**Mathematical Surveys and Monographs**

Volume: 132;
2006;
441 pp;
Hardcover

MSC: Primary 19; 55;

**Print ISBN: 978-0-8218-3922-5
Product Code: SURV/132**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-1359-0
Product Code: SURV/132.E**

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

#### You may also like

#### Supplemental Materials

# Parametrized Homotopy Theory

Share this page
*J. P. May; J. Sigurdsson*

This book develops rigorous foundations for
parametrized homotopy theory, which is the algebraic topology of
spaces and spectra that are continuously parametrized by the points of
a base space. It also begins the systematic study of parametrized
homology and cohomology theories.

The parametrized world provides the natural home for many classical
notions and results, such as orientation theory, the Thom isomorphism,
Atiyah and Poincaré duality, transfer maps, the Adams and
Wirthmüller isomorphisms, and the Serre and Eilenberg–Moore
spectral sequences. But in addition to providing a clearer conceptual
outlook on these classical notions, it also provides powerful methods
to study new phenomena, such as twisted \(K\)-theory, and to
make new constructions, such as iterated Thom spectra.

Duality theory in the parametrized setting is particularly illuminating
and comes in two flavors. One allows the construction and analysis of
transfer maps, and a quite different one relates parametrized homology
to parametrized cohomology. The latter is based formally on a new theory
of duality in symmetric bicategories that is of considerable independent
interest.

The text brings together many recent developments in homotopy
theory. It provides a highly structured theory of parametrized
spectra, and it extends parametrized homotopy theory to the
equivariant setting. The theory of topological model categories is
given a more thorough treatment than is available in the literature.
This is used, together with an interesting blend of classical methods,
to resolve basic foundational problems that have no
nonparametrized counterparts.

#### Readership

Research mathematicians interested in recent advances in algebraic topology.

#### Table of Contents

# Table of Contents

## Parametrized Homotopy Theory

Table of Contents pages: 1 2

- Contents v6 free
- Prologue 112 free
- Part I. Point-set topology, change functors, and proper actions 1122
- Introduction 1324
- Chapter 1. The point-set topology of parametrized spaces 1526
- Introduction 1526
- 1.1. Convenient categories of topological spaces 1526
- 1.2. Topologically bicomplete categories and ex-objects 1627
- 1.3. Convenient categories of ex-spaces 1930
- 1.4. Convenient categories of ex-G-spaces 2233
- 1.5. Philosophical comments on the point-set topology 2334
- 1.6. Technical point-set topological lemmas 2435
- 1.7. Appendix: nonassociativity of smash products in [omitted] op[sub(*)] 2637

- Chapter 2. Change functors and compatibility relations 2940
- Chapter 3. Proper actions, equivariant bundles and fibrations 4354

- Part II. Model categories and parametrized spaces 5768
- Introduction 5970
- Chapter 4. Topologically bicomplete model categories 6172
- Introduction 6172
- 4.1. Model theoretic philosophy: h, q, and m-model structures 6273
- 4.2. Strong Hurewicz cofibrations and fibrations 6374
- 4.3. Towards classical model structures in topological categories 6677
- 4.4. Classical model structures in general and in K and U 6980
- 4.5. Compactly generated q-type model structures 7283

- Chapter 5. Well-grounded topological model categories 7788
- Introduction 7788
- 5.1. Over and under model structures 7889
- 5.2. The specialization to over and under categories of spaces 8293
- 5.3. Well-grounded topologically bicomplete categories 8596
- 5.4. Well-grounded categories of weak equivalences 8798
- 5.5. Well-grounded compactly generated model structures 90101
- 5.6. Properties of well-grounded model categories 91102

- Chapter 6. The qf-model structure on K[sub(B)] 97108
- Introduction 97108
- 6.1. Some of the dangers in the parametrized world 98109
- 6.2. The qf model structure on the category K/B 100111
- 6.3. Statements and proofs of the thickening lemmas 102113
- 6.4. The compatibility condition for the qf-model structure 105116
- 6.5. The quasifibration and right properness properties 107118

- Chapter 7. Equivariant qf-type model structures 109120
- Chapter 8. Ex-fibrations and ex-quasifibrations 127138
- Chapter 9. The equivalence between HoGK[sub(B)] and hGW[sub(B)] 137148

- Part III. Parametrized equivariant stable homotopy theory 147158
- Introduction 149160
- Chapter 10. Enriched categories and G-categories 151162
- Chapter 11. The category of orthogonal G-spectra over B 159170
- Introduction 159170
- 11.1. The category of l[sub(G)]-spaces over B 159170
- 11.2. The category of orthogonal G-spectra over B 163174
- 11.3. Orthogonal G-spectra as diagram ex-G-spaces 166177
- 11.4. The base change functors f*, f!, and f[sub(*)] 167178
- 11.5. Change of groups and restriction to fibers 170181
- 11.6. Some problems concerning non-compact Lie groups 172183

- Chapter 12. Model structures for parametrized G-spectra 175186
- Introduction 175186
- 12.1. The level model structure on Gl[sub(B)] 176187
- 12.2. Some Quillen adjoint pairs relating level model structures 179190
- 12.3. The stable model structure on Gl[sub(B)] 180191
- 12.4. Cofiber sequences and π[sub(*)]-isomorphisms 183194
- 12.5. Proofs of the model axioms 186197
- 12.6. Some Quillen adjoint pairs relating stable model structures 190201

- Chapter 13. Adjunctions and compatibility relations 195206
- Introduction 195206
- 13.1. Brown representability and the functors f[sub(*)] and F[sub(B)] 196207
- 13.2. The category GE[sub(B)] of excellent prespectra over B 200211
- 13.3. The level ex-fibrant approximation functor P on prespectra 202213
- 13.4. The auxiliary approximation functors K and E 205216
- 13.5. The equivalence between HoGp[sub(B)] and hGE[sub(B)] 207218
- 13.6. Derived functors on homotopy categories 208219
- 13.7. Compatibility relations for smash products and base change 209220

- Chapter 14. Module categories, change of universe, and change of groups 215226

- Part IV. Parametrized duality theory 229240
- Introduction 231242
- Chapter 15. Fiberwise duality and transfer maps 233244
- Introduction 233244
- 15.1. The fiberwise duality theorem 234245
- 15.2. Duality and trace maps in symmetric monoidal categories 236247
- 15.3. Transfer maps of Hurewicz fibrations 238249
- 15.4. The bundle construction on parametrized spectra 240251
- 15.5. II-free parametrized Γ-spectra 242253
- 15.6. The fiberwise transfer for (∏;Γ)-bundles 244255

- Chapter 16. Closed symmetric bicategories 247258
- Introduction 247258
- 16.1. Recollections about bicategories 248259
- 16.2. The definition of symmetric bicategories 249260
- 16.3. The definition of closed symmetric bicategories 252263
- 16.4. Duality in closed symmetric bicategories 255266
- 16.5. Composites and naturality of dualities 259270
- 16.6. A quick review of triangulated categories 261272
- 16.7. Compatibly triangulated symmetric bicategories 262273
- 16.8. Duality in triangulated symmetric bicategories 266277

- Chapter 17. The closed symmetric bicategory of parametrized spectra 269280
- Chapter 18. Costenoble-Waner duality 285296
- Introduction 285296
- 18.1. The two notions of duality in HoGl[sub(B)] 286297
- 18.2. Costenoble-Waner dualizability of finite cell spectra 288299
- 18.3. Costenoble-Waner V-duality 290301
- 18.4. Preliminaries on unreduced relative mapping cones 292303
- 18.5. V-duality of G-ENRs 295306
- 18.6. Parametrized Atiyah duality for closed manifolds 296307
- 18.7. Parametrized Atiyah duality for manifolds with boundary 300311
- 18.8. The proof of the Costenoble-Waner duality theorem 302313

- Chapter 19. Fiberwise Costenoble-Waner duality 311322
- Introduction 311322
- 19.1. Costenoble-Waner duality and homotopical Poincaré duality 312323
- 19.2. The bicategories Ex[sub(B)] 314325
- 19.3. Comparisons of bicategories 316327
- 19.4. The bundle construction pseudo-functor 319330
- 19.5. The fiberwise Costenoble-Waner duality theorem 320331
- 19.6. Fiberwise Poincaré duality 324335
- 19.7. The Adams isomorphism 326337
- 19.8. Some background and comparisons 328339

- Part V. Homology and cohomology, Thorn spectra, and addenda 333344
- Introduction 335346
- Chapter 20. Parametrized homology and cohomology theories 337348
- Introduction 337348
- 20.1. Axioms for parametrized homology and cohomology theories 338349
- 20.2. Represented homology and cohomology theories 341352
- 20.3. Coefficient systems and restriction maps 343354
- 20.4. The Serre spectral sequence 344355
- 20.5. Poincaré duality and the Thorn isomorphism 347358
- 20.6. Relative Poincaré duality 350361
- 20.7. Products in parametrized homology and cohomology 350361
- 20.8. The represent ability of homology theories 353364

- Chapter 21. Equivariant parametrized homology and cohomology 357368
- Introduction 357368
- 21.1. Equivariant homology and cohomology theories 358369
- 21.2. Represented equivariant theories 360371
- 21.3. Change of base and equivariant cofficient systems 361372
- 21.4. Duality theorems and orientations 363374
- 21.5. Products and the represent ability of homology 366377
- 21.6. Fiberwise parametrized homology and cohomology 367378
- 21.7. Fiberwise Poincaré duality and orientations 369380

- Chapter 22. Twisted theories and spectral sequences 373384
- Introduction 373384
- 22.1. Twisted homology and cohomology theories 374385
- 22.2. Automorphism monoids of spectra and GL[sub(1)](k) 375386
- 22.3. Twisted K-theory 378389
- 22.4. The simplicial spectral sequence 380391
- 22.5. Cech type spectral sequences 384395
- 22.6. The twisted Rothenberg–Steenrod spectral sequence 386397
- 22.7. The parametrized Künneth spectral sequence 388399

- Chapter 23. Parametrized FSP's and generalized Thorn spectra 393404
- Introduction 393404
- 23.1.D-functors with products in symmetric monoidal categories 395406
- 23.2. The specialization of D-FP's to spaces and ex-spaces 397408
- 23.3. Group, monoid, and module FCP's; examples 399410
- 23.4. The two-sided bar construction on FCP's 402413
- 23.5. Examples: iterated Thorn spectra 403414
- 23.6. l[sub(c)]-FCP's and L-spaces 405416
- 23.7. Universal spherical fibration spectra 407418
- 23.8. Some historical background 408419

- Chapter 24. Epilogue: cellular philosophy and alternative approaches 411422

Table of Contents pages: 1 2