Topological Methods in Euclidean Spaces

by Gregory L. Naber

Overview

Extensive development of a number of topics central to topology, including elementary combinatorial techniques, Sperner's Lemma, the Brouwer Fixed Point Theorem, homotopy theory and the fundamental group, simplicial homology theory, the Hopf Trace Theorem, the Lefschetz Fixed Point Theorem, the Stone-Weierstrass Theorem, and Morse functions. Includes new section of solutions to selected problems.

Product Details

ISBN-13:	9780486153445
Publisher:	Dover Publications
Publication date:	08/01/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	240
File size:	46 MB
Note:	This product may take a few minutes to download.

Table of Contents

	Preface	ix
Chapter 1	Point-set topology of Euclidean spaces	1
1	Introduction	1
2	Preliminaries	6
3	Open sets, closed sets, and continuity	10
4	Compact spaces	21
5	Connectivity properties	25
6	Real-valued continuous functions	34
7	Retracts	38
8	Topological dimension	41
	Supplementary exercises	44
Chapter 2	Elementary combinatorial techniques	46
1	Introduction	46
2	Hyperplanes in R[superscript n]	46
3	Simplexes and complexes	49
4	Sample triangulations	55
5	Simplicial maps	60
6	Barycentric subdivision	63
7	The Simplicial Approximation Theorem	70
8	Sperner's Lemma	73
9	The Brouwer Fixed Point Theorem	75
10	Topological dimension of compact subsets of R[superscript n]	77
	Supplementary exercises	79
Chapter 3	Homotopy theory and the fundamental group	81
1	Introduction	81
2	The homotopy relation, nullhomotopic maps, and contractible spaces	84
3	Maps of spheres	86
4	The fundamental group	90
5	Fundamental groups of the spheres	99
	Supplementary exercises	106
Chapter 4	Simplicial homology theory	108
1	Introduction	108
2	Oriented complexes and chains	111
3	Boundary operators	116
4	Cycles, boundaries, and homology groups	118
5	Elementary examples	122
6	Cone complexes, augmented complexes, and the homology groups	125
7	Incidence numbers and the homology groups	128
8	Elementary homological algebra	130
9	The homology complex of a geometric complex	133
10	Acyclic carrier functions	136
11	Invariance of homology groups under barycentric subdivision	138
12	Homomorphisms induced by continuous maps	141
13	Homology groups of topological polyhedra	144
14	The Hopf Trace Theorem	147
15	The Lefschetz Fixed Point Theorem	150
	Supplementary exercises	150
Chapter 5	Differential techniques	154
1	Introduction	154
2	Smooth maps	161
3	The Stone--Weierstrass Theorem	163
4	Derivatives as linear transformations	166
5	Differentiable manifolds	173
6	Tangent spaces and derivatives	175
7	Regular and critical values of smooth maps	179
8	Measure zero and Sard's Theorem	183
9	Morse functions	190
10	Manifolds with boundary	201
11	One-dimensional manifolds	205
12	Topological characterization of S[superscript k]	208
13	Smooth tangent vector fields	211
	Supplementary exercises	217
	Solutions to Selected Exercises	220
	Guide to further study	238
	Bibliography	240
	List of symbols and notation	242
	Index	246