Differential Topology 2nd Edition by Amiya Mukherjee 331919044X 9783319190440

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ISBN 10: 331919044X

ISBN 13: 9783319190440

Author: Amiya Mukherjee

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.

Differential Topology 2nd Table of contents:

CHAPTER 1 BASIC CONCEPTS OF MANIFOLDS

1.1. Two definitions of a manifold and examples

1.2. Smooth maps between manifolds

1.3. Induced smooth structures

1.4. Immersions and Submersions Convention.

1.5. Submanifolds

1.6. Further examples of manifolds

1.7. Quotient manifolds

1.8. Manifolds with boundary and corner

CHAPTER 2 APPROXIMATION THEOREMS AND WHITNEY’S EMBEDDING

2.1. Smooth partition unity

2.2. Smooth approximations to continuous maps

2.3. Sard’s theorem

2.4. Approximations by immersions

2.5. Whitney’s embedding theorem

2.6. Homotopy of smooth maps

2.7. Stability of smooth maps

CHAPTER 3 LINEAR STRUCTURES ON MANIFOLDS

3.1. Tangent spaces and derivative maps

3.2. Vector Fields and Flows

3.3. Exterior algebra

3.4. Differential forms

3.5. Derivations of algebra of differential forms

3.6. Darboux-Weinstein theorems

CHAPTER 4 RIEMANNIAN MANIFOLDS

4.1. Riemannian Metric

4.2. Geodesics on a Manifold

4.3. Riemannian connection and geodesics

4.4. Exponential maps

4.5. Hopf-Rinow theorem

4.6. Totally geodesic submanifolds

CHAPTER 5 VECTOR BUNDLES ON MANIFOLDS

5.1. Vector bundles

5.2. Construction of vector bundles

5.3. Homotopy property of vector bundles

5.4. Subbundle and quotient bundle

5.5. Orientation

5.6. Reduction of structure group of a vector bundle

5.7. Homology characterisation of orientation

5.8. Integration of differential forms on manifolds

CHAPTER 6 TRANSVERSALITY

6.1. E-neighbourhood of submanifold of Euclidean space

6.2. Transversality

6.3. Compact one-manifolds and Brouwer’s theorem

6.4. Boundary and preimage orientations

6.5. Intersection numbers, and Degrees of maps

6.6. Hopf’s degree theorem

CHAPTER 7 TUBULAR NEIGHBOURHOODS

7.1. Tubular neighbourhood theorems

7.2. Collar neighbourhoods

7.3. Isotopy extension theorem

7.4. Uniqueness of tubular neighbourhoods

7.5. Manifolds with corner and straightening them

7.6. Construction of manifolds by gluing process

CHAPTER 8 SPACES OF SMOOTH MAPS

8.1. Spaces of Jets

8.2. Weak and strong topologies

8.3. Continuity of maps between spaces of smooth maps

8.4. Spaces of immersions and embeddings

8.5. Baire property of the space of smooth maps

8.6. Smooth structures on jet spaces

8.7. Thom’s Transversality Theorem

8.8. Multi-jet transversality

8.9. More results on Whitney’s immersion and embedding

CHAPTER 9 MORSE THEORY

9.1. Morse functions

9.2. Critical levels and attaching handles

9.3. Morse inequalities

9.4. Perfect Morse functions

9.5. Triangulations of manifolds

CHAPTER 10 THEORY OF HANDLE PRESENTATIONS

10.1. Existence of handle presentation

10.2. Duality theorems

10.3. Normalisation of presentation

10.4. Cancellation of handles

10.5. Classification of closed surfaces

10.6. Removal of intersection points

10.7. Addition of handles

10.8. Simplification of handle presentations

10.9. h-cobordism theorem and generalised Poincar´e conjecture

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