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Author: Tara Suzanne Holm
Publisher:
ISBN:
Category :
Languages : en
Pages : 100
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Book Description
(Cont.) Next, we describe how to weaken the hypotheses of the GKM theorem. The spaces to which the GKM theorem applies must satisfy certain dimension conditions; however, there are many manifolds M with naturally arising T-actions that do not satisfy these conditions. We allow a more general situation, which includes some of these cases. Finally, we find a theory identical to the GKM theory in a setting suggested by work of Duistermaat. As in the GKM situation, this theory applies only when the spaces involved satisfy certain dimension conditions.
Author: Tara Suzanne Holm
Publisher:
ISBN:
Category :
Languages : en
Pages : 100
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Book Description
(Cont.) Next, we describe how to weaken the hypotheses of the GKM theorem. The spaces to which the GKM theorem applies must satisfy certain dimension conditions; however, there are many manifolds M with naturally arising T-actions that do not satisfy these conditions. We allow a more general situation, which includes some of these cases. Finally, we find a theory identical to the GKM theory in a setting suggested by work of Duistermaat. As in the GKM situation, this theory applies only when the spaces involved satisfy certain dimension conditions.
Author: Constantin Leonardo Mihalcea
Publisher:
ISBN:
Category :
Languages : en
Pages : 264
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Book Description
Author: David Anderson
Publisher: Cambridge University Press
ISBN: 1009349961
Category : Mathematics
Languages : en
Pages : 464
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Book Description
Intended for first- or second-year graduate students in mathematics, as well as researchers working in algebraic geometry or combinatorics, this text introduces techniques that are essential in several areas of modern mathematics. With numerous exercises and examples, it covers the core notions and applications of equivariant cohomology.
Author: David E. Anderson
Publisher:
ISBN: 9781009349970
Category : Geometry, Algebraic
Languages : en
Pages : 0
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Book Description
Equivariant cohomology has become an indispensable tool in algebraic geometry and in related areas including representation theory, combinatorial and enumerative geometry, and algebraic combinatorics. This text introduces the main ideas of the subject for first- or second-year graduate students in mathematics, as well as researchers working in algebraic geometry or combinatorics. The first six chapters cover the basics: definitions via finite-dimensional approximation spaces, computations in projective space, and the localization theorem. The rest of the text focuses on examples - toric varieties, Grassmannians, and homogeneous spaces - along with applications to Schubert calculus and degeneracy loci. Prerequisites are kept to a minimum, so that one-semester graduate-level courses in algebraic geometry and topology should be sufficient preparation. Featuring numerous exercises, examples, and material that has not previously appeared in textbook form, this book will be a must-have reference and resource for both students and researchers for years to come.
Author: J. Peter May
Publisher: American Mathematical Soc.
ISBN: 0821803190
Category : Mathematics
Languages : en
Pages : 384
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Book Description
This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The works begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. The book then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the Segal conjecture and generalized Tate cohomology. Finally, the book gives an introduction to "brave new algebra", the study of point-set level algebraic structures on spectra and its equivariant applications. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail.
Author: Loring W. Tu
Publisher: Princeton University Press
ISBN: 0691197482
Category : Mathematics
Languages : en
Pages : 200
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Book Description
This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah-Bott and Berline-Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.
Author: Pavle V. M. Blagojević
Publisher: Springer Nature
ISBN: 3030841383
Category : Mathematics
Languages : en
Pages : 217
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Book Description
This book gives a brief treatment of the equivariant cohomology of the classical configuration space F(R^d,n) from its beginnings to recent developments. This subject has been studied intensively, starting with the classical papers of Artin (1925/1947) on the theory of braids, and progressing through the work of Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). The focus of this book is on the mod 2 equivariant cohomology algebras of F(R^d,n), whose additive structure was described by Cohen (1976) and whose algebra structure was studied in an influential paper by Hung (1990). A detailed new proof of Hung's main theorem is given, however it is shown that some of the arguments given by him on the way to his result are incorrect, as are some of the intermediate results in his paper. This invalidates a paper by three of the authors, Blagojević, Lück and Ziegler (2016), who used a claimed intermediate result in order to derive lower bounds for the existence of k-regular and l-skew embeddings. Using the new proof of Hung's main theorem, new lower bounds for the existence of highly regular embeddings are obtained: Some of them agree with the previously claimed bounds, some are weaker. Assuming only a standard graduate background in algebraic topology, this book carefully guides the reader on the way into the subject. It is aimed at graduate students and researchers interested in the development of algebraic topology in its applications in geometry.
Author: Ezra Miller
Publisher: American Mathematical Soc.
ISBN: 9780821886953
Category : Mathematics
Languages : en
Pages : 710
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Book Description
Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. This text is a compilation of expository articles at the interface between combinatorics and geometry.
Author: D.A. Timashev
Publisher: Springer Science & Business Media
ISBN: 3642183999
Category : Mathematics
Languages : en
Pages : 267
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Book Description
Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry, theory of algebraic groups, classical projective and enumerative geometry, harmonic analysis, and representation theory. By standard reasons of algebraic geometry, in order to solve various problems on a homogeneous space, it is natural and helpful to compactify it while keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of "combinatorial" nature (the Luna-Vust theory) and description of various geometric and representation-theoretic properties of these varieties based on these data. The class of spherical varieties, intensively studied during the last three decades, is of special interest in the scope of this book. Spherical varieties include many classical examples, such as Grassmannians, flag varieties, and varieties of quadrics, as well as well-known toric varieties. We have attempted to cover most of the important issues, including the recent substantial progress obtained in and around the theory of spherical varieties.
Author: Glen E. Bredon
Publisher: Springer
ISBN: 3540349731
Category : Mathematics
Languages : en
Pages : 72
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Book Description
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