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.

Geometry and Cohomology in Group Theory
Geometry and Cohomology in Group Theory

Author: Peter H. Kropholler

Publisher: Cambridge University Press

Published: 1998-05-14

Total Pages: 332

ISBN-13: 052163556X

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This volume reflects the fruitful connections between group theory and topology. It contains articles on cohomology, representation theory, geometric and combinatorial group theory. Some of the world's best known figures in this very active area of mathematics have made contributions, including substantial articles from Ol'shanskii, Mikhajlovskii, Carlson, Benson, Linnell, Wilson and Grigorchuk, which will be valuable reference works for some years to come. Pure mathematicians working in the fields of algebra, topology, and their interactions, will find this book of great interest.

Homological Group Theory
Homological Group Theory

Author: Charles Terence Clegg Wall

Publisher: Cambridge University Press

Published: 1979-12-27

Total Pages: 409

ISBN-13: 0521227291

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Eminent mathematicians have presented papers on homological and combinatorial techniques in group theory. The lectures are aimed at presenting in a unified way new developments in the area.

Cohomological Methods in Group Theory
Cohomological Methods in Group Theory

Author: Ararat Babakhanian

Publisher:

Published: 1972

Total Pages: 264

ISBN-13:

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This book is intended for students interested in learning the use of cohomology and homology theory in solving problems in group theory. Although cohomology groups of a groups were formally defined in the early 1940s, these groups in low dimensions had been studied earlier as part of the general body of theory of groups. In the last three decades cohomology of groups has played a central role in various branches of mathematics. This book provides readers with the basic tools in cohomology of groups and to illustrate their use in obtaining group theoretic results.

Cohomology of Finite Groups
Cohomology of Finite Groups

Author: Alejandro Adem

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 333

ISBN-13: 3662062828

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The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fields in mathematics, like homological algebra and algebraic K-theory. This is the first book to deal comprehensively with the cohomology of finite groups: it introduces the most important and useful algebraic and topological techniques, and describes the interplay of the subject with those of homotopy theory, representation theory and group actions. The combination of theory and examples, together with the techniques for computing the cohomology of important classes of groups including symmetric groups, alternating groups, finite groups of Lie type, and some of the sporadic simple groups, enable readers to acquire an in-depth understanding of group cohomology and its extensive applications.

Geometric Group Theory
Geometric Group Theory

Author: Clara Löh

Publisher: Springer

Published: 2017-12-19

Total Pages: 390

ISBN-13: 3319722549

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Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.

L2-Invariants: Theory and Applications to Geometry and K-Theory
L2-Invariants: Theory and Applications to Geometry and K-Theory

Author: Wolfgang Lück

Publisher: Springer Science & Business Media

Published: 2002-08-06

Total Pages: 624

ISBN-13: 9783540435662

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In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.

Continuous Bounded Cohomology of Locally Compact Groups
Continuous Bounded Cohomology of Locally Compact Groups

Author: Nicolas Monod

Publisher: Springer

Published: 2003-07-01

Total Pages: 219

ISBN-13: 3540449620

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Recent research has repeatedly led to connections between important rigidity questions and bounded cohomology. However, the latter has remained by and large intractable. This monograph introduces the functorial study of the continuous bounded cohomology for topological groups, with coefficients in Banach modules. The powerful techniques of this more general theory have successfully solved a number of the original problems in bounded cohomology. As applications, one obtains, in particular, rigidity results for actions on the circle, for representations on complex hyperbolic spaces and on Teichmüller spaces. A special effort has been made to provide detailed proofs or references in quite some generality.

Galois Cohomology and Class Field Theory
Galois Cohomology and Class Field Theory

Author: David Harari

Publisher: Springer Nature

Published: 2020-06-24

Total Pages: 336

ISBN-13: 3030439011

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This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory. Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.