Fundamental Groups and Covering Spaces.

Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- Preface -- I: Fundamental Groups -- 1: Homotopy -- 1.1 Homotopic Maps -- 1.1.1 Vector Fields on Spheres -- 1.2 Homotopy Type -- 1.3 Contractible Spaces -- 1.4 Homotopy and Map Extension -- 1.4.1 Euclidean Neighborhood Retracts -- 1.5 Trees -- 1.6 Homotopy of Pairs and Relative Homotopy -- 1.7 Exercises -- 2: The Fundamental Group -- 2.1 Path Homotopy -- 2.1.1 Operations with Paths -- 2.1.2 Homotopy and Path Decomposition -- 2.2 The Fundamental Group -- 2.3 The Induced Homomorphism -- 2.4 Other Descriptions of the Fundamental Group -- 2.4.1 Spaces with Abelian Fundamental Group -- 2.4.2 The Fundamental Group and Maps from S1 to X -- 2.5 Simply Connected Spaces -- 2.6 Some Properties of the Fundamental Group -- 2.7 Topological Groups -- 2.8 Exercises -- 3: Some Examples and Applications -- 3.1 The Fundamental Group of the Circle -- 3.2 The Isomorphism π(S1) ≈ Z -- 3.3 Real Projective Spaces -- 3.4 Fibrations and Complex Projective Spaces -- 3.5 Exercises -- 4: Classical Matrix Groups -- 4.1 Rotations in Euclidean Space -- 4.2 The Groups SU(n) and Sp(n) -- 4.3 Exercises -- 5: The Winding Number -- 5.1 The Winding Number of a Closed Plane Curve -- 5.2 The Graustein-Whitney Theorem -- 5.2.1 About Eversions -- 5.3 The Winding Number as a Curvilinear Integral -- 5.3.1 The Winding Number as a Complex Integral -- 5.4 Winding Number and Polynomial Roots -- 5.5 Exercises -- II: Covering Spaces -- 6: Covering Spaces -- 6.1 Local Homeomorphisms and Liftings -- 6.2 Covering Maps -- 6.3 Properly Discontinuous Groups -- 6.4 Path Lifting and Homotopies -- 6.4.1 An Application -- 6.5 Differentiable Coverings -- 6.6 Exercises -- 7: Covering Maps and Fundamental Groups -- 7.1 The Conjugate Class of a Covering Map -- 7.2 The Fundamental Lifting Theorem -- 7.3 Homomorphisms of Covering Spaces.

7.4 Covering Automorphisms -- 7.5 Properly Discontinuous Groups and Regular Coverings -- 7.6 Existence of Coverings -- 7.7 Fundamental Group of a Compact Surface -- 7.8 Exercises -- 8: Oriented Double Covering -- 8.1 Orientation of a Vector Space -- 8.2 Orientable Manifolds -- 8.3 Properly Discontinuous Groups of Diffeomorphisms -- 8.4 Oriented Double Covering -- 8.5 Relations with the Fundamental Group -- 8.6 Exercises -- Appendix Proper Maps -- Bibliography -- Index.

This introductory textbook describes fundamental groups and their topological soul mates, the covering spaces. The author provides several illustrative examples that touch upon different areas of mathematics, but in keeping with the books introductory aim, they are all quite elementary. Basic concepts are clearly defined, proofs are complete, and n.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.