acts freely and properly?” The hope is that this will shine some more light on the hidden internal structure of the bad quotients.

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G-equivariant embeddings of manifolds

October 24, 2007

This is my first post, and I plan on sporadically writing some in the future. I’m Peter, a third year grad student at Cornell, and I talk to Greg pretty often, so I thought I’d write down some of the things I say. This first post won’t be long or deep, but it’s kind of cute, and the trick behind it is useful in many other situations, so I decided to share it. Let’s say we have a finite group G acting on a compact manifold M. The Whitney embedding theorem says that we can embed M into \mathbb{R}^k for sufficiently large k, and what I want to show in this post is that you can do this in a G-equivariant way, i.e. there is an embedding \phi:M \to \mathbb{R}^k and an injective homomorphism f:G \to O(\mathbb{R}^k) such that \phi(g\cdot x) = f(g)\cdot \phi(x). I guess the moral of the story is that compact manifolds are really just “nice” subsets of Euclidean space, and a compact manifold with a finite group action is really nothing but a “nice” subset of Euclidean space that is preserved by the action of a finite group of permutation matrices. Also, if one were to summarize the moral of the trick used, one might say “averaging over the elements of a group makes things equivariant,” and this idea comes up almost uncountably many times in many areas of mathematics (of course, averaging takes on different meanings in different situations).

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