Algebraic Topology from a Homotopical Viewpoint

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Springer Science & Business Media, Jun 13, 2002 - Mathematics - 479 pages
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The authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology.

 

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Contents

FUNCTION SPACES
2
12 CompactOpen Topology
3
13 The Exponential Law
4
CONNECTEDNESS AND ALGEBRAIC INVARIANTS
9
22 Homotopy Classes
10
23 Topological Groups
13
24 Homotopy of Mappings of the Circle into Itself
15
25 The Fundamental Group
28
82 Projections and Vector Bundles
268
83 Grassmann Manifolds and Universal Bundles
271
84 Classification of Vector Bundles of Finite Type
276
85 Classification of Vector Bundles over Paracompact Spaces
279
𝐾THEORY
289
92 Definition of 𝐾𝐵
292
93 𝐾𝐵 and Stable Equivalence of Vector Bundles
295
94 Representations of 𝐾B and 𝐾𝐵
299

26 The fundamental Group of the Circle
41
27 𝐻Spaces
45
28 Loop Spaces
48
29 𝐻COSPACES
50
210 Suspensions
53
HOMOTOPY GROUPS
59
32 The Seifertvan Kampen Theorem
63
33 Homotopy Sequences I
72
34 Homotopy Groups
80
35 Homotopy Sequences II
84
HOMOTOPY EXTENSION AND LIFTING PROPERTIES
89
42 Some Results on Cofibrations
95
43 Fibrations
101
44 Pointed and Unpointed Homotopy Classes
119
45 Locally Trivial Bundles
125
46 Classification of Covering Maps over Paracompact Spaces
138
CWCOMPLEXES AND HOMOLOGY
149
52 Infinite Symmetric Products
167
53 Homology Groups
176
HOMOTOPY PROPERTIES OF CWCOMPLEXES
189
62 Homotopy Excision and Related Results
193
63 Homotopy Properties of the Moore Spaces
201
64 Homotopy Properties of the EilenbergMac Lane Spaces
217
COHOMOLOGY GROUPS AND RELATED TOPICS
227
72 Multiplication in Cohomology
238
73 Cellular Homology and Cohomology
243
74 Exact Sequences in Homology and Cohomology
252
VECTOR BUNDLES
259
95 Bott Periodicity and Applications
302
ADAMS OPERATIONS AND APPLICATIONS
309
102 The Splitting Principle
313
103 Normed Algebras
315
104 Division Algebras
317
105 Multiplicative Structures on ℝⁿ and on 𝕊ⁿ
319
106 The Hopf Invariant
321
RELATIONS BETWEEN COHOMOLOGY AND VECTOR BUNDLES
331
111 Contractibility of 𝕊
332
112 Description of 𝐾ℤ21
334
113 Classification of Real Line Bundles
337
114 Description of 𝐾ℤ2
340
115 Classification of Complex Line Bundles
343
116 Characteristic Classes
345
117 Thom Isomorphism and Gysin Sequence
349
118 Construction of Characteristic Classes and Applications
366
COHOMOLOGY THEORIES AND BROWN REPRESENTABILITY
383
122 Brown Representability Theorem
394
123 Spectra
406
PROOF OF THE DOLDTHOM THEOREM
421
A2 Symmetric Products
431
A3 Proof of the DoldThom Theorem
434
PROOF OF THE BOTT PERIODICITY THEOREM
437
B2 Proof of the Bott Periodicity Theorem
440
REFERENCES
457
SYMBOLS
463
INDEX
467
Copyright

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Page xxv - This slogan should be taken literally, but not naively: it means that we can apply intuitions from the category of topological spaces and continuous maps to the category Top, and hence to logic. It does not mean that any space is a topos, in the naive sense of "is".
Page 459 - JL Kelley, General topology (D. Van Nostrand Co., Inc., New York, 1955).

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About the author (2002)

CARLOS PRIETO is an internationally acclaimed concert cellist who is considered one of Latin America's premier cellists. He has performed throughout the United States, Latin America, Europe, Russia, China, India, and other countries. His many honors include the National Award for the Arts of Mexico, the French Order of Arts and Letters, the Order of Civil Merit awarded by the King of Spain, the Pushkin Medal from the President of Russia, the Eva Janzer Award ("Chevalier du Violoncelle") from the University of Indiana, the Cultural Leadership Citation from the Yale University School of Music, and the Achievement Award of the Mexican Cultural Institute of New York. He has recorded over eighty works, including the complete Bach suites for solo cello, and is the author of seven books.

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