Topological Methods in Euclidean Spaces

Name: Topological Methods in Euclidean Spaces
Brand: Dover Publications
Price: 11.49 EUR
Availability: OnlineOnly

Gregory L. Naber(Autor*in)

Dover Publications (Verlag)

Erschienen am 29. August 2012

476 Seiten

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978-0-486-15344-5 (ISBN)

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Inhalt

Cover
Title page
Dedication
Copyrights page
Contents
Chapter 1 Point-set topology of Euclidean spaces page
1 Introduction
2 Preliminaries
3 Open sets, closed sets, and continuity
4 Compact spaces
5 Connectivity properties
6 Real-valued continuous functions
7 Retracts
8 Topological dimension
Supplementary exercises
Chapter 2 Elementary combinatorial techniques
1 Introduction
2 Hyperplanes in Rn
3 Simplexes and complexes
4 Sample triangulations
5 Simplicial maps
6 Barycentric subdivision
7 The Simplicial Approximation Theorem
8 Sperner's Lemma
9 The Brouwer Fixed Point Theorem
10 Topological dimension of compact subsets of Rn
Supplementary exercises
Chapter 3 Homotopy theory and the fundamental group
1 Introduction
2 The homotopy relation, nullhomotopic maps, and contractible spaces
3 Maps of spheres
4 The fundamental group
5 Fundamental groups of the spheres
Supplementary exercises
Chapter 4 Simplicial homology theory
1 Introduction
2 Oriented complexes and chains
3 Boundary operators
4 Cycles, boundaries, and homology groups
5 Elementary examples
6 Cone complexes, augmented complexes, and the homology groups Hp(K(sn+1))
7 Incidence numbers and the homology groups Hp(K(sn+1))
8 Elementary homological algebra
9 The homology complex of a geometric complex
10 Acyclic carrier functions
11 Invariance of homology groups under barycentric subdivision
12 Homomorphisms induced by continuous maps
13 Homology groups of topological polyhedra
14 The Hopf Trace Theorem
15 The Lefschetz Fixed Point Theorem
Supplementary exercises
Chapter 5 Differential techniques
1 Introduction
2 Smooth maps
3 The Stone-Weierstrass Theorem
4 Derivatives as linear transformations
5 Differentiable manifolds
6 Tangent spaces and derivatives
7 Regular and critical values of smooth maps
8 Measure zero and Sard's Theorem
9 Morse functions
10 Manifolds with boundary
11 One-dimensional manifolds
12 Topological characterization of Sk
13 Smooth tangent vector fields
Supplementary exercises
Solutions to Selected Exercises
Guide to further study
Bibliography
List of symbols and notation
Index

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