Algebraic Topology: A First Course

Couverture
Springer Science & Business Media, 5 sept. 1997 - 430 pages
To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups
 

Table des matières

III
3
IV
7
V
10
VI
17
VII
23
VIII
27
IX
33
X
35
LXIV
217
LXV
219
LXVI
220
LXVII
222
LXVIII
225
LXIX
227
LXX
228
LXXI
231

XI
38
XII
42
XIII
43
XIV
48
XV
49
XVI
53
XVII
56
XVIII
59
XIX
63
XX
65
XXI
68
XXII
72
XXIII
78
XXIV
82
XXV
85
XXVI
89
XXVII
91
XXVIII
95
XXIX
97
XXX
101
XXXI
102
XXXII
106
XXXIII
113
XXXIV
121
XXXV
123
XXXVI
127
XXXVII
130
XXXVIII
131
XXXIX
137
XL
140
XLI
144
XLII
147
XLIII
151
XLIV
153
XLV
156
XLVI
158
XLVII
163
XLVIII
165
XLIX
170
L
173
LI
177
LII
179
LIII
182
LIV
186
LV
189
LVI
193
LVII
196
LVIII
197
LIX
201
LX
205
LXI
207
LXII
210
LXIII
213
LXXII
233
LXXIII
236
LXXIV
242
LXXV
247
LXXVI
251
LXXVII
252
LXXVIII
256
LXXIX
261
LXXX
263
LXXXI
268
LXXXII
272
LXXXIII
277
LXXXIV
281
LXXXV
284
LXXXVI
289
LXXXVII
291
LXXXVIII
295
LXXXIX
299
XC
303
XCI
306
XCII
313
XCIII
317
XCIV
320
XCV
324
XCVI
325
XCVII
328
XCVIII
332
XCIX
334
C
339
CI
343
CII
346
CIII
350
CIV
355
CV
359
CVI
365
CVII
367
CVIII
369
CIX
370
CX
371
CXI
373
CXII
375
CXIII
378
CXIV
380
CXV
385
CXVI
387
CXVII
389
CXVIII
391
CXIX
393
CXX
397
CXXI
419
CXXII
421
CXXIII
425
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Page v - The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists — without, we hope, discouraging budding topologists.

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