Lecture Notes on Elementary Topology and Geometry

Front Cover
Springer Science & Business Media, Dec 10, 1976 - Mathematics - 232 pages
At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After the calculus, he takes a course in analysis and a course in algebra. Depending upon his interests (or those of his department), he takes courses in special topics. Ifhe is exposed to topology, it is usually straightforward point set topology; if he is exposed to geom etry, it is usually classical differential geometry. The exciting revelations that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate. He must wait until he is well into graduate work to see interconnections, presumably because earlier he doesn't know enough. These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topol ogy, and group theory. (De Rham's theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces are the most note worthy examples.) In the first two chapters the bare essentials of elementary point set topology are set forth with some hint ofthe subject's application to functional analysis.
 

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Contents

II
III
3
IV
9
V
11
VI
14
VII
18
VIII
24
XI
29
XXII
88
XXIII
92
XXIV
107
XXVII
116
XXVIII
130
XXIX
151
XXXII
159
XXXIII
173

XII
32
XIII
38
XIV
41
XV
47
XVII
50
XVIII
60
XIX
76
XX
77
XXI
81
XXXVI
182
XXXVII
188
XXXVIII
196
XXXIX
205
XL
214
XLII
228
XLIII
229
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