Volume: 218; 2021; 425 pp; Softcover
MSC: Primary 58; 55; 57;
Print ISBN: 978-1-4704-6674-9
Product Code: GSM/218.S
List Price: $85.00
AMS Member Price: $68.00
MAA Member Price: $76.50
Electronic ISBN: 978-1-4704-6673-2
Product Code: GSM/218.E
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Supplemental Materials
Lectures on Differential Topology
Share this pageRiccardo Benedetti
This book gives a comprehensive introduction to the theory of smooth
manifolds, maps, and fundamental associated structures with an
emphasis on “bare hands” approaches, combining
differential-topological cut-and-paste procedures and applications of
transversality. In particular, the smooth cobordism cup-product is
defined from scratch and used as the main tool in a variety of
settings. After establishing the fundamentals, the book proceeds to a
broad range of more advanced topics in differential topology,
including degree theory, the Poincaré-Hopf index theorem,
bordism-characteristic numbers, and the Pontryagin-Thom construction.
Cobordism intersection forms are used to classify compact surfaces;
their quadratic enhancements are developed and applied to studying the
homotopy groups of spheres, the bordism group of immersed surfaces in
a 3-manifold, and congruences mod 16 for the signature of
intersection forms of 4-manifolds. Other topics include the
high-dimensional \(h\)-cobordism theorem stressing the role of the
“Whitney trick”, a determination of the singleton bordism modules in
low dimensions, and proofs of parallelizability of orientable
3-manifolds and the Lickorish-Wallace theorem. Nash manifolds and
Nash's questions on the existence of real algebraic models are also
discussed.
This book will be useful as a textbook for beginning masters and
doctoral students interested in differential topology, who have
finished a standard undergraduate mathematics curriculum. It
emphasizes an active learning approach, and exercises are included
within the text as part of the flow of ideas. Experienced readers may
use this book as a source of alternative, constructive approaches to
results commonly presented in more advanced contexts with specialized
techniques.
Readership
Graduate students interested in a graduate level introduction to differential topology.
Table of Contents
Table of Contents
Lectures on Differential Topology
- Cover Cover11
- Title page iii4
- Preface xiii14
- Introduction xv16
- Chapter 1. The smooth category of open subsets of Euclidean spaces 132
- 1.1. Basic structures on ℝⁿ 132
- 1.2. Differential calculus 435
- 1.3. An elementary division theorem 637
- 1.4. Bump functions and partitions of unity 738
- 1.5. The smooth category of open sets in Euclidean spaces 940
- 1.6. The chain rule and the tangent functor 1041
- 1.7. Tangent vector fields, Riemannian metrics, gradient fields 1344
- 1.8. Inverse function theorem and applications 1647
- 1.9. Topologies on spaces of smooth maps 2152
- 1.10. Stability of submersions and immersions at a compact set 2253
- 1.11. Morse lemma 2354
- 1.12. Homotopy, isotopy, diffeotopy 2758
- 1.13. Linearization of diffeomorphisms of ℝⁿ up to isotopy 2859
- 1.14. Homogeneity 2960
- Chapter 2. The category of embedded smooth manifolds 3162
- Chapter 3. Stiefel and Grassmann manifolds 4374
- 3.1. Stiefel manifolds 4374
- 3.2. Fibrations of Stiefel manifolds by Stiefel manifolds 4576
- 3.3. Grassmann manifolds 4879
- 3.4. Stiefel manifolds as fibre bundles over Grassmann manifolds 5081
- 3.5. A cellular decomposition of the Grassmann manifolds 5283
- 3.6. Stiefel and Grassmannian manifolds as real algebraic sets 5485
- Chapter 4. The category of smooth manifolds 5788
- 4.1. Topologies on spaces of smooth maps 6293
- 4.2. Homotopy, isotopy, diffeotopy, homogeneity 6394
- 4.3. The (abstract) tangent functor 6394
- 4.4. Principal and associated bundles with given structure group 69100
- 4.5. Tensor bundles 70101
- 4.6. Tensor fields, unitary tensor bundles 71102
- 4.7. Parallelizable, combable, and orientable manifolds 74105
- 4.8. On complex manifolds 76107
- 4.9. Manifolds with boundary, proper submanifolds 77108
- 4.10. Product, manifolds with corners, smoothing 81112
- 4.11. Embedding compact manifolds 84115
- Chapter 5. Tautological bundles and pull-back 89120
- Chapter 6. Compact embedded smooth manifolds 107138
- 6.1. Tubular neighbourhoods and collars 107138
- 6.2. The “double” of a manifold with boundary 111142
- 6.3. A fibration theorem 114145
- 6.4. Density of smooth maps among 𝒞^{𝓇}-maps 114145
- 6.5. Smooth homotopy groups: Vector bundles on spheres 115146
- 6.6. Smooth approximation of compact embedded 𝒞^{𝓇}-manifolds 116147
- 6.7. Sard-Brown theorem 118149
- 6.8. Morse functions via generic linear projections to lines 120151
- 6.9. Morse functions via distance functions 123154
- 6.10. Generic linear projections to hyperplanes 124155
- 6.11. Approximation by Nash manifolds 126157
- Chapter 7. Cut and paste compact manifolds 133164
- 7.1. Extension of isotopies to diffeotopies 133164
- 7.2. Gluing manifolds together along boundary components 136167
- 7.3. On corner smoothing 138169
- 7.4. Uniqueness of smooth disks up to diffeotopy 139170
- 7.5. Connected sum, shelling 140171
- 7.6. Attaching handles 144175
- 7.7. Strong embedding theorem, the Whitney trick 146177
- 7.8. On immersions of 𝑛-manifolds in ℝ²ⁿ⁻¹ 150181
- 7.9. Embedding 𝑚-manifolds in ℝ^{2𝕞-1} up to surgery 153184
- 7.10. Projectivized vector bundles and blowing up 156187
- Chapter 8. Transversality 163194
- Chapter 9. Morse functions and handle decompositions 177208
- Chapter 10. Bordism 191222
- 10.1. The bordism modules of a topological space 192223
- 10.2. Bordism covariant functors 194225
- 10.3. Relative bordism of topological pairs 195226
- 10.4. On Eilenberg-Steenrood axioms 196227
- 10.5. Bordism nontriviality 199230
- 10.6. Relation between bordism and homotopy group functors 201232
- 10.7. Bordism categories 203234
- 10.8. A glance at TQFT 204235
- Chapter 11. Smooth cobordism 207238
- Chapter 12. Applications of cobordism rings 219250
- Chapter 13. Line bundles, hypersurfaces, and cobordism 231262
- Chapter 14. Euler-Poincaré characteristic 241272
- 14.1. E-P characteristic via Morse functions 242273
- 14.2. The index of an isolated zero of a tangent vector field 242273
- 14.3. Index theorem 244275
- 14.4. E-P characteristic for nonoriented manifolds 244275
- 14.5. Examples and properties of 𝜒 246277
- 14.6. The relative E-P characteristic of a triad, 𝜒-additivity 247278
- 14.7. E-P characteristic of tubular neighbourhoods and the Gauss map 248279
- 14.8. Nontriviality of 𝜂_{∙} and Ω_{∙} 250281
- 14.9. Combinatorial E-P characteristic 251282
- Chapter 15. Surfaces 255286
- Chapter 16. Bordism characteristic numbers 277308
- Chapter 17. The Pontryagin-Thom construction 287318
- Chapter 18. High-dimensional manifolds 307338
- Chapter 19. On 3-manifolds 317348
- 19.1. Heegaard splitting 317348
- 19.2. Surgery equivalence 323354
- 19.3. Proofs of Ω₃=0 326357
- 19.4. Proofs of Lickorish-Wallace theorem 326357
- 19.5. On 𝜂₃=0 331362
- 19.6. Combing and framing 332363
- 19.7. The bordism group of immersed surfaces in a 3-manifold 348379
- 19.8. Tear and smooth-rational equivalences 367398
- Chapter 20. On 4-manifolds 381412
- 20.1. Symmetric unimodular ℤ-bilinear forms 382413
- 20.2. Some 4-manifold counterparts 387418
- 20.3. Ω₄ 391422
- 20.4. A classification up to odd stabilization 395426
- 20.5. On the classification up to even stabilization 396427
- 20.6. Congruences modulo 16 398429
- 20.7. On the topological classification of smooth 4-manifolds 409440
- Appendix: Baby categories 413444
- Bibliography 417448
- Index 423454
- Back Cover Back Cover1459