Posts Tagged ‘math.AT’

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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A Resolution of a Tensor Algebra

January 24, 2008

    This post is about projective resolutions of algebras, thought of as a bimodules over themselves.  As long as B is an associative, unital algebra (which it always will be in this post), there is a canonical projective resolution of B, called the bar resolution, which is sufficient for most purposes.  However, this resolution is of infinite length, and so it isn’t useful in bounding projective dimensions of modules.  For those purposes, it is natural to look for finite projective resolutions of B.

    I came across such a problem in my research, and came up with limited and ultimately unhelpful results.  My interest was in the case that B is a tensor algebra T_AM of an algebra A over a bimodule M.  Under what conditions on A and M would a nice, finite resolution of T_AM exist?  My result is as follows:

Let A be an algebra, M be a bimodule over A, and let \Omega^1A denote the kernel of the multiplication map m: A\otimes A \rightarrow A.  If \Omega^1A is projective as an A bimodule, and Tor_1^A(M_A,_AM)=0, then there is a projective resolution of T_AM of length 3.

This is kinda neat, but its not super useful unless it can be used to produce projective resolutions of T_AM modules.  Hence, the second result:

Let A, M, and \Omega^1A be defined as above.  If \Omega^1A is projective and M is flat as a right A module, then any left T_AM module has a projective resolution of length 3.

Sadly, I wanted the assumptions of the former to prove the latter, which my techniques don’t.

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Wanted: Intuition for Spectra

September 8, 2007

    I just read Greenlees’ paper, Spectra for Commutative Algebraists.  Spectra have long been one of my ‘favorite things in a subject far from my own’, though the above paper mentions techniques designed at removing that qualification.

     Spectra (singular: spectrum) are a weakening of the notion of a topological space.  The motivation is that the Freudenthal suspention theorem states that by successively suspending a finite CW complex, eventually the behavior ‘stabilizes’ and no new information is introduced.  Concretely, suspension induces a map from Hom(X,Y) to Hom(\Sigma X, \Sigma Y), and this map is an isomorphism if the suspension functor has already been applied enough times.  The idea of spectra is to describe things akin to spaces, but which capture the stable information, rather than the pre-stable information of the actual spaces.

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Sphere Eversion Video

August 30, 2007

    A friend of mine just showed me this video online of an eversion of the sphere; that is, a regular homotopy from the sphere sitting inside 3 space to itself which reverses orientation.

Sphere Eversion Video

As it is on google video, its not clear whether it is actually a public domain video or just standard internet crime, so I apolgize if I am linking to an ill gotten work (especially given that at least one credited contributor to the video has been known to frequent this site).