(Ebook) Algebraic Topology A Primer 2nd Edition by Satya Deo ISBN 9788185931685 8185931682

1 Basic Topology: a review

1.1 Introduction

1.2 Euclidean Spaces and their Subspaces

1.3 Continuous Maps and Product Spaces

1.4 Homeomorphisms and Examples

1.5 Quotient Spaces

1.6 Connected and Path-Connected Spaces

1.7 Compact Spaces and Locally Compact Spaces

1.8 Compact Surfaces

1.9 What is Algebraic Topology?

2 The Fundamental Group

2.1 Introduction

2.2 Homotopy

2.3 Contractible Spaces and Homotopy Type

2.4 Fundamental Group and its Properties

2.5 Simply-Connected Spaces

2.6 Results for Computing Fundamental Groups

3 Simplicial Complexes

3.1 Finite Simplicial Complexes

3.2 Polyhedra and Triangulations

3.3 Simplicial Approximation

3.4 Barycentric Subdivision – Simplicial Approximation

3.5 General Simplicial Complexes

4 Simplicial Homology

4.1 Introduction

4.2 Orientation of Simplicial Complexes

4.3 Simplicial Chain Complex and Homology

4.4 Some Examples

4.5 Properties of Integral Homology Groups

4.6 Induced Homomorphisms

4.7 Some Applications

4.8 Degree of a Map and its Applications

4.9 Invariance of Homology Groups

4.9.1 Subdivision Chain Map

4.9.2 Homomorphism Induced by a Continuous Map

4.9.3 Homotopy Invariance

4.9.4 Lefschetz Fixed-Point Theorem

4.9.5 The Borsuk-Ulam Theorem

4.10 Homology of General Simplicial Complexes

5 Covering Projections

5.1 Introduction

5.2 Properties of Covering Projections

5.3 Applications of Homotopy Lifting Theorem

5.4 Lifting of an arbitrary map

5.5 Covering Homomorphisms

5.6 Universal Covering Space – Applications

6 Singular Homology

6.1 Introduction

6.2 Singular Chain Complex

6.3 One-Dimensional Homology and the Fundamental Group

6.4 Homotopy Axiom for Singular Homology

6.5 Relative Homology and the Axioms

6.6 The Excision Theorem

6.7 Homology and Cohomology Theories

6.8 Singular Homology with Coefficients

6.9 Mayer-Vietoris Sequence for Singular Homology

6.10 Some Classical Applications

6.11 Singular Cohomology and Cohomology Algebra

7 Appendix

7.1 Basic Algebra – a Review

7.1.1 Groups and Homomorphisms

7.1.2 Direct Product and Direct Sum

7.1.3 The Structure of a Finite Abelian Group

7.1.4 Free Groups and Free Products

7.1.5 Modules and their Direct Sum

7.2 Categories and Functors

7.2.1 The Hom(M,N) Functor

7.2.2 Exact Sequences

7.2.3 The Tensor Product of Modules and Homomorphisms

7.2.4 Chain Complexes and Homology

7.2.5 Tensor Product of two Chain Complexes

7.2.6 Exact Homology Sequence Theorem

7.3 Topological Transformation Groups

7.3.1 Topological Transformation Groups

References

Index