Cubical homotopy theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2015
|
| Series: | New mathematical monographs
25 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xv, 631 Seiten Illustrationen |
| ISBN: | 9781107030251 1107030250 |
Staff View
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| 650 | 4 | |a Algebraic topology | |
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Record in the Search Index
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| adam_text | Contents
R
Preface page xi
PART I CUBICAL DIAGRAMS
1 Preliminaries 3
l.i Spaces and maps 3
1.2 Spaces of maps 12
1.3 Homotopy 14
1.4 Algebra: homotopy and homology 20
2 1-cubes: Homotopy fibers and cofibers 28
2.1 Fibrations 29
2.2 Homotopy fibers 40
2.3 Cofibrations 47
2.4 Homotopy cofibers 56
2.5 Algebra of fibrations and cofibrations 62
2.6 Connectivity of spaces and maps 67
2.7 Quasifibrations 80
3 2-cubes: Homotopy pullbacks and pushouts 91
3.1 Pullbacks 94
3.2 Homotopy pullbacks 100
3.3 Arithmetic of homotopy cartesian squares 111
3.4 Total homotopy fibers 127
3.5 Pushouts 133
3.6 Homotopy pushouts 137
3.7 Arithmetic of homotopy cocartesian squares 149
3.8 Total homotopy cofibers 168
3.9 Algebra of homotopy cartesian and cocartesian squares 171
vii
Contents
viii
4 2-cubes: The Blakers-Massey Theorems 183
4.1 Historical remarks 183
4.2 Statements and applications 188
4.3 Hurewicz, Whitehead, and Serre Theorems 198
4.4 Proofs of the Blakers-Massey Theorems for squares 203
4.5 Proofs of the dual B lakers---Massey Theorems for squares 213
4.6 Homotopy groups of squares 216
5 «-cubes: Generalized homotopy pullbacks and pushouts 221
5.1 Cubical and punctured cubical diagrams 221
5.2 Limits of punctured cubes 226
5.3 Homotopy limits of punctured cubes 230
5.4 Arithmetic of homotopy cartesian cubes 234
5.5 Total homotopy fibers 249
5.6 Colimits of punctured cubes 255
5.7 Homotopy colimits of punctured cubes 258
5.8 Arithmetic of homotopy cocartesian cubes 261
5.9 Total homotopy cofibers 271
5.10 Algebra of homotopy cartesian and cocartesian cubes 275
6 The Blakers-Massey Theorems for «-cubes 288
6.1 Historical remarks 288
6.2 Statements and applications 290
6.3 Proofs of the Blakers---Massey Theorems for «-cubes 299
6.4 Proofs of the dual Blakers---Massey Theorems for «-cubes 324
6.5 Homotopy groups of cubes 329
PART II GENERALIZATIONS, RELATED TOPICS, AND
APPLICATIONS
7 Some category theory 339
7.1 Categories, functors, and natural transformations 339
7.2 Products and coproducts 347
7.3 Limits and colimits 351
7.4 Models for limits and colimits 367
7.5 Algebra of limits and colimits 375
8 Homotopy limits and colimits of diagrams of spaces 379
8.1 The classifying space of a category 380
8.2 Homotopy limits and colimits 388
8.3 Homotopy invariance of homotopy limits and colimits 399
8.4 Models for homotopy limits and colimits 412
Contents ix
8.5 Algebra of homotopy limits and colimits 423
8.6 Algebra in the indexing variable 431
9 Cosimplicial spaces 443
9.1 Cosimplicial spaces and totalization 444
9.2 Examples 458
9.3 Cosimplicial replacement of a diagram 464
9.4 Cosimplicial spaces and cubical diagrams 471
9.5 Multi-cosimplicial spaces 476
9.6 Spectral sequences 479
10 Applications 502
10.1 Homotopy calculus of functors 502
10.2 Manifold calculus of functors 524
10.3 Embeddings, immersions, and disjunction 542
10.4 Spaces of knots 553
Appendix 570
A. 1 Simplicial sets 570
A.2 Smooth manifolds and transversality 578
A.3 Spectra 586
A.4 Operads and cosimplicial spaces 595
References 600
Index 613
IIV-VU i 1 lull iv. 11 lull Vwdi II
Graduate students and researchers alike will benefit from this treatment of various classical and modern topics in the homotopy theory of topological spaces, with an emphasis on cubical diagrams. The book contains more than 300 examples and provides detailed explanations of many fundamental results.
Part I focuses on foundational material on homotopy theory, viewed through the lens of cubical diagrams: fibrations and cofibrations; homotopy pullbacks and pushouts; and the Blakers-Massey Theorems. Part II includes a brief example-driven introduction to categories, limits, and colimits, and an accessible account of homotopy limits and colimits of diagrams of spaces. It also discusses cosimplicial spaces and relates this topic to the cubical theory of Part I, and provides computational tools via spectral sequences.
The book finishes with applications to some exciting new topics that use cubical diagrams: an overview of two versions of calculus of functors and an account of recent developments in the study of the topology of spaces of knots.
Brian A. Munson is an Associate Professor of Mathematics at the US Naval Academy. He has held postdoctoral and visiting positions at Stanford University, Harvard University, and Wellesley College, Massachusetts. His research area is algebraic topology, and his work spans topics such as embedding theory, knot theory, and homotopy theory.
The New Mathematical Monographs are dedicated to books containing an in-depth discussion of a substantial area of mathematics. They bring the reader to the forefront of research by presenting a synthesis of the key results, while also acknowledging the wider mathematical context. As well as being detailed, they are readable and contain the motivational material necessary for those entering a field. For established researchers they are a valuable resource. Books are edited and typeset to a high standard and published in hardback.
Ismar Voucisan Associate Professor of Mathematics at Wellesley College, Massachusetts. He has held postdoctoral and visiting positions at the University of Virginia, Massachusetts Institute of Technology, and Louvain-la-Neuve University in Belgium. His research is in algebraic topology and his articles span a wide variety of subjects such as knot theory, homotopy theory, and category theory. He is an award-winning teacher whose research has been recognized by several grants from the National Science Foundation.
Cambridge
UNITERSITY PRESS www.cambridge.org
ISBN 978-1-107-03025-1
9 781107 030251
|
| any_adam_object | 1 |
| author | Munson, Brian A. 1976- Volić, Ismar 1973- |
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| building | Verbundindex |
| bvnumber | BV042907997 |
| classification_rvk | SK 300 |
| contents | Includes bibliographical references and index |
| ctrlnum | (OCoLC)927927465 (DE-599)BVBBV042907997 |
| discipline | Mathematik |
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| spelling | Munson, Brian A. 1976- Verfasser (DE-588)1079480692 aut Cubical homotopy theory Brian A. Munson, United States Naval Academy, Maryland; Ismar Volić, Wellesley College, Massachusetts Cambridge Cambridge University Press 2015 xv, 631 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier New mathematical monographs 25 Includes bibliographical references and index Homotopy theory Cube Algebraic topology Algebraic topology fast Cube fast Homotopy theory fast 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 Volić, Ismar 1973- Verfasser (DE-588)1052653650 aut New mathematical monographs 25 (DE-604)BV035420183 25 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028335902&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028335902&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Munson, Brian A. 1976- Volić, Ismar 1973- Cubical homotopy theory New mathematical monographs Includes bibliographical references and index Homotopy theory Cube Algebraic topology Algebraic topology fast Cube fast Homotopy theory fast Algebraische Topologie (DE-588)4120861-4 gnd Homotopietheorie (DE-588)4128142-1 gnd |
| subject_GND | (DE-588)4120861-4 (DE-588)4128142-1 |
| title | Cubical homotopy theory |
| title_auth | Cubical homotopy theory |
| title_exact_search | Cubical homotopy theory |
| title_full | Cubical homotopy theory Brian A. Munson, United States Naval Academy, Maryland; Ismar Volić, Wellesley College, Massachusetts |
| title_fullStr | Cubical homotopy theory Brian A. Munson, United States Naval Academy, Maryland; Ismar Volić, Wellesley College, Massachusetts |
| title_full_unstemmed | Cubical homotopy theory Brian A. Munson, United States Naval Academy, Maryland; Ismar Volić, Wellesley College, Massachusetts |
| title_short | Cubical homotopy theory |
| title_sort | cubical homotopy theory |
| topic | Homotopy theory Cube Algebraic topology Algebraic topology fast Cube fast Homotopy theory fast Algebraische Topologie (DE-588)4120861-4 gnd Homotopietheorie (DE-588)4128142-1 gnd |
| topic_facet | Homotopy theory Cube Algebraic topology Algebraische Topologie Homotopietheorie |
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