Group Actions And Equivariant Cohomology
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Group Actions and Equivariant Cohomology
Author: Loring W. Tu
language: en
Publisher: American Mathematical Society
Release Date: 2024-11-26
This volume contains the proceedings of the virtual AMS Special Session on Equivariant Cohomology, held March 19?20, 2022. Equivariant topology is the algebraic topology of spaces with symmetries. At the meeting, ?equivariant cohomology? was broadly interpreted to include related topics in equivariant topology and geometry such as Bredon cohomology, equivariant cobordism, GKM (Goresky, Kottwitz, and MacPherson) theory, equivariant $K$-theory, symplectic geometry, and equivariant Schubert calculus. This volume offers a view of the exciting progress made in these fields in the last twenty years. Several of the articles are surveys suitable for a general audience of topologists and geometers. To be broadly accessible, all the authors were instructed to make their presentations somewhat expository. This collection should be of interest and useful to graduate students and researchers alike.
Hamiltonian Group Actions and Equivariant Cohomology
This monograph could be used for a graduate course on symplectic geometry as well as for independent study. The monograph starts with an introduction of symplectic vector spaces, followed by symplectic manifolds and then Hamiltonian group actions and the Darboux theorem. After discussing moment maps and orbits of the coadjoint action, symplectic quotients are studied. The convexity theorem and toric manifolds come next and we give a comprehensive treatment of Equivariant cohomology. The monograph also contains detailed treatment of the Duistermaat-Heckman Theorem, geometric quantization, and flat connections on 2-manifolds. Finally, there is an appendix which provides background material on Lie groups. A course on differential topology is an essential prerequisite for this course. Some of the later material will be more accessible to readers who have had a basic course on algebraic topology. For some of the later chapters, it would be helpful to have some background on representation theory and complex geometry.
Moment Maps, Cobordisms, and Hamiltonian Group Actions
Author: Victor Guillemin
language: en
Publisher: American Mathematical Soc.
Release Date: 2002
During the last 20 years, ``localization'' has been one of the dominant themes in the area of equivariant differential geometry. Typical results are the Duistermaat-Heckman theory, the Berline-Vergne-Atiyah-Bott localization theorem in equivariant de Rham theory, and the ``quantization commutes with reduction'' theorem and its various corollaries. To formulate the idea that these theorems are all consequences of a single result involving equivariant cobordisms, the authors have developed a cobordism theory that allows the objects to be non-compact manifolds. A key ingredient in this non-compact cobordism is an equivariant-geometrical object which they call an ``abstract moment map''. This is a natural and important generalization of the notion of a moment map occurring in the theory of Hamiltonian dynamics. The book contains a number of appendices that include introductions to proper group-actions on manifolds, equivariant cohomology, Spin${^\mathrm{c}}$-structures, and stable complex structures. It is geared toward graduate students and research mathematicians interested in differential geometry. It is also suitable for topologists, Lie theorists, combinatorists, and theoretical physicists. Prerequisite is some expertise in calculus on manifolds and basic graduate-level differential geometry.