A Concise Course in Algebraic Topology,9780226511832

A Concise Course in Algebraic Topology

by May, J. Peter

ISBN13: 9780226511832

ISBN10: 0226511839

Format: Paperback

Pub. Date: 1999-09-01

Publisher(s): Univ of Chicago Pr

Other versions by this Author

List Price: ~~$33.60~~

Rent Textbook

Select for Price

Add to Cart

There was a problem. Please try again later.

New Textbook

We're Sorry
Sold Out

Used Textbook

We're Sorry
Sold Out

eTextbook

We're Sorry
Not Available

Buy from our Marketplace starting at $30.04

Summary

Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.

Author Biography

J. P. May is professor of mathematics at the University of Chicago. He is author or coauthor of many books, including Simplicial Objects in Algebraic Topology and Equivalent Homotopy and Cohomology Theory.

Table of Contents

Introduction

1

(4)

The fundamental group and some of its applications

5

(8)

What is algebraic topology?

5

(1)

The fundamental group

6

(1)

Dependence on the basepoint

7

(1)

Homotopy invariance

7

(1)

Calculations: π1(R) = 0 and π (S1) = Z

8

(2)

The Brouwer fixed point theorem

10

(1)

The fundamental theorem of algebra

10

(3)

Categorical language and the van Kampen theorem

13

(8)

Categories

13

(1)

Functors

13

(1)

Natural transformations

14

(1)

Homotopy categories and homotopy equivalences

14

(1)

The fundamental groupoid

15

(1)

Limits and colimits

16

(1)

The van Kampen theorem

17

(2)

Examples of the van Kampen theorem

19

(2)

Covering spaces

21

(12)

The definition of covering spaces

21

(1)

The unique path lifting property

22

(1)

Coverings of groupoids

22

(1)

Group actions and orbit categories

23

(2)

The classification of coverings of groupoids

25

(2)

The construction of coverings of groupoids

27

(1)

The classification of coverings of spaces

28

(1)

The construction of coverings of spaces

29

(4)

Graphs

33

(4)

The definition of graphs

33

(1)

Edge paths and trees

33

(1)

The homotopy types of graphs

34

(1)

Covers of graphs and Euler characteristics

35

(1)

Applications to groups

35

(2)

Compactly generated spaces

37

(4)

The definition of compactly generated spaces

37

(1)

The category of compactly generated spaces

38

(3)

Cofibrations

41

(6)

The definition of cofibrations

41

(1)

Mapping cylinders and cofbrations

42

(1)

Replacing maps by cofibrations

43

(1)

A criterion for a map to be a cofibration

43

(1)

Cofiber homotopy equivalence

44

(3)

Fibrations

47

(8)

The definition of fibrations

47

(1)

Path lifting functions and fibrations

47

(1)

Replacing maps by fibrations

48

(1)

A criterion for a map to be a fibration

49

(1)

Fiber homotopy equivalence

50

(1)

Change of fiber

51

(4)

Based cofiber and fiber sequences

55

(8)

Based homotopy classes of maps

55

(1)

Cones, suspensions, paths, loops

55

(1)

Based cofibrations

56

(1)

Cofiber sequences

57

(2)

Based fibrations

59

(1)

Fiber sequences

59

(2)

Connections between cofiber and fiber sequences

61

(2)

Higher homotopy groups

63

(8)

The definition of homotopy groups

63

(1)

Long exact sequences associated to pairs

63

(1)

Long exact sequences associated to fibrations

64

(1)

A few calculations

64

(2)

Change of basepoint

66

(1)

n-Equivalences, weak equivalences, and a technical lemma

67

(4)

CW complexes

71

(10)

The definition and some examples of CW complexes

71

(1)

Some constructions on CW complexes

72

(1)

Help and the Whitehead theorem

73

(1)

The cellular approximation theorem

74

(1)

Approximation of spaces by CW complexes

75

(1)

Approximation of pairs by CW pairs

76

(1)

Approximation of excisive triads by CW triads

77

(4)

The homotopy excision and suspension theorems

81

(8)

Statement of the homotopy excision theorem

81

(2)

The Freudenthal suspension theorem

83

(1)

Proof of the homotopy excision theorem

84

(5)

A little homological algebra

89

(4)

Chain complexes

89

(1)

Maps and homotopies of maps of chain complexes

89

(1)

Tensor products of chain complexes

90

(1)

Short and long exact sequences

91

(2)

Axiomatic and cellular homology theory

93

(12)

Axioms for homology

93

(1)

Cellular homology

94

(4)

Verification of the axioms

98

(1)

The cellular chains of products

99

(2)

Some examples: T, K, and RPn

101

(4)

Derivations of properties from the axioms

105

(10)

Reduced homology; based versus unbased spaces

105

(1)

Cofibrations and the homology of pairs

106

(1)

Suspension and the long exact sequence of pairs

107

(1)

Axioms for reduced homology

108

(2)

Mayer-Vietoris sequences

110

(2)

The homology of colimits

112

(3)

The Hurewicz and uniqueness theorems

115

(6)

The Hurewicz theorem

115

(2)

The uniqueness of the homology of CW complexes

117

(4)

Singular homology theory

121

(8)

The singular chain complex

121

(1)

Geometric realization

122

(1)

Proofs of the theorems

123

(1)

Simplicial objects in algebraic topology

124

(2)

Classifying spaces and K(π,n)s

126

(3)

Some more homological algebra

129

(6)

Universal coefficients in homology

129

(1)

The Kunneth theorem

130

(1)

Hom functors and universal coefficients in cohomology

131

(2)

Proof of the universal coefficient theorem

133

(1)

Relations between ⨷ and Hom

133

(2)

Axiomatic and cellular cohomology theory

135

(8)

Axioms for cohomology

135

(1)

Cellular and singular cohomology

136

(1)

Cup products in cohomology

137

(1)

An example: RP2 and the Borsuk-Ulam theorem

138

(2)

Obstruction theory

140

(3)

Derivations of properties from the axioms

143

(6)

Reduced cohomology groups and their properties

143

(1)

Axioms for reduced cohomology

144

(1)

Mayer-Vietoris sequences in cohomology

145

(1)

Lim1 and the cohomology of colimits

146

(1)

The uniqueness of the cohomology of CW complexes

147

(2)

The Poincare duality theorem

149

(14)

Statement of the theorem

149

(2)

The definition of the cap product

151

(2)

Orientations and fundamental classes

153

(2)

The proof of the vanishing theorem

155

(3)

The proof of the Poincare duality theorem

158

(3)

The orientation cover

161

(2)

The index of manifolds; manifolds with boundary

163

(8)

The Euler characteristic of compact manifolds

163

(1)

The index of compact oriented manifolds

164

(2)

Manifolds with boundary

166

(1)

Poincare duality for manifolds with boundary

167

(2)

The index of manifolds that are boundaries

169

(2)

Homology, cohomology, and K(π,n)

171

(12)

K(π,n)s and homology

171

(2)

K(π,n)s and cohomology

173

(2)

Cup and cap products

175

(3)

Postnikov systems

178

(2)

Cohomology operations

180

(3)

Characteristic classes of vector bundles

183

(16)

The classification of vector bundles

183

(2)

Characteristic classes for vector bundles

185

(2)

Stiefel-Whitney classes of manifolds

187

(2)

Characteristic numbers of manifolds

189

(1)

Thom spaces and the Thom isomorphism theorem

190

(2)

The construction of the Stiefel-Whitney classes

192

(1)

Clern, Pontryagin, and Euler classes

193

(3)

A glimpse at the general theory

196

(3)

An introduction to K-theory

199

(16)

The definition of K-theory

199

(3)

The Bott periodicity theorem

202

(2)

The splitting principle and the Thom isomorphism

204

(3)

The Chern character; almost complex structures on spheres

207

(2)

The Adams operations

209

(2)

The Hopf invariant one problem and its applications

211

(4)

An introduction to cobordism

215

(16)

The cobordism groups of smooth closed manifolds

215

(1)

Sketch proof that N* is isomorphic to π*(TO)

216

(3)

Prespectra and the algebra H*(TO;Z2)

219

(3)

The Steenrod algebra and its coaction on H*(TO)

222

(2)

The relationship to Stiefel-Whitney numbers

224

(2)

Spectra and the computation of π*(TO) = π*(MO)

226

(2)

An introduction to the stable category

228

(3)

Suggestions for further reading

231

(8)

1. A classic book and historical references

231

(1)

2. Textbooks in algebraic topology and homotopy theory

231

(1)

3. Books on CW complexes

232

(1)

4. Differential forms and Morse theory

232

(1)

5. Equivariant algebraic topology

233

(1)

6. Category theory and homological algebra

233

(1)

7. Simplicial sets in algebraic topology

233

(1)

8. The Serre spectral sequence and Serre class theory

233

(1)

9. The Eilenberg-Moore spectral sequence

233

(1)

10. Cohomology operations

234

(1)

11. Vector bundles

234

(1)

12. Characteristic classes

234

(1)

13. K-theory

235

(1)

14. Hopf algebras; the Steenrod algebra, Adams spectral sequence

235

(1)

15. Cobordism

236

(1)

16. Generalized homology theory and stable homotopy theory

236

(1)

17. Quillen model categories

236

(1)

18. Localization and completion; rational homotopy theory

237

(1)

19. Infinite loop space theory

237

(1)

20. Complex cobordism and stable homotopy theory

238

(1)

21. Follow-ups to this book

238

(1)

Index

239