Cohomology Theory of Topological Transformation Groups

Cohomology Theory of Topological Transformation Groups

Author: W.Y. Hsiang

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 175

ISBN-13: 3642660525

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Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tl by more general compact groups.


Cohomological Methods in Transformation Groups

Cohomological Methods in Transformation Groups

Author: C. Allday

Publisher: Cambridge University Press

Published: 1993-07

Total Pages: 486

ISBN-13: 0521350220

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This is an account of the theory of certain types of compact transformation groups, namely those that are susceptible to study using ordinary cohomology theory and rational homotopy theory, which in practice means the torus groups and elementary abelian p-groups. The efforts of many mathematicians have combined to bring a depth of understanding to this area. However to make it reasonably accessible to a wide audience, the authors have streamlined the presentation, referring the reader to the literature for purely technical results and working in a simplified setting where possible. In this way the reader with a relatively modest background in algebraic topology and homology theory can penetrate rather deeply into the subject, whilst the book at the same time makes a useful reference for the more specialised reader.


Representation Theories and Algebraic Geometry

Representation Theories and Algebraic Geometry

Author: A. Broer

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 455

ISBN-13: 9401591318

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The 12 lectures presented in Representation Theories and Algebraic Geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, Hecke algebras, restricted Lie algebras, and their companions. This interplay has been extensively exploited during recent years, resulting in great progress in these representation theories. Conversely, a great stimulus has been given to the development of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology. The range of topics covered is wide, from equivariant Chow groups, decomposition classes and Schubert varieties, multiplicity free actions, convolution algebras, standard monomial theory, and canonical bases, to annihilators of quantum Verma modules, modular representation theory of Lie algebras and combinatorics of representation categories of Harish-Chandra modules.


Topological Methods in Group Theory

Topological Methods in Group Theory

Author: Ross Geoghegan

Publisher: Springer Science & Business Media

Published: 2007-12-17

Total Pages: 473

ISBN-13: 0387746110

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This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit.


A Concise Course in Algebraic Topology

A Concise Course in Algebraic Topology

Author: J. P. May

Publisher: University of Chicago Press

Published: 1999-09

Total Pages: 262

ISBN-13: 9780226511832

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Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.


Transformation Groups

Transformation Groups

Author: Tammo tom Dieck

Publisher: Walter de Gruyter

Published: 2011-04-20

Total Pages: 325

ISBN-13: 3110858371

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“This book is a jewel – it explains important, useful and deep topics in Algebraic Topology that you won’t find elsewhere, carefully and in detail.” Prof. Günter M. Ziegler, TU Berlin


Equivariant Poincaré Duality on G-Manifolds

Equivariant Poincaré Duality on G-Manifolds

Author: Alberto Arabia

Publisher: Springer Nature

Published: 2021-06-12

Total Pages: 383

ISBN-13: 3030704408

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This book carefully presents a unified treatment of equivariant Poincaré duality in a wide variety of contexts, illuminating an area of mathematics that is often glossed over elsewhere. The approach used here allows the parallel treatment of both equivariant and nonequivariant cases. It also makes it possible to replace the usual field of coefficients for cohomology, the field of real numbers, with any field of arbitrary characteristic, and hence change (equivariant) de Rham cohomology to the usual singular (equivariant) cohomology . The book will be of interest to graduate students and researchers wanting to learn about the equivariant extension of tools familiar from non-equivariant differential geometry.


Seminar on Transformation Groups

Seminar on Transformation Groups

Author: Armand Borel

Publisher: Princeton University Press

Published: 1960

Total Pages: 262

ISBN-13: 9780691090948

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The description for this book, Seminar on Transformation Groups. (AM-46), Volume 46, will be forthcoming.