Algebraic topology from a homotopical viewpoint:
"The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common m...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2002
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the book, no previous knowledge about category theory is expected from the reader. This book is intended for advanced undergraduate and graduate students with a basic background in point set topology as well as group theory and can be used in a two-semester course."--BOOK JACKET. |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXIX, 478 S. graph. Darst. |
| ISBN: | 0387954503 9781441930057 |
Staff View
MARC
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| 245 | 1 | 0 | |a Algebraic topology from a homotopical viewpoint |c Marcelo Aguilar ; Samuel Gitler ; Carlos Prieto |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2002 | |
| 300 | |a XXIX, 478 S. |b graph. Darst. | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Universitext | |
| 500 | |a Includes bibliographical references and index | ||
| 520 | 1 | |a "The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the book, no previous knowledge about category theory is expected from the reader. This book is intended for advanced undergraduate and graduate students with a basic background in point set topology as well as group theory and can be used in a two-semester course."--BOOK JACKET. | |
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| 700 | 1 | |a Gitler, Samuel |e Verfasser |4 aut | |
| 700 | 1 | |a Prieto de Castro, Carlos |d 1948- |e Verfasser |0 (DE-588)12402050X |4 aut | |
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Record in the Search Index
| _version_ | 1804129250808168448 |
|---|---|
| adam_text | Contents
Preface vii
Introduction xiii
Basic Concepts and Notation xvii
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 77 Spaces 45
2.8 Loop Spaces 48
2.9 // Cospaces 50
2.10 Suspensions 53
ix
x Contents
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
Contents xi
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 /^ 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 R and on S 1 319
10.6 The Hopf Invariant 321
xii Contents
11 Relations Between Cohomology and
Vector Bundles 331
11.1 Contractibility of S°° 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 Thorn 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.I 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.I A Convenient Description of BU x Z 437
B.2 Proof of the Bott Periodicity Theorem 440
References 457
Symbols 463
Index 467
|
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| author | Aguilar, Marcelo Gitler, Samuel Prieto de Castro, Carlos 1948- |
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| dewey-full | 514/.2 |
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| dewey-ones | 514 - Topology |
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| dewey-tens | 510 - Mathematics |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:09Z |
| institution | BVB |
| isbn | 0387954503 9781441930057 |
| language | English |
| lccn | 2002019556 |
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| spelling | Aguilar, Marcelo Verfasser (DE-588)124020496 aut Algebraic topology from a homotopical viewpoint Marcelo Aguilar ; Samuel Gitler ; Carlos Prieto New York [u.a.] Springer 2002 XXIX, 478 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Includes bibliographical references and index "The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the book, no previous knowledge about category theory is expected from the reader. This book is intended for advanced undergraduate and graduate students with a basic background in point set topology as well as group theory and can be used in a two-semester course."--BOOK JACKET. Homotopie Homotopie ram Topologie algébrique Topologie algébrique ram Algebraic topology Homotopy theory Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Homotopietheorie (DE-588)4128142-1 gnd rswk-swf Homotopietheorie (DE-588)4128142-1 s Algebraische Topologie (DE-588)4120861-4 s DE-604 Gitler, Samuel Verfasser aut Prieto de Castro, Carlos 1948- Verfasser (DE-588)12402050X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009844776&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Aguilar, Marcelo Gitler, Samuel Prieto de Castro, Carlos 1948- Algebraic topology from a homotopical viewpoint Homotopie Homotopie ram Topologie algébrique Topologie algébrique ram Algebraic topology Homotopy theory Algebraische Topologie (DE-588)4120861-4 gnd Homotopietheorie (DE-588)4128142-1 gnd |
| subject_GND | (DE-588)4120861-4 (DE-588)4128142-1 |
| title | Algebraic topology from a homotopical viewpoint |
| title_auth | Algebraic topology from a homotopical viewpoint |
| title_exact_search | Algebraic topology from a homotopical viewpoint |
| title_full | Algebraic topology from a homotopical viewpoint Marcelo Aguilar ; Samuel Gitler ; Carlos Prieto |
| title_fullStr | Algebraic topology from a homotopical viewpoint Marcelo Aguilar ; Samuel Gitler ; Carlos Prieto |
| title_full_unstemmed | Algebraic topology from a homotopical viewpoint Marcelo Aguilar ; Samuel Gitler ; Carlos Prieto |
| title_short | Algebraic topology from a homotopical viewpoint |
| title_sort | algebraic topology from a homotopical viewpoint |
| topic | Homotopie Homotopie ram Topologie algébrique Topologie algébrique ram Algebraic topology Homotopy theory Algebraische Topologie (DE-588)4120861-4 gnd Homotopietheorie (DE-588)4128142-1 gnd |
| topic_facet | Homotopie Topologie algébrique Algebraic topology Homotopy theory Algebraische Topologie Homotopietheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009844776&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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