nLab
Alex Hoffnung

chains. Here we will sketch the idea and eventually should formalize this on the page for Borel-Moore homology. In the meantime the Wikipedia article plus its references and the Chriss-Ginzburg book Representation Theory and Complex Geometry provide plenty of information.

H (Y)=(possiblyunbounded)ichainsinY(withcoefficientsin) H_\bullet(Y) = (possibly\; unbounded)\; i-chains\; in\; Y\;(with\; coefficients\; in \;\mathbb{C})
H i(Y)=idimsubspaceofYwithoutboundary(possiblysingularorunbounded) H_i(Y) = i-dim\; subspace\;of\; Y \;without\; boundary \;(possibly \;singular\; or\; unbounded)

The construction starts by triangulating the space and considering the vector space of formal sums of ii-simplices.

Of course, one wants independence of triangulation so we need to take a direct limit of these vectors spaces over all refinements of the triangulation. The boundary map then comes from simplicial homology and we have a chain complex. The homology of this complex is the Borel-Moore homology.

To anyone reading this: These are notes from the University of Ottawa/Fields Institute summer school which took place last week (if you are reading this before Friday July 3). Adam Katz and I are trying to make our way through the notes from several of the classes. Anyone is welcome to add comments, corrections, insights or just join our discussion. Actually it would be great if we had some more people involved. I think the plan is something like: 1) Understand Kamnitzer’s lectures on Borel-Weil, Ginzburg construction and geometric Satake, then move to Savage’s lectures relating the geometry from Kamnitzer and the combinatorics from Kang’s lectures. So Kang’s lectures will hopefully serve as background reading to understand Savage’s lectures.

Alex

Other projects:

1) Understand q-Schur algebras and relationship to representation theory of Hecke algebras

2) Learn lots of things in Ginzburg-Chriss.

3) Write my thesis!

4) Write everything else I should be writing.

5) Get a job! :)

I guess 3) and 5) are taken care of now.

If anyone wants to talk about the first item that would be great. I guess this is my new recruiting station to get people to talk to me about math.

category: people

Last revised on March 17, 2021 at 17:34:37. See the history of this page for a list of all contributions to it.