Topological library Part 3, Spectral sequences in topology / editors, S.P. Novikov, I.A. Taimanov ; translated by V.P. Golubyatnikov.

Format:

Book

Contributor:

Novikov, S. P. (Sergeĭ Petrovich)

Taĭmanov, I. A. (Iskander Asanovich), 1961-

Series:

K & E series on knots and everything ; v. 39.

K & E series on knots and everything ; v. 39

Language:

English

Russian

Subjects (All):

Cobordism theory.

Characteristic classes.

Differential topology.

Physical Description:

1 online resource (592 p.)

Place of Publication:

Hackensack, N.J. : World Scientific, 2012.

Summary:

The final volume of the three-volume edition, this book features classical papers on algebraic and differential topology published in the 1950s-1960s. The partition of these papers among the volumes is rather conditional. The original methods and constructions from these works are properly documented for the first time in this book. No existing book covers the beautiful ensemble of methods created in topology starting from approximately 1950. That is, from Serre's celebrated "singular homologies of fiber spaces.". Sample Chapter(s). Chapter 1: Singular homology of fiber spaces - Introduction (

Contents:

TOPOLOGICAL LIBRARY; Contents; Foreword; 1. J.-P. Serre. Singular homology of fiber spaces (Translated by V. P. Golubyatnikov); Introduction; Chapter I. The notion of spectral sequence; 1. Spectral sequence of a differential group with increasing filtration; 2. The case of graded group; 3. Transgression and suspension; 4. Exact sequence; 5. The cohomology spectral sequence; 6. Spectral sequence of universal covering; Chapter II. Singular homology and cohomology of fiber spaces; 1. Singular cubic homology; 2. Fiber spaces. Definitions and simple properties

3. Local family composed by homology of fiber4. Filtration of singular complex of the space E; 5. Calculation of the term E1; 6. Calculation of the term E2; 7. Properties of the homology spectral sequences; 8. Cohomology spectral sequence; 9. Properties of the cohomology spectral sequence; 10. Transformation of second terms of homology and cohomology spectral sequences; 11. Proof of Lemma 4; 12. Proof of Lemma 5; 13. Proof of Lemma 3; Chapter III. Applications of spectral sequences of fiber spaces; 1. First application; 2. Euler-Poincare characteristic of fiber spaces

3. Fibrations of Euclidean spaces4. Exact sequence; 5. Gysin exact sequence; 6. Wang exact sequence; 7. Leray-Hirsch theorem; Chapter IV. Loop spaces; 1. Loop spaces; 2. Hopf theorem; 3. Simplicity of H-spaces; 4. The loop spaces fibrations; 5. Fibration of a path space with fixed origin; 6. Some general results on homology of loop spaces; 7. Applications to variations calculus (Morse theory); 8. Applications to variations calculus: geodesics transversal to two sub-manifolds; 9. The homology and cohomology of loop space on a sphere; Chapter V. Homotopy groups; 1. General method

2. First results3. Finiteness of homotopy groups of odd-dimensional spheres; 4. Auxiliary calculations; 5. The first nontrivial modulo p homotopy group of an odd-dimensional sphere; 6. Stiefel manifolds and even-dimensional spheres; Chapter VI. Groups of Eilenberg-MacLane; 1. Introduction; 2. General results; 3. The Hopf theorem; Appendix. On homology of some coverings; References; 2. J.-P. Serre. Homotopy groups and classes of abelian groups (Translated by V. P. Golubyatnikov); Introduction; Chapter I. The notion of a class; Notations; 1. Definition of classes; 2. e-notions

3. Torsion product4. Two axioms on classes; 5. A new axiom; 6. Examples of classes satisfying axioms (IIA) and (III); 7. Examples of classes satisfying the axioms (lIB) and (III); Chapter II. Fiber spaces; 1. Relative fiber spaces; 2. The homology spectral sequence of a relative fiber space; 3. The cohomology spectral sequence of a relative fibration; 4. The main theorems; 5. Applications; 6. The loop spaces and the Eilenberg-MacLane groups; Chapter III. The theorems of Hurewicz and J. H. C. Whitehead; 1. Hurewicz theorem; 2. Hurewicz theorem: the second proof; 3. Relative Hurewicz theorem

4. Theorem of J. H. C. Whitehead

Notes:

"Translattion of the Russian edition published in 2005".

Includes bibliographical references and index.

ISBN:

981-4401-31-5