Differential Manifolds (eBook)

Antoni A. Kosinski (Autor)

eBook Download: PDF

1992 | 1. Auflage
248 Seiten
Elsevier Science (Verlag)
978-0-08-087458-6 (ISBN)

Infos

Differential Manifolds is a modern graduate-level introduction to the important field of differential topology. The concepts of differential topology lie at the heart of many mathematical disciplines such as differential geometry and the theory of lie groups. The book introduces both the h-cobordism theorem and the classification of differential structures on spheres. The presentation of a number of topics in a clear and simple fashion make this book an outstanding choice for a graduate course in differential topology as well as for individual study.

Key Features
* Presents the study and classification of smooth structures on manifolds
* It begins with the elements of theory and concludes with an introduction to the method of surgery
* Chapters 1-5 contain a detailed presentation of the foundations of differential topology--no knowledge of algebraic topology is required for this self-contained section
* Chapters 6-8 begin by explaining the joining of manifolds along submanifolds, and ends with the proof of the h-cobordism theory
* Chapter 9 presents the Pontriagrin construction, the principle link between differential topology and homotopy theory, The final chapter introduces the method of surgery and applies it to the classification of smooth structures on spheres

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Differential Manifolds is a modern graduate-level introduction to the important field of differential topology. The concepts of differential topology lie at the heart of many mathematical disciplines such as differential geometry and the theory of lie groups. The book introduces both the h-cobordism theorem and the classification of differential structures on spheres. The presentation of a number of topics in a clear and simple fashion make this book an outstanding choice for a graduate course in differential topology as well as for individual study. - Presents the study and classification of smooth structures on manifolds- It begins with the elements of theory and concludes with an introduction to the method of surgery- Chapters 1-5 contain a detailed presentation of the foundations of differential topology--no knowledge of algebraic topology is required for this self-contained section- Chapters 6-8 begin by explaining the joining of manifolds along submanifolds, and ends with the proof of the h-cobordism theory- Chapter 9 presents the Pontriagrin construction, the principle link between differential topology and homotopy theory; The final chapter introduces the method of surgery and applies it to the classification of smooth structures on spheres

Inhalt

Front Cover 1
Differential Manifolds 4
Copyright Page 5
Contents 8
Introduction 12
Notational Conventions 16
Chapter I. Differentiable Structures 18
1. Smooth Manifolds and Maps 18
2. Partitions of Unity 23
3. Smooth Vector Bundles 25
4. Tangent Space 29
5. Vector Fields 33
6. Differential Equations on a Smooth Manifold 35
7. Collars 38
Chapter II. Immersions, Imbeddings, Submanifolds 42
1. Local Equivalence of Maps 42
2. Submanifolds 43
3. Imbeddings in Rn 49
4. Isotopies 50
5. Ambient Isotopies 53
6. Historical Remarks 55
Chapter III. Normal Bundle, Tubular Neighborhoods 58
1. Exponential Map 58
2. Normal Bundle and Tubular Neighborhoods 61
3. Uniqueness of Tubular Neighborhoods 66
4. Submanifolds of the Boundary 69
5. Inverse Image of a Regular Value 72
6. The group Gm 73
7. Remarks 74
Chapter IV. Transversality 76
1. Transversal Maps and Manifolds 76
2. Transversality Theorem 80
3. Morse Functions 83
4. Neighborhood of a Critical Point 85
5. Intersection Numbers 87
6. Historical Remarks 90
Chapter V. Foliations 92
1. d-Fields 93
2. Foliations 95
3. Frobenius Theorem 97
4. Leaves of a Foliation 99
5. Examples 101
Chapter VI. Operations on Manifolds 106
1. Connected Sum 107
2. # and Homotopy Spheres 111
3. Boundary Connected Sum 114
4. Joining Manifolds along Submanifolds 116
5. Joining Manifolds along Submanifolds of the Boundary 117
6. Attaching Handles 120
7. Cancellation Lemma 123
8. Combinatorial Attachment 127
9. Surgery 129
10. Homology and Intersections in a Handle 130
11. (m, k)-Handlebodies, m > 2k
12. (2k, k)-Handlebodies Plumbing
Chapter VII. Handle Presentation Theorem 142
1. Elementary Cobordisms 142
2. Handle Presentation Theorem 144
3. Homology Data of a Cobordism 148
4. Morse Inequalities 152
5. Poincark Duality 153
6. 0-Dimensional Handles 154
7. Heegaard Diagrams 155
8. Historical Remarks 158
Chapter VIII. The h-Cobordism Theorem 160
1. Elementary Row Operations 161
2. Cancellation of Handles 165
3. 1-Handles 168
4. Minimal Presentation Main Theorems
5. h-cobordism The Group .m
6. Highly Connected Manifolds 176
7. Remarks 178
Chapter IX. Framed Manifolds 184
1. Framings 185
2. Framed Submanifolds 188
3. Ok(Mm) 191
4. Oo(Mm) 194
5. The Pontriagin Construction 196
6. Operations on Framed Submanifolds and Homotopy Theory 200
7. p-Manifolds 203
8. Almost Parallelizable Manifolds 206
9. Historical Remarks 209
Chapter X. Surgery 212
1. Effect of Surgery on Homology 214
2. Framing a Surgery Surgery below Middle Dimension
3. Surgery on 4n-Dimensional Manifolds 219
4. Surgery on (4n + 2)-Dimensional Manifolds 223
5. Surgery on Odd-Dimensional Manifolds 227
6. Computation of .n 232
7. Historical Note 236
Appendix 240
1. Implicit Function Theorem 240
2. A Lemma of M. Morse 243
3. Brown-Sard Theorem 243
4. Orthonormalization 244
5. Homotopy Groups of SO(k) 247
Bibliography 250
Index 258

Produkt-Details

• Erscheint lt. Verlag: 3.12.1992
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-087458-4 / 0080874584
• ISBN-13: 978-0-08-087458-6 / 9780080874586