Introductory lectures on equivariant cohomology:
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| Language: | English |
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Princeton and Oxford
Princeton University Press
2020
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| Series: | Annals of mathematics studies
number 204 |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Enthält Literaturverzeichnis (Seite [303] - 307) und Index |
| Physical Description: | xx, 315 Seiten Illustrationen, Diagramme |
| ISBN: | 9780691191751 9780691191744 |
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| adam_text | Introductory Lectures on
Equivariant Cohomology
Loring W Tu
With Appendices by Loring W Tu
and Alberto Arabia
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
© 2020 Princeton University Press
Requests for permission to reproduce material from this
work should be sent to permissions@press princeton edu
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire 0X20 1TR
press princeton edu
All Rights Reserved
ISBN: 978-0-691-19174-4
ISBN (pbk): 978-0-691-19175-1
ISBN (e-book): 978-0-691-19748-7
British Library Cataloging-in-Publication Data is available
Editorial: Susannah Shoemaker and Lauren Bucca
Production Editorial: Nathan Carr
Text Design: Leslie Flis
Jacket/Cover Design: Leslie Flis
Production: Jacquie Poirier
Publicity: Matthew Taylor and Katie Lewis
Copyeditor: Lor Campbell Gehret
The publisher would like to acknowledge the author of this volume for providing
the camera-ready copy from which this book was printed
This book has been composed in LaTeX
Printed on acid-free paper
Printed in the United States of America
10 987654321
20’’0Z12Z
Contents
List of Figures xv
Preface xvii
Acknowledgments xix
I Equivariant Cohomology in the Continuous Category 1
1 Overview 5
1 1 Actions of a Group 5
1 2 Orbits, Stabilizers, and Fixed Point Sets 6
1 3 Homogeneous Spaces 7
1 4 Equivariant Cohomology 8
Problems 10
2 Homotopy Groups and CW Complexes 11
2 1 Homotopy Groups 11
2 2 Fiber Bundles 13
2 3 Homotopy Exact Sequence of a Fiber Bundle 14
2 4 Attaching Cells 15
2 5 CW Complexes 16
2 6 Manifolds and CW Complexes 17
2 7 The Infinite Sphere 18
Problems 19
3 Principal Bundles 21
3 1 Principal Bundles 21
3 2 The Pullback of a Fiber Bundle 24
Problems 26
4 Homotopy Quotients and Equivariant Cohomology 29
41A First Candidate for Equivariant Cohomology 29
4 2 Homotopy Quotients 30
4 3 Cartan’s Mixing Space and Cartan’s Mixing Diagram 31
4 4 Equivariant Cohomology Is Well-Defined 35
4 5 Algebraic Structure of Equivariant Cohomology
Problems
5 Universell Bundles and Classifying Spaces
5 1 Universal Bundles
5 2 Uniqueness of a CW Classifying Space
5 3 Milnor’s Construction
5 4 Equivariant Cohomology of a Point
6 Spectral Sequences
6 1 Leray’s Theorem
6 2 Leray’s Theorem on the Product Structure
6 3 Example: The Cohomology of CP2
Problems
7 Equivariant Cohomology of S2 Under Rotation
7 1 Homotopy Quotient as a Fiber Bundle
7 2 Equivariant Cohomology of S2 Under Rotation
Problems
8 A Universal Bundle for a Compact Lie Group
8 1 The Stiefel Variety
82A Principal 0(fc)-Bundle
8 3 Homotopy Groups of a Stiefel Variety
8 4 Closed Subgroups of a Lie Group
8 5 Universal Bundle for a Compact Lie Group
8 6 Universal Bundle for a Direct Product
8 7 Infinite-Dimensional Manifolds
Problems
9 General Properties of Equivariant Cohomology
9 1 Functorial Properties
9 2 Free Actions
9 3 Coefficient Ring of Equivariant Cohomology
9 4 Equivariant Cohomology of a Disjoint Union
Problems
II Differential Geometry of a Principal Bundle
10 The Lie Derivative and Interior Multiplication
10 1 The Lie Derivative of a Vector Field
10 2 The Lie Derivative of a Differential Form
10 3 Interior Multiplication
CONTENTS ix
Problems 85
11 Fundamental Vector Fields 87
11 1 Fundamental Vector Fields 87
11 2 Zeros of a Fundamental Vector Field 90
11 3 Vertical Vectors on a Principal Bundle 92
11 4 Translate of a Fundamental Vector Field 93
11 5 The Lie Bracket of Fundamental Vector Fields 94
Problems 95
12 Basic Forms 97
12 1 Basic Forms on M2 97
12 2 Invariant Forms 98
12 3 Horizontal Forms 99
12 4 Basic Forms 100
Problems 101
13 Integration on a Compact Connected Lie Group 103
13 1 Bi-Invariant Forms on a Compact Connected Lie Group 103
13 2 Integration over a Compact Connected Lie Group 105
13 3 Invariance of the Integral 106
13 4 The Pullback of an Integral 109
13 5 Differentiation Under the Integral Sign 110
13 6 Cohomology Does Not Commute with Invariants 113
Problems 114
14 Vector-Valued Forms 115
14 1 Vector-Valued Forms 115
14 2 Lie Algebra Valued Forms 116
14 3 Matrix-Valued Forms 118
Problems 119
15 The Maurer-Cartan Form 121
15 1 The Lie Algebra 0 of a Lie Group and Its Dual gv 121
15 2 Maurer-Cartan Equation with Respect to a Basis 123
15 3 The Maurer-Cartan Form 124
16 Connections on a Principal Bundle 127
16 1 Maps of Vector Bundles 127
16 2 Vertical and Horizontal Subbundles 128
16 3 Connections on a Principal Bundle 129
16 4 The Maurer-Cartan Form Is a Connection 131
16 5 Existence of a Connection on a Principal Bundle 132
X
CONTENTS
17 Curvature on a Principal Bundle 135
17 1 Curvature 135
17 2 Properties of Curvature 136
III The Cartan Model 141
18 Differential Graded Algebras 145
18 1 Differential Graded Algebras 145
18 2 Tensor Product of Differential Graded Algebras 147
18 3 The Basic Subcomplex of a g-Differential Graded Algebra 148
Problems 150
19 The Weil Algebra and the Weil Model 151
19 1 The Weil Algebra and the Weil Map 151
19 2 The Weil Map Relative to a Basis 152
19 3 The Weil Algebra as a g-DGA 153
19 4 The Cohomolog) of the Weil Algebra 156
19 5 An Algebraic Model for the Universal Bundle 158
19 6 An Algebraic Model for the Homotopy Quotient 159
19 7 Functoriality of the Weil Model 159
Problems 160
20 Circle Actions 161
20 1 The Weil Algebra for a Circle Action 161
20 2 The Weil Model for a Circle Action 161
20 3 The Cartan Model for a Circle Action 163
20 4 The Cartan Differential for a Circle Action 164
20 5 Example: The Action of a Circle on a Point 165
21 The Cartan Model in General 167
21 1 The Weil-Cartan Isomorphism 167
21 2 The Cartan Differential 171
21 3 Intrinsic Description of the Cartan Differential 172
21 4 Pullback of Equivariant Forms 174
21 5 The Equivariant de Rham Theorem 175
21 6 Equivariant Forms for a Torus Action 175
21 7 Example: The Action of a Torus on a Point 176
21 8 Equivariantly Closed Extensions 177
Problems 179
22 Outline of a Proof of the Equivariant de Rham Theorem 181
22 1 The Cohomology of the Base 181
22 2 Equivariant de Rham Theorem for a Free Action 183
CONTENTS xi
22 3 Equivariant de Rham Theorem in General 183
22 4 Cohomology of a Classifying Space 184
IV Borel Localization 187
23 Localization in Algebra 189
23 1 Localization with Respect to a Variable 189
23 2 Localization and Torsion 192
23 3 Antiderivations Under Localization 192
23 4 Localization and Exactness 193
Problems 196
24 Free and Locally Free Actions 197
24 1 Equivariant Cohomology of a Free Action 197
24 2 Locally Free Actions 198
24 3 Equivariant Cohomology of a Locally Free Circle Action 202
Problems 204
25 The Topology of a Group Action 205
25 1 Smooth Actions of a Compact Lie Group 205
25 2 Equivariant Vector Bundles 206
25 3 The Normal Bundle of a Submanifold 207
25 4 Equivariant Tubular Neighborhoods 209
25 5 Equivariant Mayer-Vietoris Sequence 211
26 Borel Localization for a Circle Action 213
26 1 Borel Localization 213
26 2 Example: The Ring Structure of (S2) 216
Problems 219
V The Equivariant Localization Formula 221
27 A Crash Course in Representation Theory 223
27 1 Representations of a Group 223
27 2 Local Data at a Fixed Point 225
Problems 227
28 Integration of Equivariant Forms 229
28 1 Manifolds with Boundary 229
28 2 Integration of Equivariant Forms 230
28 3 Stokes’s Theorem for Equivariant Integration 231
28 4 Integration of Equivariant Forms for a Circle Action 232
Xll
CONTENTS
Problems 232
29 Rationale for a Localization Formula 233
29 1 Circle Actions with Finitely Many Fixed Points 233
29 2 The Spherical Blow-Up 235
Problems 238
30 Localization Formulas 239
30 1 Equivariant Localization Formula for a Circle Action 239
30 2 Application: The Area of a Sphere 241
30 3 Equivariant Characteristic Classes 242
30 4 Localization Formula for a Torus Action 243
Problems 244
31 Proof of the Localization Formula for a Circle Action 245
31 1 On the Spherical Blow-Up 245
31 2 Surface Area of a Sphere 249
32 Some Applications 253
32 1 Integration of Invariant Forms 253
32 2 Rational Cohomology of a Homogeneous Space 253
32 3 Topological Invariants of a Homogeneous Space 254
32 4 Symplectic Geometry and Classical Mechanics 255
32 5 Stationary Phase Approximation 256
32 6 Exponents at Fixed Points 256
32 7 Gvsin Maps 257
Appendices 259
A Proof of the Equivariant de Rham Theorem
by Loring W Tu and Alberto Arabia 261
A l The Weil Algebra 262
A 2 The Weil Map 263
A 3 Cohomology of Basic Subcomplexes 263
A 4 Cartan s Operator K 265
A5A Degree-Lowering Property 266
A 6 The Weil-Cartan Isomorphism 268
A 7 Equivariant de Rham Theorem for a Free Action 268
A 8 Approximation Theorems 269
A 9 Proof of the Equivariant de Rham Theorem in General 269
A 10 Approximations of EG 271
A ll Approximations of the Homotopv Quotient Mq 272
A 12 A Spectra! Sequence for the Cartan Model 274
CONTENTS
xiii
A 13 Ordinary Cohomology and the Cohomology of the Cartan Model 275
B A Comparison Theorem for Spectral Sequences
by Alberto Arabia 279
C Commutativity of Cohomology with Invariants
by Alberto Arabia 283
Hints and Solutions to Selected End-of-Section Problems 289
List of Notations 297
Bibliography 303
Index
|
| adam_txt |
Introductory Lectures on
Equivariant Cohomology
Loring W Tu
With Appendices by Loring W Tu
and Alberto Arabia
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
© 2020 Princeton University Press
Requests for permission to reproduce material from this
work should be sent to permissions@press princeton edu
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire 0X20 1TR
press princeton edu
All Rights Reserved
ISBN: 978-0-691-19174-4
ISBN (pbk): 978-0-691-19175-1
ISBN (e-book): 978-0-691-19748-7
British Library Cataloging-in-Publication Data is available
Editorial: Susannah Shoemaker and Lauren Bucca
Production Editorial: Nathan Carr
Text Design: Leslie Flis
Jacket/Cover Design: Leslie Flis
Production: Jacquie Poirier
Publicity: Matthew Taylor and Katie Lewis
Copyeditor: Lor Campbell Gehret
The publisher would like to acknowledge the author of this volume for providing
the camera-ready copy from which this book was printed
This book has been composed in LaTeX
Printed on acid-free paper
Printed in the United States of America
10 987654321
20’’0Z12Z
Contents
List of Figures xv
Preface xvii
Acknowledgments xix
I Equivariant Cohomology in the Continuous Category 1
1 Overview 5
1 1 Actions of a Group 5
1 2 Orbits, Stabilizers, and Fixed Point Sets 6
1 3 Homogeneous Spaces 7
1 4 Equivariant Cohomology 8
Problems 10
2 Homotopy Groups and CW Complexes 11
2 1 Homotopy Groups 11
2 2 Fiber Bundles 13
2 3 Homotopy Exact Sequence of a Fiber Bundle 14
2 4 Attaching Cells 15
2 5 CW Complexes 16
2 6 Manifolds and CW Complexes 17
2 7 The Infinite Sphere 18
Problems 19
3 Principal Bundles 21
3 1 Principal Bundles 21
3 2 The Pullback of a Fiber Bundle 24
Problems 26
4 Homotopy Quotients and Equivariant Cohomology 29
41A First Candidate for Equivariant Cohomology 29
4 2 Homotopy Quotients 30
4 3 Cartan’s Mixing Space and Cartan’s Mixing Diagram 31
4 4 Equivariant Cohomology Is Well-Defined 35
4 5 Algebraic Structure of Equivariant Cohomology
Problems
5 Universell Bundles and Classifying Spaces
5 1 Universal Bundles
5 2 Uniqueness of a CW Classifying Space
5 3 Milnor’s Construction
5 4 Equivariant Cohomology of a Point
6 Spectral Sequences
6 1 Leray’s Theorem
6 2 Leray’s Theorem on the Product Structure
6 3 Example: The Cohomology of CP2
Problems
7 Equivariant Cohomology of S2 Under Rotation
7 1 Homotopy Quotient as a Fiber Bundle
7 2 Equivariant Cohomology of S2 Under Rotation
Problems
8 A Universal Bundle for a Compact Lie Group
8 1 The Stiefel Variety
82A Principal 0(fc)-Bundle
8 3 Homotopy Groups of a Stiefel Variety
8 4 Closed Subgroups of a Lie Group
8 5 Universal Bundle for a Compact Lie Group
8 6 Universal Bundle for a Direct Product
8 7 Infinite-Dimensional Manifolds
Problems
9 General Properties of Equivariant Cohomology
9 1 Functorial Properties
9 2 Free Actions
9 3 Coefficient Ring of Equivariant Cohomology
9 4 Equivariant Cohomology of a Disjoint Union
Problems
II Differential Geometry of a Principal Bundle
10 The Lie Derivative and Interior Multiplication
10 1 The Lie Derivative of a Vector Field
10 2 The Lie Derivative of a Differential Form
10 3 Interior Multiplication
CONTENTS ix
Problems 85
11 Fundamental Vector Fields 87
11 1 Fundamental Vector Fields 87
11 2 Zeros of a Fundamental Vector Field 90
11 3 Vertical Vectors on a Principal Bundle 92
11 4 Translate of a Fundamental Vector Field 93
11 5 The Lie Bracket of Fundamental Vector Fields 94
Problems 95
12 Basic Forms 97
12 1 Basic Forms on M2 97
12 2 Invariant Forms 98
12 3 Horizontal Forms 99
12 4 Basic Forms 100
Problems 101
13 Integration on a Compact Connected Lie Group 103
13 1 Bi-Invariant Forms on a Compact Connected Lie Group 103
13 2 Integration over a Compact Connected Lie Group 105
13 3 Invariance of the Integral 106
13 4 The Pullback of an Integral 109
13 5 Differentiation Under the Integral Sign 110
13 6 Cohomology Does Not Commute with Invariants 113
Problems 114
14 Vector-Valued Forms 115
14 1 Vector-Valued Forms 115
14 2 Lie Algebra Valued Forms 116
14 3 Matrix-Valued Forms 118
Problems 119
15 The Maurer-Cartan Form 121
15 1 The Lie Algebra 0 of a Lie Group and Its Dual gv 121
15 2 Maurer-Cartan Equation with Respect to a Basis 123
15 3 The Maurer-Cartan Form 124
16 Connections on a Principal Bundle 127
16 1 Maps of Vector Bundles 127
16 2 Vertical and Horizontal Subbundles 128
16 3 Connections on a Principal Bundle 129
16 4 The Maurer-Cartan Form Is a Connection 131
16 5 Existence of a Connection on a Principal Bundle 132
X
CONTENTS
17 Curvature on a Principal Bundle 135
17 1 Curvature 135
17 2 Properties of Curvature 136
III The Cartan Model 141
18 Differential Graded Algebras 145
18 1 Differential Graded Algebras 145
18 2 Tensor Product of Differential Graded Algebras 147
18 3 The Basic Subcomplex of a g-Differential Graded Algebra 148
Problems 150
19 The Weil Algebra and the Weil Model 151
19 1 The Weil Algebra and the Weil Map 151
19 2 The Weil Map Relative to a Basis 152
19 3 The Weil Algebra as a g-DGA 153
19 4 The Cohomolog)' of the Weil Algebra 156
19 5 An Algebraic Model for the Universal Bundle 158
19 6 An Algebraic Model for the Homotopy Quotient 159
19 7 Functoriality of the Weil Model 159
Problems 160
20 Circle Actions 161
20 1 The Weil Algebra for a Circle Action 161
20 2 The Weil Model for a Circle Action 161
20 3 The Cartan Model for a Circle Action 163
20 4 The Cartan Differential for a Circle Action 164
20 5 Example: The Action of a Circle on a Point 165
21 The Cartan Model in General 167
21 1 The Weil-Cartan Isomorphism 167
21 2 The Cartan Differential 171
21 3 Intrinsic Description of the Cartan Differential 172
21 4 Pullback of Equivariant Forms 174
21 5 The Equivariant de Rham Theorem 175
21 6 Equivariant Forms for a Torus Action 175
21 7 Example: The Action of a Torus on a Point 176
21 8 Equivariantly Closed Extensions 177
Problems 179
22 Outline of a Proof of the Equivariant de Rham Theorem 181
22 1 The Cohomology of the Base 181
22 2 Equivariant de Rham Theorem for a Free Action 183
CONTENTS xi
22 3 Equivariant de Rham Theorem in General 183
22 4 Cohomology of a Classifying Space 184
IV Borel Localization 187
23 Localization in Algebra 189
23 1 Localization with Respect to a Variable 189
23 2 Localization and Torsion 192
23 3 Antiderivations Under Localization 192
23 4 Localization and Exactness 193
Problems 196
24 Free and Locally Free Actions 197
24 1 Equivariant Cohomology of a Free Action 197
24 2 Locally Free Actions 198
24 3 Equivariant Cohomology of a Locally Free Circle Action 202
Problems 204
25 The Topology of a Group Action 205
25 1 Smooth Actions of a Compact Lie Group 205
25 2 Equivariant Vector Bundles 206
25 3 The Normal Bundle of a Submanifold 207
25 4 Equivariant Tubular Neighborhoods 209
25 5 Equivariant Mayer-Vietoris Sequence 211
26 Borel Localization for a Circle Action 213
26 1 Borel Localization 213
26 2 Example: The Ring Structure of (S2) 216
Problems 219
V The Equivariant Localization Formula 221
27 A Crash Course in Representation Theory 223
27 1 Representations of a Group 223
27 2 Local Data at a Fixed Point 225
Problems 227
28 Integration of Equivariant Forms 229
28 1 Manifolds with Boundary 229
28 2 Integration of Equivariant Forms 230
28 3 Stokes’s Theorem for Equivariant Integration 231
28 4 Integration of Equivariant Forms for a Circle Action 232
Xll
CONTENTS
Problems 232
29 Rationale for a Localization Formula 233
29 1 Circle Actions with Finitely Many Fixed Points 233
29 2 The Spherical Blow-Up 235
Problems 238
30 Localization Formulas 239
30 1 Equivariant Localization Formula for a Circle Action 239
30 2 Application: The Area of a Sphere 241
30 3 Equivariant Characteristic Classes 242
30 4 Localization Formula for a Torus Action 243
Problems 244
31 Proof of the Localization Formula for a Circle Action 245
31 1 On the Spherical Blow-Up 245
31 2 Surface Area of a Sphere 249
32 Some Applications 253
32 1 Integration of Invariant Forms 253
32 2 Rational Cohomology of a Homogeneous Space 253
32 3 Topological Invariants of a Homogeneous Space 254
32 4 Symplectic Geometry and Classical Mechanics 255
32 5 Stationary Phase Approximation 256
32 6 Exponents at Fixed Points 256
32 7 Gvsin Maps 257
Appendices 259
A Proof of the Equivariant de Rham Theorem
by Loring W Tu and Alberto Arabia 261
A l The Weil Algebra 262
A 2 The Weil Map 263
A 3 Cohomology of Basic Subcomplexes 263
A 4 Cartan's Operator K 265
A5A Degree-Lowering Property 266
A 6 The Weil-Cartan Isomorphism 268
A 7 Equivariant de Rham Theorem for a Free Action 268
A 8 Approximation Theorems 269
A 9 Proof of the Equivariant de Rham Theorem in General 269
A 10 Approximations of EG 271
A ll Approximations of the Homotopv Quotient Mq 272
A 12 A Spectra! Sequence for the Cartan Model 274
CONTENTS
xiii
A 13 Ordinary Cohomology and the Cohomology of the Cartan Model 275
B A Comparison Theorem for Spectral Sequences
by Alberto Arabia 279
C Commutativity of Cohomology with Invariants
by Alberto Arabia 283
Hints and Solutions to Selected End-of-Section Problems 289
List of Notations 297
Bibliography 303
Index |
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| bvnumber | BV046618000 |
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| discipline | Mathematik |
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| oclc_num | 1154011052 |
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| owner | DE-11 DE-19 DE-BY-UBM DE-20 |
| owner_facet | DE-11 DE-19 DE-BY-UBM DE-20 |
| physical | xx, 315 Seiten Illustrationen, Diagramme |
| publishDate | 2020 |
| publishDateSearch | 2020 |
| publishDateSort | 2020 |
| publisher | Princeton University Press |
| record_format | marc |
| series | Annals of mathematics studies |
| series2 | Annals of mathematics studies |
| spelling | Tu, Loring W. 1952- Verfasser (DE-588)110090322 aut Introductory lectures on equivariant cohomology Loring W. Tu Princeton and Oxford Princeton University Press 2020 xx, 315 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies number 204 Enthält Literaturverzeichnis (Seite [303] - 307) und Index Erscheint auch als Online-Ausgabe 978-0-691-19748-7 Annals of mathematics studies number 204 (DE-604)BV000000991 204 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032029774&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tu, Loring W. 1952- Introductory lectures on equivariant cohomology Annals of mathematics studies |
| title | Introductory lectures on equivariant cohomology |
| title_auth | Introductory lectures on equivariant cohomology |
| title_exact_search | Introductory lectures on equivariant cohomology |
| title_exact_search_txtP | Introductory lectures on equivariant cohomology |
| title_full | Introductory lectures on equivariant cohomology Loring W. Tu |
| title_fullStr | Introductory lectures on equivariant cohomology Loring W. Tu |
| title_full_unstemmed | Introductory lectures on equivariant cohomology Loring W. Tu |
| title_short | Introductory lectures on equivariant cohomology |
| title_sort | introductory lectures on equivariant cohomology |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032029774&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000991 |
| work_keys_str_mv | AT tuloringw introductorylecturesonequivariantcohomology |