equivariant ordinary homology and cohomology lecture notes in mathematics

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Equivariant Ordinary Homology And Cohomology

Author : Steven R. Costenoble
ISBN : 9783319504483
Genre : Mathematics
File Size : 64. 27 MB
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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.

The Ro G Graded Equivariant Ordinary Homology Of G Cell Complexes With Even Dimensional Cells For G Z P

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ISBN : 9780821834619
Genre :
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Equivariant Homotopy And Cohomology Theory

Author : J. Peter May
ISBN : 9780821803196
Genre : Science
File Size : 82. 92 MB
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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.

Parametrized Homotopy Theory

Author : J. Peter May
ISBN : 9780821839225
Genre : Mathematics
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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.

Symposium On Algebraic Topology In Honor Of Jose Adem

Author : Samuel Gitler
ISBN : 9780821850107
Genre : Mathematics
File Size : 42. 90 MB
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Encyclopaedia Of Mathematics

Author : Michiel Hazewinkel
ISBN : 9789401512886
Genre : Mathematics
File Size : 39. 90 MB
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This is the first Supplementary volume to Kluwer's highly acclaimed Encyclopaedia of Mathematics. This additional volume contains nearly 600 new entries written by experts and covers developments and topics not included in the already published 10-volume set. These entries have been arranged alphabetically throughout. A detailed index is included in the book. This Supplementary volume enhances the existing 10-volume set. Together, these eleven volumes represent the most authoritative, comprehensive up-to-date Encyclopaedia of Mathematics available.

Homotopy Theory And Related Topics

Author : Mamoru Mimura
ISBN : 9783540469384
Genre : Mathematics
File Size : 26. 66 MB
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Homotopy Methods In Algebraic Topology

Author : John Patrick Campbell Greenlees
ISBN : 9780821826218
Genre : Mathematics
File Size : 62. 42 MB
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This volume presents the proceedings from the AMS-IMS-SIAM Summer Research Conference on Homotopy Methods in Algebraic Topology held at the University of Colorado (Boulder). The conference coincided with the sixtieth birthday of J. Peter May. An article is included reflecting his wide-ranging and influential contributions to the subject area. Other articles in the book discuss the ordinary, elliptic and real-oriented Adams spectral sequences, mapping class groups, configuration spaces, extended powers, operads, the telescope conjecture, $p$-compact groups, algebraic K theory, stable and unstable splittings, the calculus of functors, the $E_{\infty}$ tensor product, and equivariant cohomology theories. The book offers a compendious source on modern aspects of homotopy theoretic methods in many algebraic settings.

Generalized Tate Cohomology

Author : John Patrick Campbell Greenlees
ISBN : 9780821826034
Genre : Mathematics
File Size : 76. 89 MB
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This book presents a systematic study of a new equivariant cohomology theory $t(k_G)^*$ constructed from any given equivariant cohomology theory $k^*_G$, where $G$ is a compact Lie group. Special cases include Tate-Swan cohomology when $G$ is finite and a version of cyclic cohomology when $G = S^1$. The groups $t(k_G)^*(X)$ are obtained by suitably splicing the $k$-homology with the $k$-cohomology of the Borel construction $EG\times _G X$, where $k^*$ is the nonequivariant cohomology theory that underlies $k^*_G$. The new theories play a central role in relating equivariant algebraic topology with current areas of interest in nonequivariant algebraic topology. Their study is essential to a full understanding of such ``completion theorems'' as the Atiyah-Segal completion theorem in $K$-theory and the Segal conjecture in cohomotopy. When $G$ is finite, the Tate theory associated to equivariant $K$-theory is calculated completely, and the Tate theory associated to equivariant cohomotopy is shown to encode a mysterious web of connections between the Tate cohomology of finite groups and the stable homotopy groups of spheres.

Local Cohomology And Its Applications

Author : Gennady Lybeznik
ISBN : 0824707419
Genre : Mathematics
File Size : 47. 68 MB
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This volume collects presentations from the international workshop on local cohomology held in Guanajuato, Mexico, including expanded lecture notes of two minicourses on applications in equivariant topology and foundations of duality theory, and chapters on finiteness properties, D-modules, monomial ideals, combinatorial analysis, and related topics. Featuring selected papers from renowned experts around the world, Local Cohomology and Its Applications is a provocative reference for algebraists, topologists, and upper-level undergraduate and graduate students in these disciplines.

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