Author: Steven R. Costenoble,Stefan Waner

Publisher: Springer

ISBN: 3319504487

Category: Mathematics

Page: 294

View: 6783

Filling a gap in the literature, this book takes the reader to the frontiers of equivariant topology, the study of objects with specified symmetries. The discussion is motivated by reference to a list of instructive “toy” examples and calculations in what is a relatively unexplored field. The authors also provide a reading path for the first-time reader less interested in working through sophisticated machinery but still desiring a rigorous understanding of the main concepts. The subject’s classical counterparts, ordinary homology and cohomology, dating back to the work of Henri Poincaré in topology, are calculational and theoretical tools which are important in many parts of mathematics and theoretical physics, particularly in the study of manifolds. Similarly powerful tools have been lacking, however, in the context of equivariant topology. Aimed at advanced graduate students and researchers in algebraic topology and related fields, the book assumes knowledge of basic algebraic topology and group actions.
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Author: J. Peter May,Johann Sigurdsson

Publisher: American Mathematical Soc.

ISBN: 0821839225

Category: Mathematics

Page: 441

View: 3281

This book develops rigorous foundations for parametrized homotopy theory, which is the algebraic topology of spaces and spectra that are continuously parametrized by the points of a base space. It also begins the systematic study of parametrized homology and cohomology theories. The parametrized world provides the natural home for many classical notions and results, such as orientation theory, the Thom isomorphism, Atiyah and Poincare duality, transfer maps, the Adams and Wirthmuller isomorphisms, and the Serre and Eilenberg-Moore spectral sequences. But in addition to providing a clearer conceptual outlook on these classical notions, it also provides powerful methods to study new phenomena, such as twisted $K$-theory, and to make new constructions, such as iterated Thom spectra. Duality theory in the parametrized setting is particularly illuminating and comes in two flavors. One allows the construction and analysis of transfer maps, and a quite different one relates parametrized homology to parametrized cohomology. The latter is based formally on a new theory of duality in symmetric bicategories that is of considerable independent interest. The text brings together many recent developments in homotopy theory. It provides a highly structured theory of parametrized spectra, and it extends parametrized homotopy theory to the equivariant setting. The theory of topological model categories is given a more thorough treatment than is available in the literature. This is used, together with an interesting blend of classical methods, to resolve basic foundational problems that have no nonparametrized counterparts.
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Dedicated to the Memory of Robert J. Piacenza

Author: J. Peter May,Robert John Piacenza,M. Cole

Publisher: American Mathematical Soc.

ISBN: 0821803190

Category: Science

Page: 366

View: 8925

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 book 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. It 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. Features: Introduces many of the fundamental ideas and concepts of modern algebraic topology. Presents comprehensive material not found in any other book on the subject. Provides a coherent overview of many areas of current interest in algebraic topology. Surveys a great deal of material, explaining main ideas without getting bogged down in details.
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An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia

Author: Michiel Hazewinkel

Publisher: Springer Science & Business Media

ISBN: 9780792347095

Category: Mathematics

Page: 588

View: 9790

The Encyclopaedia of Mathematics is the most up-to-date, authoritative and comprehensive English-language work of reference in mathematics and is now available on CD-ROM. The CD-ROM edition contains all the articles published in the 10-Volume set and the first Supplementary volume. All articles are hyperlinked and some animation and 3D objects have been added. Presentation and navigation is managed by Dynatext and an advanced search engine allows for full text search, searching with wildcards, Boolean search, proximity search and structure-aware searches. All text can be printed and exported. The CD-ROM is available in two versions: a network version for PC and Unix platforms, and a stand-alone version for PC and Mac platforms only.
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Author: Jean-Claude Hausmann

Publisher: Springer

ISBN: 3319093541

Category: Mathematics

Page: 535

View: 5220

Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has many pedagogical advantages: 1. It leads more quickly to the essentials of the subject, 2. An absence of signs and orientation considerations simplifies the theory, 3. Computations and advanced applications can be presented at an earlier stage, 4. Simple geometrical interpretations of (co)chains. Mod 2 (co)homology was developed in the first quarter of the twentieth century as an alternative to integral homology, before both became particular cases of (co)homology with arbitrary coefficients. The first chapters of this book may serve as a basis for a graduate-level introductory course to (co)homology. Simplicial and singular mod 2 (co)homology are introduced, with their products and Steenrod squares, as well as equivariant cohomology. Classical applications include Brouwer's fixed point theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith theory, Kervaire invariant, etc. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the relationship between Stiefel-Whitney classes and Schubert calculus. More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst others. Each chapter ends with exercises, with some hints and answers at the end of the book.
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Author: J. P. May

Publisher: University of Chicago Press

ISBN: 9780226511832

Category: Mathematics

Page: 243

View: 9016

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.
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Author: William G. Dwyer,Hans-Werner Henn

Publisher: Birkhäuser

ISBN: 3034883560

Category: Mathematics

Page: 98

View: 8589

This book consists essentially of notes which were written for an Advanced Course on Classifying Spaces and Cohomology of Groups. The course took place at the Centre de Recerca Mathematica (CRM) in Bellaterra from May 27 to June 2, 1998 and was part of an emphasis semester on Algebraic Topology. It consisted of two parallel series of 6 lectures of 90 minutes each and was intended as an introduction to new homotopy theoretic methods in group cohomology. The first part of the book is concerned with methods of decomposing the classifying space of a finite group into pieces made of classifying spaces of appropriate subgroups. Such decompositions have been used with great success in the last 10-15 years in the homotopy theory of classifying spaces of compact Lie groups and p-compact groups in the sense of Dwyer and Wilkerson. For simplicity the emphasis here is on finite groups and on homological properties of various decompositions known as centralizer resp. normalizer resp. subgroup decomposition. A unified treatment of the various decompositions is given and the relations between them are explored. This is preceeded by a detailed discussion of basic notions such as classifying spaces, simplicial complexes and homotopy colimits.
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Author: Joseph Bernstein,Valery Lunts

Publisher: Springer

ISBN: 3540484302

Category: Mathematics

Page: 146

View: 9812

The equivariant derived category of sheaves is introduced. All usual functors on sheaves are extended to the equivariant situation. Some applications to the equivariant intersection cohomology are given. The theory may be useful to specialists in representation theory, algebraic geometry or topology.
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Author: Norman Earl Steenrod,D. B. A. Epstein

Publisher: Princeton University Press

ISBN: 9780691079240

Category: Mathematics

Page: 138

View: 2647

Written and revised by D. B. A. Epstein.
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Author: Alejandro Adem,R.James Milgram

Publisher: Springer Science & Business Media

ISBN: 3662062828

Category: Mathematics

Page: 331

View: 4983

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Author: Jonathan A. Barmak

Publisher: Springer Science & Business Media

ISBN: 3642220029

Category: Mathematics

Page: 170

View: 7652

This volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.
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Author: I.M. James

Publisher: Elsevier

ISBN: 9780080532981

Category: Mathematics

Page: 1324

View: 2117

Algebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who have contributed to the Handbook. Written for the reader who already has a grounding in the subject, the volume consists of 27 expository surveys covering the most active areas of research. They provide the researcher with an up-to-date overview of this exciting branch of mathematics.
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Author: Mark Goresky,Robert MacPherson

Publisher: Springer Science & Business Media

ISBN: 3642717144

Category: Mathematics

Page: 272

View: 6097

Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, including its historical development. Some further important topics in the book are: Morse theory, singularities, transversality theory, complex analytic varieties, Lefschetz theorems, connectivity theorems, intersection homology, complements of affine subspaces and combinatorics. The book is designed for all interested students or professionals in this area.
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