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Author: Victor P. Snaith
Publisher: Springer Science & Business Media
ISBN: 376439904X
Category : Mathematics
Languages : en
Pages : 250
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Book Description
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 .
Author: Victor P. Snaith
Publisher: Springer Science & Business Media
ISBN: 376439904X
Category : Mathematics
Languages : en
Pages : 250
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Book Description
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 .
Author: Victor P. Snaith
Publisher: Birkhäuser
ISBN: 9783764399344
Category : Mathematics
Languages : en
Pages : 239
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Book Description
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 .
Author: Michael A. Hill
Publisher: Cambridge University Press
ISBN: 1108831443
Category : Mathematics
Languages : en
Pages : 881
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Book Description
A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.
Author: Michael A. Hill
Publisher: Cambridge University Press
ISBN: 1108912907
Category : Mathematics
Languages : en
Pages : 882
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Book Description
The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper. The proof is long and involved, using many sophisticated tools of modern (equivariant) stable homotopy theory that are unfamiliar to non-experts. This book presents the proof together with a full development of all the background material to make it accessible to a graduate student with an elementary algebraic topology knowledge. There are explicit examples of constructions used in solving the problem. Also featuring a motivating history of the problem and numerous conceptual and expository improvements on the proof, this is the definitive account of the resolution of the Kervaire invariant problem.
Author: Douglas C. Ravenel
Publisher: American Mathematical Society
ISBN: 1470472937
Category : Mathematics
Languages : en
Pages : 417
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Book Description
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: Rob de Jeu
Publisher: American Mathematical Soc.
ISBN: 0821844946
Category : Mathematics
Languages : en
Pages : 354
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Book Description
Spencer J. Bloch has, and continues to have, a profound influence on the subject of Algebraic $K$-Theory, Cycles and Motives. This book, which is comprised of a number of independent research articles written by leading experts in the field, is dedicated in his honour, and gives a snapshot of the current and evolving nature of the subject. Some of the articles are written in an expository style, providing a perspective on the current state of the subject to those wishing to learn more about it. Others are more technical, representing new developments and making them especially interesting to researchers for keeping abreast of recent progress.
Author: Samuel Gitler
Publisher: American Mathematical Soc.
ISBN: 0821850105
Category : Mathematics
Languages : en
Pages : 374
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Book Description
Author: Stanley O. Kochman
Publisher: Springer
ISBN: 3540469931
Category : Mathematics
Languages : en
Pages : 338
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Book Description
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: C. T. C. Wall
Publisher: American Mathematical Society
ISBN: 1470479133
Category : Mathematics
Languages : en
Pages : 321
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Book Description
The publication of this book in 1970 marked the culmination of a particularly exciting period in the history of the topology of manifolds. The world of high-dimensional manifolds had been opened up to the classification methods of algebraic topology by Thom's work in 1952 on transversality and cobordism, the signature theorem of Hirzebruch in 1954, and by the discovery of exotic spheres by Milnor in 1956. In the 1960s, there had been an explosive growth of interest in the surgery method of understanding the homotopy types of manifolds (initially in the differentiable category), including results such as the $h$-cobordism theory of Smale (1960), the classification of exotic spheres by Kervaire and Milnor (1962), Browder's converse to the Hirzebruch signature theorem for the existence of a manifold in a simply connected homotopy type (1962), the $s$-cobordism theorem of Barden, Mazur, and Stallings (1964), Novikov's proof of the topological invariance of the rational Pontrjagin classes of differentiable manifolds (1965), the fibering theorems of Browder and Levine (1966) and Farrell (1967), Sullivan's exact sequence for the set of manifold structures within a simply connected homotopy type (1966), Casson and Sullivan's disproof of the Hauptvermutung for piecewise linear manifolds (1967), Wall's classification of homotopy tori (1969), and Kirby and Siebenmann's classification theory of topological manifolds (1970). The original edition of the book fulfilled five purposes by providing: • a coherent framework for relating the homotopy theory of manifolds to the algebraic theory of quadratic forms, unifying many of the previous results; • a surgery obstruction theory for manifolds with arbitrary fundamental group, including the exact sequence for the set of manifold structures within a homotopy type, and many computations; • the extension of surgery theory from the differentiable and piecewise linear categories to the topological category; • a survey of most of the activity in surgery up to 1970; • a setting for the subsequent development and applications of the surgery classification of manifolds. This new edition of this classic book is supplemented by notes on subsequent developments. References have been updated and numerous commentaries have been added. The volume remains the single most important book on surgery theory.
Author: Arthur James Wells
Publisher:
ISBN:
Category : Bibliography, National
Languages : en
Pages : 1922
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Book Description
