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Differential Topology 2nd ed. 2015 [Hardback]

  • Formāts: Hardback, 349 pages, height x width: 235x155 mm, weight: 6682 g, 25 Illustrations, black and white; XIII, 349 p. 25 illus., 1 Hardback
  • Izdošanas datums: 26-Aug-2015
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 331919044X
  • ISBN-13: 9783319190440
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Differential Topology 2nd ed. 2015Palielināt
  • Formāts: Hardback, 349 pages, height x width: 235x155 mm, weight: 6682 g, 25 Illustrations, black and white; XIII, 349 p. 25 illus., 1 Hardback
  • Izdošanas datums: 26-Aug-2015
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 331919044X
  • ISBN-13: 9783319190440
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783319190440.html
Keywords:

This book presents a systematic and comprehensive account of the theory of differentiable manifolds and provides the necessary background for the use of fundamental differential topology tools. The text includes, in particular, the earlier works of Stephen Smale, for which he was awarded the Fields Medal. Explicitly, the topics covered are Thom transversality, Morse theory, theory of handle presentation, h-cobordism theorem and the generalised Poincaré conjecture. The material is the outcome of lectures and seminars on various aspects of differentiable manifolds and differential topology given over the years at the Indian Statistical Institute in Calcutta, and at other universities throughout India.

The book will appeal to graduate students and researchers interested in these topics. An elementary knowledge of linear algebra, general topology, multivariate calculus, analysis and algebraic topology is recommended.

Recenzijas

The book presented by the author consists of ten chapters. it may serve as the first source of information on Differential Topology for all mathematics major students. (Andrew Bucki, zbMATH 1332.57001, 2016)

Preface vii
Chapter 1 Basic Concepts Of Manifolds
1(42)
1.1 Two definitions of a manifold and examples
1(9)
1.2 Smooth maps between manifolds
10(4)
1.3 Induced smooth structures
14(2)
1.4 Immersions and Submersions
16(6)
1.5 Submanifolds
22(4)
1.6 Further examples of manifolds
26(4)
1.7 Quotient manifolds
30(6)
1.8 Manifolds with boundary and corner
36(7)
Chapter 2 Approximation Theorems And Whitney's Embedding
43(26)
2.1 Smooth partition unity
43(8)
2.2 Smooth approximations to continuous maps
51(3)
2.3 Sard's theorem
54(5)
2.4 Approximations by immersions
59(3)
2.5 Whitney's embedding theorem
62(1)
2.6 Homotopy of smooth maps
63(2)
2.7 Stability of smooth maps
65(4)
Chapter 3 Linear Structures on Manifolds
69(36)
3.1 Tangent spaces and derivative maps
69(7)
3.2 Vector Fields and Flows
76(6)
3.3 Exterior algebra
82(6)
3.4 Differential forms
88(2)
3.5 Derivations of algebra of differential forms
90(5)
3.6 Darboux-Weinstein theorems
95(10)
Chapter 4 Riemannian Manifolds
105(28)
4.1 Riemannian Metric
105(7)
4.2 Geodesics on a Manifold
112(4)
4.3 Riemannian connection and geodesics
116(6)
4.4 Exponential maps
122(4)
4.5 Hopf-Rinow theorem
126(3)
4.6 Totally geodesic submanifolds
129(4)
Chapter 5 Vector Bundles On Manifolds
133(36)
5.1 Vector bundles
133(9)
5.2 Construction of vector bundles
142(3)
5.3 Homotopy property of vector bundles
145(3)
5.4 Subbundle and quotient bundle
148(3)
5.5 Orientation
151(6)
5.6 Reduction of structure group of a vector bundle
157(2)
5.7 Homology characterisation of orientation
159(4)
5.8 Integration of differential forms on manifolds
163(6)
Chapter 6 Transversality
169(30)
6.1 ε-neighbourhood of submanifold of Euclidean space
169(4)
6.2 Transversality
173(6)
6.3 Compact one-manifolds and Brouwer's theorem
179(3)
6.4 Boundary and preimage orientations
182(3)
6.5 Intersection numbers, and Degrees of maps
185(6)
6.6 Hopf's degree theorem
191(8)
Chapter 7 Tubular Neighbourhoods
199(26)
7.1 Tubular neighbourhood theorems
199(3)
7.2 Collar neighbourhoods
202(4)
7.3 Isotopy extension theorem
206(6)
7.4 Uniqueness of tubular neighbourhoods
212(4)
7.5 Manifolds with corner and straightening them
216(3)
7.6 Construction of manifolds by gluing process
219(6)
Chapter 8 Spaces of Smooth Maps
225(42)
8.1 Spaces of Jets
225(8)
8.2 Weak and strong topologies
233(7)
8.3 Continuity of maps between spaces of smooth maps
240(4)
8.4 Spaces of immersions and embeddings
244(4)
8.5 Baire property of the space of smooth maps
248(2)
8.6 Smooth structures on jet spaces
250(4)
8.7 Thom's Transversality Theorem
254(6)
8.8 Multi-jet transversality
260(2)
8.9 More results on Whitney's immersion and embedding
262(5)
Chapter 9 Morse Theory
267(32)
9.1 Morse functions
267(5)
9.2 Critical levels and attaching handles
272(14)
9.3 Morse inequalities
286(6)
9.4 Perfect Morse functions
292(2)
9.5 Triangulations of manifolds
294(5)
Chapter 10 Theory Of Handle Presentations
299(42)
10.1 Existence of handle presentation
300(5)
10.2 Duality theorems
305(7)
10.3 Normalisation of presentation
312(2)
10.4 Cancellation of handles
314(5)
10.5 Classification of closed surfaces
319(2)
10.6 Removal of intersection points
321(9)
10.7 Addition of handles
330(4)
10.8 Simplification of handle presentations
334(3)
10.9 h-cobordism theorem and generalised Poincare conjecture
337(4)
Bibliography 341(4)
Index 345