Bonnie Berger thesis advisorDaniel Kleitman, Dr.
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.
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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.
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.
We also derive conditions under which the formulas we get are integral.
Using the above results, we are able to prove, inter alia, positive integral formulas for the equivariant Schubert classes on a complete flag variety of type An,Bn,Cn and Dn.
These formulas are new, except in type An.
More generally, we obtain positive integral formulas for the equivariant Schubert classes using fibrations of the complete flag variety over partial flag varieties, and when this fibration is a CP1-bundle one gets from these formulas the calculus of divided difference operators.
In Geometry, mechanics, and dynamics, pagesSpringer NY, This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. Includes bibliographical references p.
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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.
The topology of GKM spaces and GKM ﬁbrations by Silvia Sabatini Laurea in Matematica, Universita’ di Roma “La Sapienza“ () distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Balanced Fiber Bundles and GKM Theory.
Silvia Sabatini; This thesis primarily consists of results which can be used to simplify the computation of the equivariant cohomology of a GKM space.