Combinatorial descriptions of equivariant cohomology
I began working on projects to give combinatorial descriptions of
equivariant cohomology in my thesis. The motivation for these
projects comes from the work of Goresky, Kottwitz and MacPherson.
They consider a projective variety X
endowed with an algebraic action
of a complex torus. They assume that there are finitely many
zero- and one-dimensional orbits. In this case, the information
necessary to compute the equivariant cohomology of X can be encoded in
a labeled graph G.
I have worked on several projects generalizing this work:
- Act globally, compute locally:
group actions, fixed points, and localization
Contemp. Math., to appear.
Preprint arXiv:0710.5295
This is an expository article based on a couple of overview talks I
gave at the Every-10-Year-Algebraic-Geometry-Conference in Seattle in
2005, and at the Toric Topology conference in Osaka in 2006.
- A GKM description of the
equivarian cohomology ring of a homogeneous space
(with Victor Guillemin and Catalin Zara) J. Alg. Combin. 23 (2006) no. 1, 21-41.
Preprint math.SG/0112184
We explore homogeneous spaces and compute their cohomology
combinatorially.
- Computation of generalized
equivariant cohomologies of Kac-Moody flag varieties
(with Megumi Harada and Andre Henriques) Adv. Math. 197 (2005) No. 1, 198-221.
This is a substantial revision of math.DG/0402079 in which we allow
more general groups, more general spaces, and more general cohomology
theories.
Preprint math.AT/0409305.
T-equivariant cohomology of cell
complexes and the case of infinite Grassmannians
(with Megumi Harada and Andre Henriques)
Preprint math.DG/0402079
We allow infinite dimensional examples. In particular, we compute
the cohomology of the based loop space.
- The equivariant cohomology of
hypertoric varieties and their real loci
(with Megumi Harada) Commun. Anal.
and Geom. 13 (2005) No.
3, 645-677.
Preprint math.DG/0405422
We describe the cohomology of hyperkahler reductions of affine
quaternionic space by a torus in the spirit of the work of Goresky,
Kottwitz and MacPherson. We also describe the cohomology of their
real loci.
- The mod 2 equivariant cohomology
of real loci
(with Daniel Biss and Victor Guillemin) Adv. Math. 185 (2004) 370--399.
Preprint math.SG/0107151
This paper gives a combinatorial description of the equivariant
cohomology of the real points of the space X.
- GKM theory for torus actions
with non-isolated fixed points
(with Victor Guillemin) International
Math. Res. Notices 40
(2004) 2105--2124.
Preprint math.SG/038008
This paper generalizes the assumption on the zero-dimensional orbits.
- The equivariant cohomology of
Hamiltonian G-spaces from residual S^1 actions
(with Rebecca Goldin)
Math. Research Letters 8
(2001) 67--78.
Preprint math.SG/0107131
This paper generalizes the assumptions on
the one-dimensional orbits.
- Equivariant
cohomology, homogeneous spaces, and graphs
PhD Thesis, MIT 2002
Available in PDF