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E-grāmata: Mod Two Homology and Cohomology

  • Formāts: PDF+DRM
  • Sērija : Universitext
  • Izdošanas datums: 08-Jan-2015
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319093543
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Mod Two Homology and CohomologyPalielināt
  • Formāts: PDF+DRM
  • Sērija : Universitext
  • Izdošanas datums: 08-Jan-2015
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319093543
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Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has many pedagogical advantages:

1. It leads more quickly to the essentials of the subject,
2. An absence of signs and orientation considerations simplifies the theory,
3. Computations and advanced applications can be presented at an earlier stage,
4. Simple geometrical interpretations of (co)chains.

Mod 2 (co)homology was developed in the first quarter of the twentieth century as an alternative to integral homology, before both became particular cases of (co)homology with arbitrary coefficients.

The first chapters of this book may serve as a basis for a graduate-level introductory course to (co)homology. Simplicial and singular mod 2 (co)homology are introduced, with their products and Steenrod squares, as well as equivariant cohomology. Classical applications include Brouwer's fixed point theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith theory, Kervaire invariant, etc. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the relationship between Stiefel-Whitney classes and Schubert calculus. More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst others. Each chapter ends with exercises, with some hints and answers at the end of the book.

Recenzijas

The resulting book offers a good introduction to the way the subject works, including coverage of classical topics such as duality as well as the use of Steenrod operations in stable and unstable homotopy theory. The coverage of the subject is wide and some topics were probably not available previously in an introductory text. Overall this is a welcome addition to the literature and provides a good introduction to many aspects of the subject . (Andrew J. Baker, Mathematical Reviews, January, 2016)

1 Introduction
1(4)
2 Simplicial (Co)homology
5(54)
2.1 Simplicial Complexes
5(6)
2.2 Definitions of Simplicial (Co)homology
11(3)
2.3 Kronecker Pairs
14(6)
2.4 First Computations
20(10)
2.4.1 Reduction to Components
20(1)
2.4.2 0-Dimensional (Co)homology
20(1)
2.4.3 Pseudomanifolds
21(2)
2.4.4 Poincare Series and Polynomials
23(1)
2.4.5 (Co)homology of a Cone
23(2)
2.4.6 The Euler Characteristic
25(1)
2.4.7 Surfaces
26(4)
2.5 The Homomorphism Induced by a Simplicial Map
30(5)
2.6 Exact Sequences
35(6)
2.7 Relative (Co)homology
41(6)
2.8 Mayer-Vietoris Sequences
47(2)
2.9 Appendix A: An Acyclic Carrier Result
49(1)
2.10 Appendix B: Ordered Simplicial (Co)homology
50(5)
2.11 Exercises for
Chapter 2
55(4)
3 Singular and Cellular (Co)homologies
59(68)
3.1 Singular (Co)homology
59(29)
3.1.1 Definitions
59(7)
3.1.2 Relative Singular (Co)homology
66(7)
3.1.3 The Homotopy Property
73(2)
3.1.4 Excision
75(3)
3.1.5 Well Cofibrant Pairs
78(9)
3.1.6 Mayer-Vietoris Sequences
87(1)
3.2 Spheres, Disks, Degree
88(6)
3.3 Classical Applications of the mod 2 (Co)homology
94(2)
3.4 CW-Complexes
96(6)
3.5 Cellular (Co)homology
102(6)
3.6 Isomorphisms Between Simplicial and Singular (Co)homology
108(4)
3.7 CW-Approximations
112(6)
3.8 Eilenberg-MacLane Spaces
118(5)
3.9 Generalized Cohomology Theories
123(1)
3.10 Exercises for
Chapter 3
124(3)
4 Products
127(74)
4.1 The Cup Product
127(7)
4.1.1 The Cup Product in Simplicial Cohomology
127(4)
4.1.2 The Cup Product in Singular Cohomology
131(3)
4.2 Examples
134(5)
4.2.1 Disjoint Unions
134(1)
4.2.2 Bouquets
134(1)
4.2.3 Connected Sum(s) of Closed Topological Manifolds
135(2)
4.2.4 Cohomology Algebras of Surfaces
137(2)
4.3 Two-Fold Coverings
139(8)
4.3.1 H1, Fundamental Group and 2-Fold Coverings
139(3)
4.3.2 The Characteristic Class
142(2)
4.3.3 The Transfer Exact Sequence of a 2-Fold Covering
144(2)
4.3.4 The Cohomology Ring of RPn
146(1)
4.4 Nilpotency, Lusternik-Schnirelmann Categories and Topological Complexity
147(3)
4.5 The Cap Product
150(6)
4.6 The Cross Product and the Kunneth Theorem
156(9)
4.7 Some Applications of the Kunneth Theorem
165(33)
4.7.1 Poincare Series and Euler Characteristic of a Product
165(1)
4.7.2 Slices
165(1)
4.7.3 The Cohomology Ring of a Product of Spheres
166(1)
4.7.4 Smash Products and Joins
167(5)
4.7.5 The Theorem of Leray-Hirsch
172(8)
4.7.6 The Thorn Isomorphism
180(9)
4.7.7 Bundles Over Spheres
189(5)
4.7.8 The Face Space of a Simplicial Complex
194(2)
4.7.9 Continuous Multiplications on K(Z2,m)
196(2)
4.8 Exercises for
Chapter 4
198(3)
5 Poincare Duality
201(38)
5.1 Algebraic Topology and Manifolds
201(1)
5.2 Poincare Duality in Polyhedral Homology Manifolds
202(9)
5.3 Other Forms of Poincare Duality
211(10)
5.3.1 Relative Manifolds
211(4)
5.3.2 Manifolds with Boundary
215(2)
5.3.3 The Intersection Form
217(2)
5.3.4 Non Degeneracy of the Cup Product
219(1)
5.3.5 Alexander Duality
220(1)
5.4 Poincare Duality and Submanifolds
221(15)
5.4.1 The Poincare Dual of a Submanifold
221(4)
5.4.2 The Gysin Homomorphism
225(2)
5.4.3 Intersections of Submanifolds
227(4)
5.4.4 The Linking Number
231(5)
5.5 Exercises for
Chapter 5
236(3)
6 Projective Spaces
239(20)
6.1 The Cohomology Ring of Projective Spaces---Hopf Bundles
239(6)
6.2 Applications
245(4)
6.2.1 The Borsuk-Ulam Theorem
245(1)
6.2.2 Non-singular and Axial Maps
246(3)
6.3 The Hopf Invariant
249(8)
6.3.1 Definition
249(1)
6.3.2 The Hopf Invariant and Continuous Multiplications
250(2)
6.3.3 Dimension Restrictions
252(1)
6.3.4 Hopf Invariant and Linking Numbers
253(4)
6.4 Exercises for
Chapter 6
257(2)
7 Equivariant Cohomology
259(66)
7.1 Spaces with Involution
259(15)
7.2 The General Case
274(9)
7.3 Localization Theorems and Smith Theory
283(5)
7.4 Equivariant Cross Products and Kunneth Theorems
288(10)
7.5 Equivariant Bundles and Euler Classes
298(11)
7.6 Equivariant Morse-Bott Theory
309(12)
7.7 Exercises for
Chapter 7
321(4)
8 Steenrod Squares
325(30)
8.1 Cohomology Operations
326(4)
8.2 Properties of Steenrod Squares
330(3)
8.3 Construction of Steenrod Squares
333(6)
8.4 Adem Relations
339(7)
8.5 The Steenrod Algebra
346(6)
8.6 Applications
352(2)
8.7 Exercises for
Chapter 8
354(1)
9 Stiefel-Whitney Classes
355(78)
9.1 Trivializations and Structures on Vector Bundles
355(8)
9.2 The Class w1---Orientability
363(4)
9.3 The Class w2---Spin Structures
367(5)
9.4 Definition and Properties of Stiefel-Whitney Classes
372(3)
9.5 Real Flag Manifolds
375(25)
9.5.1 Definitions and Morse Theory
376(5)
9.5.2 Cohomology Rings
381(8)
9.5.3 Schubert Cells and Stiefel-Whitney Classes
389(11)
9.6 Splitting Principles
400(4)
9.7 Complex Flag Manifolds
404(7)
9.8 The Wu Formula
411(14)
9.8.1 Wu's Classes and Formula
411(4)
9.8.2 Orientability and Spin Structures
415(3)
9.8.3 Applications to 3-Manifolds
418(2)
9.8.4 The Universal Class for Double Points
420(5)
9.9 Thorn's Theorems
425(7)
9.9.1 Representing Homology Classes by Manifolds
425(4)
9.9.2 Cobordism and Stiefel-Whitney Numbers
429(3)
9.10 Exercises for
Chapter 9
432(1)
10 Miscellaneous Applications and Developments
433(68)
10.1 Actions with Scattered or Discrete Fixed Point Sets
433(3)
10.2 Conjugation Spaces
436(8)
10.3 Chain and Polygon Spaces
444(26)
10.3.1 Definitions and Basic Properties
444(5)
10.3.2 Equivariant Cohomology
449(9)
10.3.3 Non-equivariant Cohomology
458(6)
10.3.4 The Inverse Problem
464(4)
10.3.5 Spatial Polygon Spaces and Conjugation Spaces
468(2)
10.4 Equivariant Characteristic Classes
470(5)
10.5 The Equivariant Cohomology of Certain Homogeneous Spaces
475(10)
10.6 The Kervaire Invariant
485(13)
10.7 Exercises for
Chapter 10
498(3)
11 Hints and Answers for Some Exercises
501(16)
11.1 Exercises for
Chapter 2
501(1)
11.2 Exercises for
Chapter 3
502(1)
11.3 Exercises for
Chapter 4
503(3)
11.4 Exercises for
Chapter 5
506(1)
11.5 Exercises for
Chapter 6
507(1)
11.6 Exercises for
Chapter 7
508(4)
11.7 Exercises for
Chapter 8
512(1)
11.8 Exercises for
Chapter 9
513(1)
11.9 Exercises for
Chapter 10
514(3)
References 517(10)
Index 527
Jean-Claude Hausmann is honorary professor of Mathematics at the University of Geneva. His domains of interest include topology of high-dimensional manifolds, with applications to configuration spaces inspired by topological robotics and Hamiltonian geometry.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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