Stable Homotopy Over The Steenrod Algebra

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Stable Homotopy over the Steenrod Algebra

John Harold Palmieri
Stable Homotopy over the Steenrod Algebra Book PDF

Author : John Harold Palmieri
Publisher : American Mathematical Soc.
Page : 193 pages
File Size : 47,9 Mb
Release : 2001
Category : Homotopy theory
ISBN : 9780821826683

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Stable Homotopy over the Steenrod Algebra by John Harold Palmieri Pdf

This title applys the tools of stable homotopy theory to the study of modules over the mod $p$ Steenrod algebra $A DEGREES{*}$. More precisely, let $A$ be the dual of $A DEGREES{*}$; then we study the category $\mathsf{stable}(A)$ of unbounded cochain complexes of injective comodules over $A$, in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, Brown-Comenetz duality, and other homotopy-theoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of $A$, $\mathrm{Ext}_A DEGREES{**}(\mathbf{F}_p, \mathbf{F}_p)$. This title also has nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a nu

Spectra and the Steenrod Algebra

H.R. Margolis
Spectra and the Steenrod Algebra Book PDF

Author : H.R. Margolis
Publisher : Elsevier
Page : 511 pages
File Size : 40,6 Mb
Release : 2011-08-18
Category : Mathematics
ISBN : 9780080960173

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Spectra and the Steenrod Algebra by H.R. Margolis Pdf

I have intended this book to be more than just the sum of its chapters, and the introduction is, in part, an attempt to spell out what the more is. Algebraic topology is the study of topological problems by algebraic means. More precisely, this has come to be framed as the study of topological categories by means of functors to algebraic categories. Beyond the basic definitions and structure, the focus is often on particular problems, for example, Adams’ use of K-theory to solve the vector fields on spheres problem. On the other hand, there are contributions of a more global nature yielding insight into the overall structure of some topological category, for example, Quillen’s work on rational homotopy type. This book is intended primarily as a contribution of this latter sort. So while there will be a variety of particular examples and computations, and although the structure being developed has significant application to many specific problems (some of which are considered here), the major thrust of the text is toward understanding the global structure and linkage of the topological and algebraic categories considered: the stable homotopy category and the category of modules over the Steenrod algebra.

Nilpotence and Periodicity in Stable Homotopy Theory. (AM-128), Volume 128

Douglas C. Ravenel
Nilpotence and Periodicity in Stable Homotopy Theory AM 128 Volume 128 Book PDF

Author : Douglas C. Ravenel
Publisher : Princeton University Press
Page : 224 pages
File Size : 48,9 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882489

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Nilpotence and Periodicity in Stable Homotopy Theory. (AM-128), Volume 128 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.

Complex Cobordism and Stable Homotopy Groups of Spheres

Douglas C. Ravenel
Complex Cobordism and Stable Homotopy Groups of Spheres Book PDF

Author : Douglas C. Ravenel
Publisher : American Mathematical Society
Page : 417 pages
File Size : 53,5 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.

Stable and Unstable Homotopy

William G. Dwyer
Stable and Unstable Homotopy Book PDF

Author : William G. Dwyer
Publisher : American Mathematical Soc.
Page : 328 pages
File Size : 43,7 Mb
Release : 1998-01-01
Category : Mathematics
ISBN : 0821871269

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Stable and Unstable Homotopy by William G. Dwyer Pdf

This volume presents the proceedings of workshops on stable homotopy theory and on unstable homotopy theory held at The Fields Institute as part of the homotopy program during the year 1996. The papers in the volume describe current research in the subject, and all included works were refereed. Rather than being a summary of work to be published elsewhere, each paper is the unique source for the new material it contains. The book contains current research from international experts in the subject area, and presents open problems with directions for future research.

Stable Homotopy Around the Arf-Kervaire Invariant

Victor P. Snaith
Stable Homotopy Around the Arf Kervaire Invariant Book PDF

Author : Victor P. Snaith
Publisher : Springer Science & Business Media
Page : 250 pages
File Size : 53,6 Mb
Release : 2009-03-28
Category : Mathematics
ISBN : 9783764399047

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Stable Homotopy Around the Arf-Kervaire Invariant by Victor P. Snaith Pdf

Were I to take an iron gun, And ?re it o? towards the sun; I grant ‘twould reach its mark at last, But not till many years had passed. But should that bullet change its force, And to the planets take its course, ‘Twould never reach the nearest star, Because it is so very far. from FACTS by Lewis Carroll [55] Let me begin by describing the two purposes which prompted me to write this monograph. This is a book about algebraic topology and more especially about homotopy theory. Since the inception of algebraic topology [217] the study of homotopy classes of continuous maps between spheres has enjoyed a very exc- n n tional, central role. As is well known, for homotopy classes of maps f : S ?? S with n? 1 the sole homotopy invariant is the degree, which characterises the homotopy class completely. The search for a continuous map between spheres of di?erent dimensions and not homotopic to the constant map had to wait for its resolution until the remarkable paper of Heinz Hopf [111]. In retrospect, ?nding 3 an example was rather easy because there is a canonical quotient map from S to 3 1 1 2 theorbitspaceofthe freecircleactionS /S =CP = S .

Complex Cobordism and Stable Homotopy Groups of Spheres

Douglas C. Ravenel
Complex Cobordism and Stable Homotopy Groups of Spheres Book PDF

Author : Douglas C. Ravenel
Publisher : American Mathematical Soc.
Page : 418 pages
File Size : 55,6 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.

Stable Homotopy and Generalised Homology

John Frank Adams
Stable Homotopy and Generalised Homology Book PDF

Author : John Frank Adams
Publisher : University of Chicago Press
Page : 384 pages
File Size : 45,6 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.

Bordism, Stable Homotopy and Adams Spectral Sequences

Stanley O. Kochman
Bordism Stable Homotopy and Adams Spectral Sequences Book PDF

Author : Stanley O. Kochman
Publisher : American Mathematical Soc.
Page : 294 pages
File Size : 45,7 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.

Cohomology Operations and Applications in Homotopy Theory

Robert E. Mosher,Martin C. Tangora
Cohomology Operations and Applications in Homotopy Theory Book PDF

Author : Robert E. Mosher,Martin C. Tangora
Publisher : Courier Corporation
Page : 226 pages
File Size : 52,5 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.

Stable Stems

Daniel C. Isaksen
Stable Stems Book PDF

Author : Daniel C. Isaksen
Publisher : Unknown
Page : 159 pages
File Size : 52,8 Mb
Release : 2019
Category : Algebraic topology
ISBN : 1470455129

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Stable Stems by Daniel C. Isaksen Pdf

We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, eta, and nu, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. We also compute the motivic stable homotopy groups of the cofiber of the motivic element tau. This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of tau are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.

Steenrod Squares in Spectral Sequences

William M. Singer
Steenrod Squares in Spectral Sequences Book PDF

Author : William M. Singer
Publisher : American Mathematical Soc.
Page : 170 pages
File Size : 47,6 Mb
Release : 2006
Category : Spectral sequences (Mathematics).
ISBN : 9780821841419

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Steenrod Squares in Spectral Sequences by William M. Singer Pdf

This book develops a general theory of Steenrod operations in spectral sequences. It gives special attention to the change-of-rings spectral sequence for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces and homotopy orbit spaces. In treating the change-of-rings spectral sequence, the book develops from scratch the necessary properties of extensions of Hopf algebras and constructs the spectral sequence in a form particularly suited to the introduction of Steenrod squares. The resulting theory can be used effectively for the computation of the cohomology rings of groups and Hopf algebras, and of the Steenrod algebra in particular, and so should play a useful role in stable homotopy theory. Similarly the book offers a self-contained construction of the Eilenberg-Moore spectral sequence, in a form suitable for the introduction of Steenrod operations. The corresponding theory is an effective tool for the computation of t

Handbook of Homotopy Theory

Haynes Miller
Handbook of Homotopy Theory Book PDF

Author : Haynes Miller
Publisher : CRC Press
Page : 1043 pages
File Size : 41,8 Mb
Release : 2020-01-23
Category : Mathematics
ISBN : 9781351251600

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Handbook of Homotopy Theory by Haynes Miller Pdf

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Homotopy Theory via Algebraic Geometry and Group Representations

Mark E. Mahowald
Homotopy Theory via Algebraic Geometry and Group Representations Book PDF

Author : Mark E. Mahowald
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 41,5 Mb
Release : 1998
Category : Geometry, Algebraic
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.

Stable Homotopy Theory

J.F. Adams
Stable Homotopy Theory Book PDF

Author : J.F. Adams
Publisher : Springer
Page : 80 pages
File Size : 48,5 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662159422

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Stable Homotopy Theory by J.F. Adams Pdf

Before I get down to the business of exposition, I'd like to offer a little motivation. I want to show that there are one or two places in homotopy theory where we strongly suspect that there is something systematic going on, but where we are not yet sure what the system is. The first question concerns the stable J-homomorphism. I recall that this is a homomorphism J: ~ (SQ) ~ ~S = ~ + (Sn), n large. r r r n It is of interest to the differential topologists. Since Bott, we know that ~ (SO) is periodic with period 8: r 6 8 r = 1 2 3 4 5 7 9· . · Z o o o z On the other hand, ~S is not known, but we can nevertheless r ask about the behavior of J. The differential topologists prove: 2 Th~~: If I' = ~ - 1, so that 'IT"r(SO) ~ 2, then J('IT"r(SO)) = 2m where m is a multiple of the denominator of ~/4k th (l\. being in the Pc Bepnoulli numher.) Conject~~: The above result is best possible, i.e. J('IT"r(SO)) = 2m where m 1s exactly this denominator. status of conJectuI'e ~ No proof in sight. Q9njecture Eo If I' = 8k or 8k + 1, so that 'IT"r(SO) = Z2' then J('IT"r(SO)) = 2 , 2 status of conjecture: Probably provable, but this is work in progl'ess.