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Hardcover ISBN:  9780821839225 
Product Code:  SURV/132 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413590 
Product Code:  SURV/132.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821839225 
eBook ISBN:  9781470413590 
Product Code:  SURV/132.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 132; 2006; 441 ppMSC: Primary 19; 55
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.
ReadershipResearch mathematicians interested in recent advances in algebraic topology.

Table of Contents

Chapters

1. The pointset topology of parametrized spaces

2. Change functors and compatibility relations

3. Proper actions, equivariant bundles and fibrations

4. Topologically bicomplete model categories

5. Wellgrounded topological model categories

6. The $qf$model structure on $\mathscr {K}_B$

7. Equivariant $qf$type model structures

8. Exfibrations and exquasifibrations

9. The equivalence between Ho $G\mathscr {K}_B$ and $hG\mathscr {W}_B$

10. Enriched categories and $G$categories

11. The category of orthogonal $G$spectra over $B$

12. Model structures for parametrized $G$spectra

13. Adjunctions and compatibility relations

14. Module categories, change of universe, and change of groups

15. Fiberwise duality and transfer maps

16. Closed symmetric bicategories

17. The closed symmetric bicategory of parametrized spectra

18. CostenobleWaner duality

19. Fiberwise CostenobleWaner duality

20. Parametrized homology and cohomology theories

21. Equivariant parametrized homology and cohomology

22. Twisted theories and spectral sequences

23. Parametrized FSP’s and generalized Thom spectra

24. Epilogue: cellular philosophy and alternative approaches


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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.
Research mathematicians interested in recent advances in algebraic topology.

Chapters

1. The pointset topology of parametrized spaces

2. Change functors and compatibility relations

3. Proper actions, equivariant bundles and fibrations

4. Topologically bicomplete model categories

5. Wellgrounded topological model categories

6. The $qf$model structure on $\mathscr {K}_B$

7. Equivariant $qf$type model structures

8. Exfibrations and exquasifibrations

9. The equivalence between Ho $G\mathscr {K}_B$ and $hG\mathscr {W}_B$

10. Enriched categories and $G$categories

11. The category of orthogonal $G$spectra over $B$

12. Model structures for parametrized $G$spectra

13. Adjunctions and compatibility relations

14. Module categories, change of universe, and change of groups

15. Fiberwise duality and transfer maps

16. Closed symmetric bicategories

17. The closed symmetric bicategory of parametrized spectra

18. CostenobleWaner duality

19. Fiberwise CostenobleWaner duality

20. Parametrized homology and cohomology theories

21. Equivariant parametrized homology and cohomology

22. Twisted theories and spectral sequences

23. Parametrized FSP’s and generalized Thom spectra

24. Epilogue: cellular philosophy and alternative approaches