Topology of Surfaces

Forsideomslag
Springer Science & Business Media, 26. sep. 1997 - 281 sider
" . . . that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore. " Michael Spivak This text was written as an antidote to topology courses such as Spivak It is meant to provide the student with an experience in geomet describes. ric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average stu dent would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an in tegral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed.
 

Indhold

Introduction to topology
1
Pointset topology in Rⁿ
7
22 Relative neighborhoods
16
23 Continuity
19
24 Compact sets
26
25 Connected sets
29
26 Applications
31
Pointset topology
37
61 The algebra of chains
126
62 Simplicial Complexes
135
63 Homology
139
64 More computations
151
65 Betti numbers and the euler characteristic
154
Cellular functions
157
72 Homology and cellular functions
165
73 Examples
170

32 Continuity connectedness and compactness
43
33 Separation axioms
47
34 Product spaces
48
35 Quotient spaces
52
Surfaces
56
42 Cell complexes
60
43 Surfaces
67
44 Triangulations
75
45 Classification of surfaces
81
46 Surfaces with boundaryn
91
The euler characteristic
94
52 Graphs and trees
95
53 The euler characteristic and the sphere
101
54 The euler characteristic and surfaces
108
55 Mapcoloring problems
113
56 Graphs revisited
119
Homology
125
74 Covering spaces
175
Invariance of homology
183
82 The Simplicial Approximation Theorem
187
Homotopy
197
92 The fundamental group
203
Miscellany
212
102 The Jordan Curve Theorem
221
103 3Manifolds
226
Topology and calculus
235
112 Differentiable manifolds
248
113 Vector fields on manifolds
250
114 Integration on manifolds
257
Groups
263
References
272
Index
274
Copyright

Andre udgaver - Se alle

Almindelige termer og sætninger

Bibliografiske oplysninger