The topology of GKM spaces and GKM fibrations

This thesis primarily consists of results which can be used to simplify the computation of the equivariant cohomology of a GKM space. In particular we investigate the role that equivariant maps play in the computation of these cohomology rings. In the first part of the thesis, we describe some implications of the existence of an equivariant map p between an equivariantly formal T-manifold M and a GKM space fM. In particular we generalize the Chang-Skjelbred Theorem to this setting and derive some of its consequences. Then we consider the abstract setting of GKM graphs and define a category of objects which we refer to as GKM fiber bundles. For this class of bundles we prove a graph theoretical version of the Serre-Leray theorem. As an example, we study the projection maps from complete flag varieties to partial flag varieties from this combinatorial perspective. In the second part of the thesis we focus on GKM manifolds M which are also T-Hamiltonian manifolds. For these spaces, Guillemin and Zara ([GZ]), and Goldin and Tolman ([GT]), introduced a special basis for H* T (M), associated to a particular choice of a generic component ? of the moment map, the elements of this basis being called canonical classes. Since, for Hamiltonian T spaces, HT (M) can be viewed as a subring of the equivariant cohomology ring of the fixed point set, it is important to be able to compute the restriction of the elements of this basis to the fixed point set, and we investigate how one can use the existence of an equivariant map to simplify this computation.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2009

Author dc:creator

• Sabatini, Silvia

Advisor dc:contributor.advisor

• Victor Guillemin.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/50269

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/50269