Topology; A Geometric Approach : Terry Lawson: 9780199202485: Book: Computers & Technology: eCampus.com

This new-in-paperback introduction to topology emphasizes a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style that encourages the student to be an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students. Part I is appropriate for a one-semester or two-quarter course, and Part II (which is problem based) allows the book to be used for a year-long course which supports a variety of syllabuses.
The over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix, with solutions to all exercises available to the instructor on a companion website.

List of Figures

I A Geometric Introduction to Topology

1 Basic point set topology

(59)

1.1 Topology in Rn

(4)

1.2 Open sets and topological spaces

(8)

1.3 Geometric constructions of planar homeomorphisms

(7)

1.4 Compactness

(4)

1.5 The product topology and compactness in Rn

(4)

1.6 Connectedness

(7)

1.7 Quotient spaces

(7)

1.8 The Jordan curve theorem and the Sch�nflies theorem

(5)

1.9 Supplementary exercises

(13)

2 The classification of surfaces

(91)

2.1 Definitions and construction of the models

(6)

2.2 Handle decompositions and more basic surfaces

(9)

2.3 Isotopy and attaching handles

(11)

2.4 Orientation

(10)

2.5 Connected sums

(8)

2.6 The classification theorem

106

(13)

2.7 Euler characteristic and the identification of surfaces

119

(7)

2.8 Simplifying handle decompositions

126

(7)

2.9 Supplementary exercises

133

(20)

3 The fundamental group and its applications

153

(90)

3.1 The main idea of algebraic topology

153

(7)

3.2 The fundamental group

160

(7)

3.3 The fundamental group of the circle

167

(5)

3.4 Applications to surfaces

172

(7)

3.5 Applications of the fundamental group

179

(6)

3.6 Vector fields in the plane

185

(9)

3.7 Vector fields on surfaces

194

(12)

3.8 Homotopy equivalences and π1

206

(9)

3.9 Seifert-van Kampen theorem and its application to surfaces

215

(11)

3.10 Dependence on the base point

226

(4)

3.11 Supplementary exercises

230

(13)

II Covering Spaces, CW Complexes and Homology

4 Covering spaces

243

(17)

4.1 Basic examples and properties

243

(5)

4.2 Conjugate subgroups of π1 and equivalent covering spaces

248

(6)

4.3 Covering transformations

254

(2)

4.4 The universal covering space and quotient covering spaces

256

(4)

5 CW complexes

260

(21)

5.1 Examples of CW complexes

260

(6)

5.2 The Fundamental group of a CW complex

266

(3)

5.3 Homotopy type and CW complexes

269

(6)

5.4 The Seifert-van Kampen theorem for CW complexes

275

(1)

5.5 Simplicial complexes and Δ-complexes

276

(5)

6 Homology

281

(74)

6.1 Chain complexes and homology

281

(2)

6.2 Homology of a Δ-complex

283

(3)

6.3 Singular homology Hi(X) and the isomorphism &pia;b1(X, x) H1(X)

286

(6)

6.4 Cellular homology of a two-dimensional CW complex

292

(2)

6.5 Chain maps and homology

294

(6)

6.6 Axioms for singular homology

300

(4)

6.7 Reformulation of excision and the Mayer -Vietoris exact sequence

304

(4)

6.8 Applications of singular homology

308

(2)

6.9 The degree of a map � : Sn -> Sn

310

(3)

6.10 Cellular homology of a CW complex

313

(7)

6.11 Cellular homology, singular homology, and Euler characteristic

320

(3)

6.12 Applications of the Mayer-Vietoris sequence

323

(5)

6.13 Reduced homology

328

(1)

6.14 The Jordan curve theorem and its generalizations

329

(4)

6.15 Orientation and homology

333

(12)

6.16 Proof of homotopy invariance of homology

345

(5)

6.17 Proof of the excision property

350

(5)

Appendix Selected solutions

355

(28)

References

383

(2)

Index

385




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