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 in terms of the cohomology of some big infinite-dimensional space . This is good on a conceptual level, but unless 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 .
The idea is to start by pining for the existance of a nice de Rham complex on . We’ll say “Oh, if only it existed, it would look like this, and this…”. Since 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 . Such DGAs will be called ‘locally-free, acyclic -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 -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).