Topology and Geometry

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Springer Science & Business Media, Jun 24, 1993 - Mathematics - 557 pages
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The golden age of mathematics-that was not the age of Euclid, it is ours. C. J. KEYSER This time of writing is the hundredth anniversary of the publication (1892) of Poincare's first note on topology, which arguably marks the beginning of the subject of algebraic, or "combinatorial," topology. There was earlier scattered work by Euler, Listing (who coined the word "topology"), Mobius and his band, Riemann, Klein, and Betti. Indeed, even as early as 1679, Leibniz indicated the desirability of creating a geometry of the topological type. The establishment of topology (or "analysis situs" as it was often called at the time) as a coherent theory, however, belongs to Poincare. Curiously, the beginning of general topology, also called "point set topology," dates fourteen years later when Frechet published the first abstract treatment of the subject in 1906. Since the beginning of time, or at least the era of Archimedes, smooth manifolds (curves, surfaces, mechanical configurations, the universe) have been a central focus in mathematics. They have always been at the core of interest in topology. After the seminal work of Milnor, Smale, and many others, in the last half of this century, the topological aspects of smooth manifolds, as distinct from the differential geometric aspects, became a subject in its own right.
 

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Contents

Acknowledgments 1X Chapter
1
Topological Spaces
3
Subspaces
8
Connectivity and Components
10
Separation Axioms
12
Nets MooreSmith Convergence
14
Compactness
18
Products
22
Singular Homology
219
The Cross Product
220
Subdivision
223
The MayerVietoris Sequence
228
The Generalized Jordan Curve Theorem
230
The BorsukUlam Theorem
240
Simplicial Complexes
245
Simplicial Maps
250

Metric Spaces Again
25
Existence of Real Valued Functions
29
Locally Compact Spaces
31
Paracompact Spaces
35
Quotient Spaces
39
Homotopy
44
Topological Groups
51
Convex Bodies
56
The Baire Category Theorem
57
Chapter II
60
Differentiable Manifolds
63
Differentiable Manifolds
68
Local Coordinates
71
Induced Structures and Examples
72
Tangent Vectors and Differentials
76
Sards Theorem and Regular Values
80
Local Properties of Immersions and Submersions
82
Vector Fields and Flows
86
Tangent Bundles
88
Embedding in Euclidean Space
89
Tubular Neighborhoods and Approximations
92
Classical Lie Groups ft
101
Fiber Bundles ft
106
Induced Bundles and Whitney Sums ft Ill 15 Transversality ft
114
ThomPon try agin Theory ft
118
Chapter III
126
Fundamental Group
127
The Fundamental Group
132
Covering Spaces
138
The Lifting Theorem
143
The Action of 7tj on the Fiber
146
Deck Transformations
147
Properly Discontinuous Actions
150
Classification of Covering Spaces
154
The SeifertVan Kampen Theorem ft
158
Remarks on SO3 ft
164
Homology Theory 18
168
The Zeroth Homology Group
172
Functorial Properties
175
Homological Algebra
177
Axioms for Homology
182
Computation of Degrees
190
CWComplexes
197
Conventions for CWComplexes
198
Cellular Homology
200
Cellular Maps
207
Products of CWComplexes ft
211
Eulers Formula
215
Homology of Real Projective Space
217
The LefschetzHopf Fixed Point Theorem
253
Chapter V
255
Cohomology
260
Differential Forms
261
Integration of Forms
265
Stokes Theorem
267
Relationship to Singular Homology
269
More Homological Algebra
271
Universal Coefficient Theorems
281
Excision and Homotopy
285
de Rhams Theorem
286
The de Rham Theory of CP
292
Hopfs Theorem on Maps to Spheres
297
Differential Forms on Compact Lie Groups Q
304
Chapter VI
315
A Sign Convention
321
The Cup Product
326
The Cap Product
334
Classical Outlook on Duality
338
The Orientation Bundle
340
Duality Theorems
348
Duality on Compact Manifolds with Boundary
355
Applications of Duality
359
Intersection Theory Q
366
The Euler Class Lefschetz Numbers and Vector Fields
378
The Gysin Sequence
390
Lefschetz Coincidence Theory
393
Steenrod Operations
404
Construction of the Steenrod Squares
412
StiefelWhitney Classes
420
Plumbing
426
Chapter VII
430
The CompactOpen Topology
437
HSpaces HGroups and HCogroups
441
Homotopy Groups
443
The Homotopy Sequence of a Pair
445
Fiber Spaces
450
Free Homotopy
457
Classical Groups and Associated Manifolds
463
The Hurewicz Theorem
475
EilenbergMac Lane Spaces
488
Obstruction Theory
497
Obstruction Cochains and Vector Bundles
511
Appendices
519
Critical Values
531
Bibliography
541
Index
549
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