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E-raamat: Homotopical Topology

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  • Formaat: PDF+DRM
  • Sari: Graduate Texts in Mathematics 273
  • Ilmumisaeg: 24-Jun-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319234885
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Homotopical TopologySuurem pilt
  • Formaat: PDF+DRM
  • Sari: Graduate Texts in Mathematics 273
  • Ilmumisaeg: 24-Jun-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319234885
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This textbook on algebraic topology updates a popular textbook from the golden era of the Moscow school of I. M. Gelfand. The first English translation, done many decades ago, remains very much in demand, although it has been long out-of-print and is difficult to obtain. Therefore, this updated English edition will be much welcomed by the mathematical community. Distinctive features of this book include: a concise but fully rigorous presentation, supplemented by a plethora of illustrations of a high technical and artistic caliber; a huge number of nontrivial examples and computations done in detail; a deeper and broader treatment of topics in comparison to most beginning books on algebraic topology; an extensive, and very concrete, treatment of the machinery of spectral sequences. The second edition contains an entirely new chapter on K-theory and the Riemann-Roch theorem (after Hirzebruch and Grothendieck).

Introduction.- Homotopy.- Homology.- Spectral Sequences of Fibrations.- Cohomology Operations.- The Adams Spectral Sequence.- K-Theory and Other Extraordinary Cohomology Theories.

Arvustused

This book is a treasure trove for every mathematician who has to deal with classical algebraic topology and homotopy theory on the research level. Its style is refreshing and informative, and the reader can feel the authors joy at sharing their insight into algebraic topology. will be a useful addition to any mathematical bookshelf. (Thomas Hüttemann, Mathematical Reviews, March, 2017)

This book covers all the basic material necessary for complete understanding of the fundamentals of algebraic topology . This increase in the number of topics has made the book more convenient for serious students not only to extend their knowledge but also to gain insight into the interplay between these three subjects. This book is designed to help students to select the level of learning subjects they want to reach . (Haruo Minami, zbMATH 1346.55001, 2016)

Introduction: The Most Important Topological Spaces 1(24)
Lecture 1 Classical Spaces
1(15)
Lecture 2 Basic Operations over Topological Spaces
16(9)
Chapter 1 Homotopy
25(118)
Lecture 3 Homotopy and Homotopy Equivalence
25(8)
Lecture 4 Natural Group Structures in the Sets π (X, Y)
33(5)
Lecture 5 CW Complexes
38(22)
Lecture 6 The Fundamental Group and Coverings
60(19)
Lecture 7 Van Kampen's Theorem and Fundamental Groups of CW Complexes
79(14)
Lecture 8 Homotopy Groups
93(11)
Lecture 9 Fibrations
104(17)
Lecture 10 The Suspension Theorem and Homotopy Groups of Spheres
121(8)
Lecture 11 Homotopy Groups and CW Complexes
129(14)
Chapter 2 Homology
143(162)
Lecture 12 Main Definitions and Constructions
143(15)
Lecture 13 Homology of CW Complexes
158(20)
Lecture 14 Homology and Homotopy Groups
178(5)
Lecture 15 Homology with Coefficients and Cohomology
183(20)
Lecture 16 Multiplications
203(12)
Lecture 17 Homology and Manifolds
215(38)
Lecture 18 The Obstruction Theory
253(17)
Lecture 19 Vector Bundles and Their Characteristic Classes
270(35)
Chapter 3 Spectral Sequences of Fibrations
305(84)
Lecture 20 An Algebraic Introduction
305(16)
Lecture 21 Spectral Sequences of a Filtered Topological Space
321(3)
Lecture 22 Spectral Sequences of Fibrations: Definitions and Basic Properties
324(12)
Lecture 23 Additional Properties of Spectral Sequences of Fibrations
336(13)
Lecture 24 A Multiplicative Structure in a Cohomological Spectral Sequence
349(15)
Lecture 25 Killing Spaces Method for Computing Homotopy Groups
364(3)
Lecture 26 Rational Cohomology of K(π, n) and Ranks of Homotopy Groups
367(12)
Lecture 27 Odd Components of Homotopy Groups
379(10)
Chapter 4 Cohomology Operations
389(40)
Lecture 28 General Theory
389(6)
Lecture 29 Steenrod Squares
395(6)
Lecture 30 The Steenrod Algebra
401(13)
Lecture 31 Applications of Steenrod Squares
414(15)
Chapter 5 The Adams Spectral Sequence
429(66)
Lecture 32 General Idea
429(5)
Lecture 33 The Necessary Algebraic Material
434(10)
Lecture 34 The Construction of the Adams Spectral Sequence
444(19)
Lecture 35 Multiplicative Structures
463(9)
Lecture 36 An Application of the Adams Spectral Sequence to Stable Homotopy Groups of Spheres
472(15)
Lecture 37 Partial Cohomology Operation
487(8)
Chapter 6 Theory and Other Extraordinary Cohomology Theories
495(110)
Lecture 38 General Theory
495(21)
Lecture 39 Calculating A'-Functor: Atiyah--Hirzebruch Spectral Sequence
516(9)
Lecture 40 The Adams Operations
525(8)
Lecture 41 J-functor
533(18)
Lecture 42 The Riemann--Roch Theorem
551(26)
Lecture 43 The Atiyah--Singer Formula: A Sketch
577(7)
Lecture 44 Cobordisms
584(21)
Captions for the Illustrations 605(8)
References 613(6)
Name Index 619(4)
Subject Index 623
Anatoly Timofeevich Fomenko is Chair of Differential Geometry and Applications in the Department of Mathematics and Mechanics at Lomonosov Moscow State University. He is a full member of the Russian Academy of Sciences, and a member of the Moscow Mathematical Society. He is the author of several books, including Visual Geometry and Topology, Modeling for Visualization (with T.L. Kunii), and Modern Geometry: Methods and Applications (with B.A. Dubrovin and S.P. Novikov).

Dmitry Borisovich Fuchs is Professor Emeritus of Mathematics at the University of California, Davis. He earned his C.Sc. from Moscow State University, and his D.Sc. at Tblisi State University. His research interests include topology and the theory of foliations, homological algebra, and representation theory. His main body of work deals with representations and cohomology of infinite-dimensional Lie algebras. This work has consequences in string theory and conformal quantum field theory as codified in the mathematical theory of vertex operator algebras. He is the author of over 25 articles, and has served as thesis advisor to several well-known mathematicians, including Boris Feigin, Fedor Malikov, and Vladimir Rokhlin.
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