Stable Homotopy Groups Of Spheres

Author : Douglas C. Ravenel
Publisher : American Mathematical Society
Page : 417 pages
File Size : 50,9 Mb
Release : 2023-02-09
Category : Mathematics
ISBN : 9781470472931

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Complex Cobordism and Stable Homotopy Groups of Spheres by Douglas C. Ravenel Pdf

Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres. The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids. The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.

Author : Daniel C. Isaksen
Publisher : American Mathematical Soc.
Page : 159 pages
File Size : 49,6 Mb
Release : 2020-02-13
Category : Education
ISBN : 9781470437886

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Stable Stems by Daniel C. Isaksen Pdf

The author presents a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. He uses the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. He then uses the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. He also describes the complete calculation to the 65-stem, but defers the proofs in this range to forthcoming publications. In addition to finding all Adams differentials, the author also resolves all hidden extensions by 2, η, and ν through the 59-stem, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. The author also computes the motivic stable homotopy groups of the cofiber of the motivic element τ. This computation is essential for resolving hidden extensions in the Adams spectral sequence. He shows that the homotopy groups of the cofiber of τ are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows him to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.

Author : M a Kervaire,John W Milnor
Publisher : Legare Street Press
Page : 0 pages
File Size : 44,9 Mb
Release : 2023-07-18
Category : Electronic
ISBN : 1019386339

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Groups of Homotopy Spheres, I by M a Kervaire,John W Milnor Pdf

This book is a groundbreaking work in the field of topology, exploring the properties of homotopy spheres and the various groups that can be derived from them. With detailed proofs and rigorous analysis, this book is a must-read for anyone interested in topology or higher mathematics. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Author : Douglas C. Ravenel
Publisher : American Mathematical Soc.
Page : 418 pages
File Size : 55,5 Mb
Release : 2003-11-25
Category : Mathematics
ISBN : 9780821829677

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Complex Cobordism and Stable Homotopy Groups of Spheres by Douglas C. Ravenel Pdf

Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres. The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids. The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.

Author : Stanley O. Kochman
Publisher : Springer
Page : 338 pages
File Size : 46,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540469933

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Stable Homotopy Groups of Spheres by Stanley O. Kochman Pdf

A central problem in algebraic topology is the calculation of the values of the stable homotopy groups of spheres +*S. In this book, a new method for this is developed based upon the analysis of the Atiyah-Hirzebruch spectral sequence. After the tools for this analysis are developed, these methods are applied to compute inductively the first 64 stable stems, a substantial improvement over the previously known 45. Much of this computation is algorithmic and is done by computer. As an application, an element of degree 62 of Kervaire invariant one is shown to have order two. This book will be useful to algebraic topologists and graduate students with a knowledge of basic homotopy theory and Brown-Peterson homology; for its methods, as a reference on the structure of the first 64 stable stems and for the tables depicting the behavior of the Atiyah-Hirzebruch and classical Adams spectral sequences through degree 64.

Author : Douglas C. Ravenel
Publisher : Princeton University Press
Page : 228 pages
File Size : 40,6 Mb
Release : 1992-11-08
Category : Mathematics
ISBN : 069102572X

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Nilpotence and Periodicity in Stable Homotopy Theory by Douglas C. Ravenel Pdf

Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

Author : Stanley O. Kochman
Publisher : Unknown
Page : 344 pages
File Size : 42,6 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 366216728X

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Stable Homotopy Groups of Spheres by Stanley O. Kochman Pdf

Author : Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 54,7 Mb
Release : 2007-07-11
Category : Mathematics
ISBN : 9783540458975

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Motivic Homotopy Theory by Bjorn Ian Dundas,Marc Levine,P.A. Østvær,Oliver Röndigs,Vladimir Voevodsky Pdf

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.

Author : Hiroshi Toda
Publisher : Princeton University Press
Page : 208 pages
File Size : 47,7 Mb
Release : 1963-01-20
Category : Mathematics
ISBN : 0691095868

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Composition methods in homotopy groups of spheres by Hiroshi Toda Pdf

Hiroshi Toda's classic treatment of composition methods in homotopy groups of spheres from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.

Author : Robert E. Mosher,Martin C. Tangora
Publisher : Courier Corporation
Page : 226 pages
File Size : 53,9 Mb
Release : 2008-01-01
Category : Mathematics
ISBN : 9780486466644

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Cohomology Operations and Applications in Homotopy Theory by Robert E. Mosher,Martin C. Tangora Pdf

Cohomology operations are at the center of a major area of activity in algebraic topology. This treatment explores the single most important variety of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation. 1968 edition.

Author : Stanley O. Kochman
Publisher : American Mathematical Soc.
Page : 294 pages
File Size : 47,5 Mb
Release : 1996
Category : Mathematics
ISBN : 0821806009

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Bordism, Stable Homotopy and Adams Spectral Sequences by Stanley O. Kochman Pdf

This book is a compilation of lecture notes that were prepared for the graduate course ``Adams Spectral Sequences and Stable Homotopy Theory'' given at The Fields Institute during the fall of 1995. The aim of this volume is to prepare students with a knowledge of elementary algebraic topology to study recent developments in stable homotopy theory, such as the nilpotence and periodicity theorems. Suitable as a text for an intermediate course in algebraic topology, this book provides a direct exposition of the basic concepts of bordism, characteristic classes, Adams spectral sequences, Brown-Peterson spectra and the computation of stable stems. The key ideas are presented in complete detail without becoming encyclopedic. The approach to characteristic classes and some of the methods for computing stable stems have not been published previously. All results are proved in complete detail. Only elementary facts from algebraic topology and homological algebra are assumed. Each chapter concludes with a guide for further study.

Author : Mark Behrens
Publisher : American Mathematical Soc.
Page : 90 pages
File Size : 54,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869024

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The Goodwillie Tower and the EHP Sequence by Mark Behrens Pdf

The author studies the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime $2$. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. He relates the Goodwillie filtration to the $P$ map, and the Goodwillie differentials to the $H$ map. Furthermore, he studies an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. He shows that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. He uses his theory to recompute the $2$-primary unstable stems through the Toda range (up to the $19$-stem). He also studies the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod $2$ stable homology of the Goodwillie layers of any functor from spaces to spaces.

Author : Michael A. Hill,Michael J. Hopkins,Douglas C. Ravenel
Publisher : Cambridge University Press
Page : 881 pages
File Size : 46,9 Mb
Release : 2021-07-29
Category : Mathematics
ISBN : 9781108831444

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Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem by Michael A. Hill,Michael J. Hopkins,Douglas C. Ravenel Pdf

A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.

Author : James F. Davis,Paul Kirk
Publisher : American Mathematical Society
Page : 385 pages
File Size : 46,8 Mb
Release : 2023-05-22
Category : Mathematics
ISBN : 9781470473686

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Lecture Notes in Algebraic Topology by James F. Davis,Paul Kirk Pdf

The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems. To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated. Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book. The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem. A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars. The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.

Author : John Frank Adams
Publisher : University of Chicago Press
Page : 384 pages
File Size : 47,7 Mb
Release : 1974
Category : Mathematics
ISBN : 9780226005249

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Stable Homotopy and Generalised Homology by John Frank Adams Pdf

J. Frank Adams, the founder of stable homotopy theory, gave a lecture series at the University of Chicago in 1967, 1970, and 1971, the well-written notes of which are published in this classic in algebraic topology. The three series focused on Novikov's work on operations in complex cobordism, Quillen's work on formal groups and complex cobordism, and stable homotopy and generalized homology. Adams's exposition of the first two topics played a vital role in setting the stage for modern work on periodicity phenomena in stable homotopy theory. His exposition on the third topic occupies the bulk of the book and gives his definitive treatment of the Adams spectral sequence along with many detailed examples and calculations in KU-theory that help give a feel for the subject.