Essential topology 1st edition by Martin Crossley – Ebook PDF Instant Download/Delivery: 1852337826, 9781852337827
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ISBN 10: 1852337826
ISBN 13: 9781852337827
Author: Martin D. Crossley
Taking a direct route, “Essential Topology” brings the most important aspects of modern topology within reach of a second-year undergraduate student. It begins with a discussion of continuity and, by way of many examples, leads to the celebrated “Hairy Ball theorem” and on to homotopy and homology: the cornerstones of contemporary algebraic topology. While containing all the key results of basic topology, Essential Topology never allows itself to get mired in details. Instead, the focus throughout is on providing interesting examples that clarify the ideas and motivate the student. With chapters on: continuity and topological spaces; deconstructionist topology; the Euler number; homotopy groups including the fundamental group; simplicial and singular homology, and fibre bundles. “Essential Topology” contains enough material for two semester-long courses, and offers a one-stop-shop for undergraduate-level topology, leaving students motivated for postgraduate study in the field, and well-prepared for it.
Essential topology 1st Table of contents:
1. Introduction
2. Continuous Functions
2.1 Naïve Continuity
2.2 Rigorous Continuity
2.3 Open Sets
2.4 Continuity by Open Sets
3. Topological Spaces
3.1 Topological Spaces
3.2 More Examples of Topological Spaces
3.3 Continuity in the Subspace Topology
3.4 Bases
Interlude
4. Topological Properties
4.1 Connectivity
4.2 Compactness
4.3 The Hausdorff Property
5. Deconstructionist Topology
5.1 Homeomorphisms
5.2 Disjoint Unions
5.3 Product Spaces
5.4 Quotient Spaces
Interlude
6. Homotopy
6.1 Homotopy
6.2 Homotopy Equivalence
6.3 The Circle
6.4 Brouwer’s Fixed-Point Theorem
6.5 Vector Fields
7. The Euler Number
7.1 Simplicial Complexes
7.2 The Euler Number
7.3 The Euler Characteristic and Surfaces
8. Homotopy Groups
8.1 Homotopy Groups
8.2 Induced Homomorphisms
8.3 The Fundamental Group
8.4 Path Connectivity and 0
8.5 The Van Kampen Theorem
9. Simplicial Homology
9.1 Simplicial Homology Modulo 2
9.2 Limitations of Homology Modulo 2
9.3 Integral Simplicial Homology
10. Singular Homology
10.1 Singular Homology
10.2 Homology and Continuous Maps
10.3 Homology Respects Homotopies
10.4 Barycentric Subdivision
10.5 The Mayer–Vietoris Sequence
10.6 Homology and Homotopy Groups
10.7 Comparison of Singular and Simplicial Homology
11. More Deconstructionism
11.1 Wedge Products
11.2 Suspensions and Loop Spaces
11.3 Fibre Bundles
11.4 Vector Bundles
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