Complex Cobordism and Stable Homotopy Groups of Spheres

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.

Author: Douglas C. Ravenel

Publisher: American Mathematical Soc.

ISBN: 9780821829677

Category: Mathematics

Page: 395

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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.
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Stable Homotopy Groups of Spheres

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.

Author: Stanley O. Kochman

Publisher: Springer

ISBN: 9783540469933

Category: Mathematics

Page: 334

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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.
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Composition Methods in Homotopy Groups of Spheres AM 49 Volume 49

... ( sh the k - th stable homotopy group of the sphere , and let Eo ( sn ) - GK " n + k be the projection . By ( 3.2 ) , ( 3.3 ) E " : Inth ( s ) anth ( sh ) - GK is an isomorphism onto if n > k + 1 and a homomorphism onto if n = k + 1 ...

Author: Hiroshi Toda

Publisher: Princeton University Press

ISBN: 9781400882625

Category: Mathematics

Page: 193

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The description for this book, Composition Methods in Homotopy Groups of Spheres. (AM-49), Volume 49, will be forthcoming.
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Homotopy Exponents and Stable Homotopy Groups of Spheres

We all know that determining the homotopy groups of a topological space is one of the central problems in homotopy theory.

Author: Hao Zhao

Publisher: LAP Lambert Academic Publishing

ISBN: 3659557684

Category:

Page: 104

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We all know that determining the homotopy groups of a topological space is one of the central problems in homotopy theory. There is always a deep relation between the homotopy groups and other mathematical invariants which have wide applications in many fields such as geometric topology, algebra, algebraic K-theory and algebraic geometry. Via the view of homotopy theory, we try to approach the homotopy groups from two different sides: (i) Consider the general property of homotopy groups such as homotopy exponent; (ii) Consider the computation of stable homotopy groups of spheres.
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Algebraic Topology

One of the striking results is the famous theorem of Serre that the homotopy groups of spheres are finite groups , except in the few cases already known to Hopf ; in particular the stable homotopy groups of spheres are finite ( except ...

Author: Tammo tom Dieck

Publisher: European Mathematical Society

ISBN: 3037190485

Category: Mathematics

Page: 567

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This book is written as a textbook on algebraic topology. The first part covers the material for two introductory courses about homotopy and homology. The second part presents more advanced applications and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). The author recommends starting an introductory course with homotopy theory. For this purpose, classical results are presented with new elementary proofs. Alternatively, one could start more traditionally with singular and axiomatic homology. Additional chapters are devoted to the geometry of manifolds, cell complexes and fibre bundles. A special feature is the rich supply of nearly 500 exercises and problems. Several sections include topics which have not appeared before in textbooks as well as simplified proofs for some important results. Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (master's) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included.
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