Table of contents for Algebraic topology from a homotopical viewpoint / Marcelo Aguilar, Samuel Gitler, Carlos Prieto.


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1 FUNCTION SPACES                                 1
1.1  Admissible Topologies  ....................  1
1.2  Compact-Open Topology ...................  2
1.3  The Exponential Law  .....................  3
2 CONNECTEDNESS AND
ALGEBRAIC INVARIANTS                           9
2.1  Path Connectedness  .....................  9
2.2  Homotopy Classes .......................  10
2.3  Topological Groups ......................  13
2.4 Homotopy of Mappings of the Circle into Itself .......  15
2.5  The Fundamental Group  ......... ..........  28
2.6 The fundamental Group of the Circle ............  41
2.7  H-Spaces  .............. .............    45
2.8  Loop Spaces ................... .......   48
2.9  H-Cospaces  ................... .......   50
2.10 Suspensions  ................... .......  53



3 HOMOTOPY GROUPS                                             59
3.1 Attaching Spaces; Cylinders and Cones ....  .......      59
3.2 The Seifert-van Kampen Theorem   ....     .........      63
3.3  Homotopy Sequences I ....................               72
3.4  Homotopy Groups ......................                 80
3.5 Homotopy Sequences II ....................              84
4 HOMOTOPY EXTENSION AND
LIFTING PROPERTIES                                         89
4.1  Cofibrations ..........................                 89
4.2 Some Results on Cofibrations ................            95
4.3  Fibrations ...........................                 101
4.4  Pointed and Unpointed Homotopy Classes .........       119
4.5  Locally Trivial Bundles ....................           125
4.6  Classification of Covering Maps over Paracompact Spaces .  138
5 CW-COMPLEXES AND HOMOLOGY                                  149
5.1  CW-Complexes ..      ................                  149
5.2 Infinite Symmetric Products .   ...........             167
5.3 Homology Groups ....................                    176
6 HOMOTOPY PROPERTIES OF
CW-COMPLEXES                                              189
6.1  Eilenberg-Mac Lane and Moore Spaces ...........        189
6.2  Homotopy Excision and Related Results ..........       193
6.3  Homotopy Properties of the Moore spaces .........      201
6.4  Homotopy Properties of the Eilenberg-Mac Lane spaces .  217



7 COHOMOLOGY GROUPS AND
RELATED TOPICS                                            227
7.1 Cohomology Groups ...       .............         ..    227
7.2 Multiplication in Cohomology ...............            238
7.3  Cellular Homology and Cohomology ............          243
7.4  Exact Sequences in Homology and Cohomology ......      252
8 VECTOR BUNDLES                                             259
8.1 Vector Bundles ..................            .....      259
8.2 Projections and Vector Bundles ..................       268
8.3 Grassmann Manifolds and Universal Bundles . .......     271
8.4 Classification of Vector Bundles of Finite Type ......  276
8.5 Classification of Vector Bundles over Paracompact Spaces .  279
9 K-THEORY                                                   289
9.1 Grothendieck Construction ...............         .     289
9.2  Definition  of K(B)  ......................            292
9.3  K(B) and Stable Equivalence of Vector Bundles ......   295
9.4 Representations of K(B) and K(B) ............           299
9.5 Bott Periodicity and Applications .............    .    302
10 ADAMS OPERATIONS AND APPLICATIONS                        309
10.1 Definition of the Adams Operations .... ........      309
10.2 The Splitting Principle .................. .313
10.3  Normed Algebras  ......................              315
10.4 Division Algebras .......................             317
10.5 Multiplicative Structures on Rn and on S-  .......  319
10.6 The Hopf Invariant ................            .      321



11 RELATIONS BETWEEN COHOMOLOGY AND
VECTOR BUNDLES                                       331
11.1 Contractibility of S00 ...................       332
11.2 Description of K(Z/2, 1) ..................      334
11.3 Classification of Real Line Bundles ...........  337
11.4 Description of K(Z, 2) ..................        340
11.5 Classification of Complex Line Bundles .........  343
11.6 Characteristic Classes ...................       345
11.7 Thom Isomorphism and Gysin Sequence .........    349
11.8 Construction of Characteristic Classes and Applications  366
12 COHOMOLOGY THEORIES AND
BROWN REPRESENTABILITY                               383
12.1 Generalized Cohomology Theories ............     383
12.2 Brown Representability Theorem .............     394
12.3 Spectra ...........................              406
A PROOF OF THE DOLD-THOM THEOREM                         421
A.1 Criteria for Quasifibrations ..................     421
A.2 Symmetric Products .....................            431
A.3 Proof of the Dold-Thom Theorem .............   .    434
B PROOF OF THE
BOTT PERIODICITY THEOREM                              437
B.1 A Convenient Description of BU x Z ........... .    437
B.2 Proof of the Bott Periodicity Theorem  .......... .  440








Library of Congress subject headings for this publication: Algebraic topology, Homotopy theory