有理ホモトピー理論と微分形式(第2版)<br>Rational Homotopy Theory and Differential Forms (Progress in Mathematics) (2ND)

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有理ホモトピー理論と微分形式(第2版)
Rational Homotopy Theory and Differential Forms (Progress in Mathematics) (2ND)

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  • 製本 Hardcover:ハードカバー版/ページ数 224 p.
  • 言語 ENG
  • 商品コード 9781461484677
  • DDC分類 514

Full Description


This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham's theorem on simplicial complexes. In addition, Sullivan's results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma*Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theoryWith its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.

Table of Contents

1 Introduction                                     1  (4)
2 Basic Concepts 5 (16)
2.1 CW Complexes 5 (3)
2.2 First Notions from Homotopy Theory 8 (5)
2.3 Homology 13 (6)
2.4 Categories and Functors 19 (2)
3 CW Homology Theorem 21 (6)
3.1 The Statement 21 (1)
3.2 The Proof 22 (2)
3.3 Examples 24 (3)
4 The Whitehead Theorem and the Hurewicz Theorem 27 (14)
4.1 Definitions and Elementary Properties of 27 (2)
Homotopy Groups
4.2 The Whitehead Theorem 29 (2)
4.3 Completion of the Computation of πn(Sn) 31 (2)
4.4 The Hurewicz Theorem 33 (1)
4.5 Corollaries of the Hurewicz Theorem 34 (4)
4.6 Homotopy Theory of a Fibration 38 (1)
4.7 Applications of the Exact Homotopy 39 (2)
Sequence
5 Spectral Sequence of a Fibration 41 (12)
5.1 Introduction 41 (1)
5.2 Fibrations- over a Cell 42 (1)
5.3 Generalities on Spectral Sequences 43 (2)
5.4 The Leray-Serre Spectral Sequence of a 45 (3)
Fibration
5.5 Examples 48 (5)
6 Obstruction Theory 53 (10)
6.1 Introduction 53 (1)
6.2 Definition and Properties of the 54 (3)
Obstruction Cocycle
6.3 Further Properties 57 (1)
6.4 Obstruction to the Existence of a Section 58 (1)
of a Fibration
6.5 Examples 58 (5)
7 Eilenberg-MacLane Spaces, Cohomology, and 63 (6)
Principal Fibrations
7.1 Relation of Cohomology and 63 (1)
Eilenberg-MacLane Spaces
7.2 Principal K(π, n)-Fibrations 64 (5)
8 Postnikov Towers and Rational Homotopy Theory 69 (14)
8.1 Rational Homotopy Theory for Simply 73 (6)
Connected Spaces
8.2 Construction of the Localization of a 79 (4)
Space
9 deRham's Theorem for Simplicial Complexes 83 (12)
9.1 Piecewise Linear Forms 83 (2)
9.2 Lemmas About Piecewise Linear Forms 85 (3)
9.3 Naturality Under Subdivision 88 (1)
9.4 Multiplicativity of the deRham Isomorphism 89 (1)
9.5 Connection with the Cinfinity deRham 90 (2)
Theorem
9.6 Generalizations of the Construction 92 (3)
10 Differential Graded Algebras 95 (8)
10.1 Introduction 95 (2)
10.2 Hirsch Extensions 97 (2)
10.3 Relative Cohomology 99 (1)
10.4 Construction of the Minimal Model 100(3)
11 Homotopy Theory of DGAs 103(10)
11.1 Homotopies 103(1)
11.2 Obstruction Theory 104(3)
11.3 Applications of Obstruction Theory 107(2)
11.4 Uniqueness of the Minimal Model 109(4)
12 DGAs and Rational Homotopy Theory 113(6)
12.1 Transgression in the Serre Spectral 113(1)
Sequence and the Duality
12.2 Hirsch Extensions and Principal 114(1)
Fibrations
12.3 Minimal Models and Postnikov Towers 115(2)
12.4 The Minimal Model of the deRham Complex 117(2)
13 The Fundamental Group 119(8)
13.1 1-Minimal Models 119(1)
13.2 π1 Q 120(3)
13.3 Functorality 123(2)
13.4 Examples 125(2)
14 Examples and Computations 127(14)
14.1 Spheres and Projective Spaces 127(1)
14.2 Graded Lie Algebras 128(1)
14.3 The Borromean Rings 129(2)
14.4 Symmetric Spaces and Formality 131(1)
14.5 The Third Homotopy Group of a Simply 132(2)
Connected Space
14.6 Homotopy Theory of Certain 4-Dimensional 134(1)
Complexes
14.7 Q-Homotopy Type of BUn and Un 135(2)
14.8 Products 137(1)
14.9 Massey Products 138(3)
15 Functorality 141(10)
15.1 The Functorial Correspondence 141(3)
15.2 Bijectivity of Homotopy Classes of Maps 144(4)
15.3 Equivalence of Categories 148(3)
16 The Hirsch Lemma 151(14)
16.1 The Cubical Complex and Cubical Forms 151(3)
16.2 Hirsch Extensions and Spectral Sequences 154(2)
16.3 Polynomial Forms for a Serre Fibration 156(3)
16.4 Serre Spectral Sequence for Polynomial 159(4)
Forms
16.5 Proof of Theorem 12.1 163(2)
17 Quillen's Work on Rational Homotopy Theory 165(12)
17.1 Differential Graded Lie Algebras 165(1)
17.2 Differential Graded Co-algebras 166(1)
17.3 The Bar Construction 167(2)
17.4 Relationship Between Quillen's 169(1)
Construction and Sullivan's
17.5 Quillen's Construction 169(8)
18 Ainfinity-Structures and Cinfinity-Structures 177(10)
18.1 Operads, Rooted Trees, and Stasheff's 177(4)
Associahedron
18.2 Ainfinity-Algebras and 181(2)
Ainfinity-Categories
18.3 Cinfinity-Algebras and DGAs 183(4)
19 Exercises 187(36)
References 223