On Finite Groups And Homotopy Theory

Author : Ran Levi
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 50,8 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821804018

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On Finite Groups and Homotopy Theory by Ran Levi Pdf

Let $p$ be a fixed prime number. Let $G$ denote a finite $p$-perfect group. This book looks at the homotopy type of the $p$-completed classifying space $BG_p$, where $G$ is a finite $p$-perfect group. The author constructs an algebraic analog of the Quillen's ``plus'' construction for differential graded coalgebras. This construction is used to show that given a finite $p$-perfect group $G$, the loop spaces $BG_p$ admits integral homology exponents. Levi gives examples to show that in some cases our bound is best possible. It is shown that in general $B\ast _p$ admits infinitely many non-trivial $k$-invariants. The author presents examples where homotopy exponents exist. Classical constructions in stable homotopy theory are used to show that the stable homotopy groups of these loop spaces also have exponents.

Author : Alejandro Adem,R.James Milgram
Publisher : Springer Science & Business Media
Page : 333 pages
File Size : 46,8 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662062821

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Cohomology of Finite Groups by Alejandro Adem,R.James Milgram Pdf

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.

Author : Donald M. Davis
Publisher : American Mathematical Soc.
Page : 424 pages
File Size : 41,9 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821828014

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Recent Progress in Homotopy Theory by Donald M. Davis Pdf

This volume presents the proceedings from the month-long program held at Johns Hopkins University (Baltimore, MD) on homotopy theory, sponsored by the Japan-U.S. Mathematics Institute (JAMI). The book begins with historical accounts on the work of Professors Peter Landweber and Stewart Priddy. Central among the other topics are the following: 1. classical and nonclassical theory of $H$-spaces, compact groups, and finite groups, 2. classical and chromatic homotopy theory andlocalization, 3. classical and topological Hochschild cohomology, 4. elliptic cohomology and its relation to Moonshine and topological modular forms, and 5. motivic cohomology and Chow rings. This volume surveys the current state of research in these areas and offers an overview of futuredirections.

Author : Alejandro Adem,R. James Milgram,Douglas C. Ravenel
Publisher : American Mathematical Soc.
Page : 250 pages
File Size : 55,6 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821803059

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Homotopy Theory and Its Applications by Alejandro Adem,R. James Milgram,Douglas C. Ravenel Pdf

This book is the result of a conference held to examine developments in homotopy theory in honor of Samuel Gitler in July 1993 (Cocoyoc, Mexico). It includes several research papers and three expository papers on various topics in homotopy theory. The research papers discuss the following: BL application of homotopy theory to group theory BL fiber bundle theory BL homotopy theory The expository papers consider the following topics: BL the Atiyah-Jones conjecture (by C. Boyer) BL classifying spaces of finite groups (by J. Martino) BL instanton moduli spaces (by J. Milgram) Homotopy Theory and Its Applications offers a distinctive account of how homotopy theoretic methods can be applied to a variety of interesting problems.

Author : Jaume Aguade,Manuel Castellet,Frederick R. Cohen
Publisher : Springer
Page : 339 pages
File Size : 54,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540467724

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Algebraic Topology by Jaume Aguade,Manuel Castellet,Frederick R. Cohen Pdf

The papers in this collection, all fully refereed, original papers, reflect many aspects of recent significant advances in homotopy theory and group cohomology. From the Contents: A. Adem: On the geometry and cohomology of finite simple groups.- D.J. Benson: Resolutions and Poincar duality for finite groups.- C. Broto and S. Zarati: On sub-A*-algebras of H*V.- M.J. Hopkins, N.J. Kuhn, D.C. Ravenel: Morava K-theories of classifying spaces and generalized characters for finite groups.- K. Ishiguro: Classifying spaces of compact simple lie groups and p-tori.- A.T. Lundell: Concise tables of James numbers and some homotopyof classical Lie groups and associated homogeneous spaces.- J.R. Martino: Anexample of a stable splitting: the classifying space of the 4-dim unipotent group.- J.E. McClure, L. Smith: On the homotopy uniqueness of BU(2) at the prime 2.- G. Mislin: Cohomologically central elements and fusion in groups.

Author : Bertram A. F. Wehrfritz
Publisher : World Scientific
Page : 138 pages
File Size : 53,7 Mb
Release : 1999
Category : Mathematics
ISBN : 9810238746

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Finite Groups by Bertram A. F. Wehrfritz Pdf

The theory of groups, especially of finite groups, is one of the most delightful areas of mathematics. Its proofs often have elegance and crystalline beauty. This textbook is intended for the reader who has been exposed to about three years of serious mathematics. The notion of a group appears widely in mathematics and even further afield in physics and chemistry, and the fundamental idea should be known to all mathematicians. In this textbook a purely algebraic approach is taken and the choice of material is based upon the notion of conjugacy. The aim is not only to cover basic material, but also to present group theory as a living, vibrant and growing discipline, by including references and discussion of some work up to the present day.

Author : Mark E. Mahowald
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 47,6 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821808054

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Homotopy Theory via Algebraic Geometry and Group Representations by Mark E. Mahowald Pdf

The academic year 1996-97 was designated as a special year in Algebraic Topology at Northwestern University (Evanston, IL). In addition to guest lecturers and special courses, an international conference was held entitled "Current trends in algebraic topology with applications to algebraic geometry and physics". The series of plenary lectures included in this volume indicate the great breadth of the conference and the lively interaction that took place among various areas of mathematics. Original research papers were submitted, and all submissions were refereed to the usual journal standards.

Author : William G. Dwyer,Hans-Werner Henn
Publisher : Birkhäuser
Page : 106 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883566

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Homotopy Theoretic Methods in Group Cohomology by William G. Dwyer,Hans-Werner Henn Pdf

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.

Author : Stefan Schwede
Publisher : Cambridge University Press
Page : 847 pages
File Size : 40,7 Mb
Release : 2018-09-06
Category : Mathematics
ISBN : 9781108425810

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Global Homotopy Theory by Stefan Schwede Pdf

A comprehensive, self-contained approach to global equivariant homotopy theory, with many detailed examples and sample calculations.

Author : Paul Selick
Publisher : American Mathematical Soc.
Page : 220 pages
File Size : 55,9 Mb
Release : 2008
Category : Mathematics
ISBN : 0821844369

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Introduction to Homotopy Theory by Paul Selick Pdf

Offers a summary for students and non-specialists who are interested in learning the basics of algebraic topology. This book covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, and generalized homology and cohomology operations.

Author : David J. Benson,Srikanth Iyengar,Henning Krause
Publisher : Springer Science & Business Media
Page : 115 pages
File Size : 52,9 Mb
Release : 2011-11-15
Category : Mathematics
ISBN : 9783034802604

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Representations of Finite Groups: Local Cohomology and Support by David J. Benson,Srikanth Iyengar,Henning Krause Pdf

The seminar focuses on a recent solution, by the authors, of a long standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on ideas from commutative algebra, cohomology of groups, and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. The prototype for this has been Daniel Quillen’s description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Jon Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings and over cocommutative Hopf algebras. One of the threads in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Mike Hopkins’ classification of thick subcategories of the perfect complexes over a commutative Noetherian ring, followed by a classification of localizing subcategories of its full derived category, due to Amnon Neeman. The authors have been developing an approach to address such classification problems, based on a construction of local cohomology functors and support for triangulated categories with ring of operators. The book serves as an introduction to this circle of ideas.

Author : Jonathan A. Barmak
Publisher : Springer Science & Business Media
Page : 184 pages
File Size : 55,9 Mb
Release : 2011-08-24
Category : Mathematics
ISBN : 9783642220029

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Algebraic Topology of Finite Topological Spaces and Applications by Jonathan A. Barmak Pdf

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.

Author : F.E.A. Johnson
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 46,7 Mb
Release : 2011-11-17
Category : Mathematics
ISBN : 9781447122944

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Syzygies and Homotopy Theory by F.E.A. Johnson Pdf

The most important invariant of a topological space is its fundamental group. When this is trivial, the resulting homotopy theory is well researched and familiar. In the general case, however, homotopy theory over nontrivial fundamental groups is much more problematic and far less well understood. Syzygies and Homotopy Theory explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents, for the first time, a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory. The first part of the book is theoretical, formulated to allow a general finitely presented group as a fundamental group. The innovation here is to regard syzygies as stable modules rather than minimal modules. Inevitably this forces a reconsideration of the problems of noncancellation; these are confronted in the second, practical, part of the book. In particular, the second part of the book considers how the theory works out in detail for the specific examples Fn ́F where Fn is a free group of rank n and F is finite. Another innovation is to parametrize the first syzygy in terms of the more familiar class of stably free modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares. The theory developed within this book has potential applications in various branches of algebra, including homological algebra, ring theory and K-theory. Syzygies and Homotopy Theory will be of interest to researchers and also to graduate students with a background in algebra and algebraic topology.

Author : Paul Gregory Goerss,Stewart Priddy
Publisher : American Mathematical Soc.
Page : 520 pages
File Size : 50,6 Mb
Release : 2004
Category : Mathematics
ISBN : 9780821832851

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Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory by Paul Gregory Goerss,Stewart Priddy Pdf

As part of its series of Emphasis Years in Mathematics, Northwestern University hosted an International Conference on Algebraic Topology. The purpose of the conference was to develop new connections between homotopy theory and other areas of mathematics. This proceedings volume grew out of that event. Topics discussed include algebraic geometry, cohomology of groups, algebraic $K$-theory, and $\mathbb{A 1$ homotopy theory. Among the contributors to the volume were Alejandro Adem,Ralph L. Cohen, Jean-Louis Loday, and many others. The book is suitable for graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.

Author : Michael Artin,Barry Mazur
Publisher : Springer
Page : 173 pages
File Size : 51,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540361428

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Etale Homotopy by Michael Artin,Barry Mazur Pdf