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El. knyga: Cubical Homotopy Theory

5.00/5 (2 ratings by Goodreads)
(United States Naval Academy, Maryland), (Wellesley College, Massachusetts)
  • Formatas: EPUB+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 06-Oct-2015
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781316348932
Kitos knygos pagal šią temą:
Cubical Homotopy TheoryDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 06-Oct-2015
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781316348932
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Graduate students and researchers alike will benefit from this treatment of classical and modern topics in homotopy theory of topological spaces with an emphasis on cubical diagrams. The book contains 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 BlakersMassey Theorem. Part II includes a brief example-driven introduction to categories, limits and colimits, an accessible account of homotopy limits and colimits of diagrams of spaces, and a treatment of cosimplicial spaces. 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.

Recenzijos

' this volume can serve as a good point of reference for the machinery of homotopy pullbacks and pushouts of punctured n-cubes, with all the associated theory that comes with it, and shows with clarity the interest these methods have in helping to solve current, general problems in homotopy theory. Chapter 10, in particular, proves that what is presented here goes beyond the simple development of a new language to deal with old problems, and rather shows promise and power that should be taken into account.' Miguel Saramago, MathSciNet

Daugiau informacijos

A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.
Preface xi
PART I CUBICAL DIAGRAMS
1 Preliminaries
3(25)
1.1 Spaces and maps
3(9)
1.2 Spaces of maps
12(2)
1.3 Homotopy
14(6)
1.4 Algebra: homotopy and homology
20(8)
2 1-cubes: Homotopy fibers and cofibers
28(63)
2.1 Fibrations
29(11)
2.2 Homotopy fibers
40(7)
2.3 Cofibrations
47(9)
2.4 Homotopy cofibers
56(6)
2.5 Algebra of fibrations and cofibrations
62(5)
2.6 Connectivity of spaces and maps
67(13)
2.7 Quasifibrations
80(11)
3 2-cubes: Homotopy pullbacks and pushouts
91(92)
3.1 Pullbacks
94(6)
3.2 Homotopy pullbacks
100(11)
3.3 Arithmetic of homotopy cartesian squares
111(16)
3.4 Total homotopy fibers
127(6)
3.5 Pushouts
133(4)
3.6 Homotopy pushouts
137(12)
3.7 Arithmetic of homotopy cocartesian squares
149(19)
3.8 Total homotopy cofibers
168(3)
3.9 Algebra of homotopy cartesian and cocartesian squares
171(12)
4 2-cubes: The Blakers--Massey Theorems
183(38)
4.1 Historical remarks
183(5)
4.2 Statements and applications
188(10)
4.3 Hurewicz, Whitehead, and Serre Theorems
198(5)
4.4 Proofs of the Blakers--Massey Theorems for squares
203(10)
4.5 Proofs of the dual Blakers--Massey Theorems for squares
213(3)
4.6 Homotopy groups of squares
216(5)
5 n-cubes: Generalized homotopy pullbacks and pushouts
221(67)
5.1 Cubical and punctured cubical diagrams
221(5)
5.2 Limits of punctured cubes
226(4)
5.3 Homotopy limits of punctured cubes
230(4)
5.4 Arithmetic of homotopy cartesian cubes
234(15)
5.5 Total homotopy fibers
249(6)
5.6 Colimits of punctured cubes
255(3)
5.7 Homotopy colimits of punctured cubes
258(3)
5.8 Arithmetic of homotopy cocartesian cubes
261(10)
5.9 Total homotopy cofibers
271(4)
5.10 Algebra of homotopy cartesian and cocartesian cubes
275(13)
6 The Blakers-Massey Theorems for n-cubes
288(51)
6.1 Historical remarks
288(2)
6.2 Statements and applications
290(9)
6.3 Proofs of the Blakers--Massey Theorems for n-cubes
299(25)
6.4 Proofs of the dual Blakers--Massey Theorems for n-cubes
324(5)
6.5 Homotopy groups of cubes
329(10)
PART II GENERALIZATIONS, RELATED TOPICS, AND APPLICATIONS
7 Some category theory
339(40)
7.1 Categories, functors, and natural transformations
339(8)
7.2 Products and coproducts
347(4)
7.3 Limits and colimits
351(16)
7.4 Models for limits and colimits
367(8)
7.5 Algebra of limits and colimits
375(4)
8 Homotopy limits and colimits of diagrams of spaces
379(64)
8.1 The classifying space of a category
380(8)
8.2 Homotopy limits and colimits
388(11)
8.3 Homotopy invariance of homotopy limits and colimits
399(13)
8.4 Models for homotopy limits and colimits
412(11)
8.5 Algebra of homotopy limits and colimits
423(8)
8.6 Algebra in the indexing variable
431(12)
9 Cosimplicial spaces
443(59)
9.1 Cosimplicial spaces and totalization
444(14)
9.2 Examples
458(6)
9.3 Cosimplicial replacement of a diagram
464(7)
9.4 Cosimplicial spaces and cubical diagrams
471(5)
9.5 Multi-cosimplicial spaces
476(3)
9.6 Spectral sequences
479(23)
10 Applications
502(68)
10.1 Homotopy calculus of functors
502(22)
10.2 Manifold calculus of functors
524(18)
10.3 Embeddings, immersions, and disjunction
542(11)
10.4 Spaces of knots
553(17)
Appendix
570(30)
A.1 Simplicial sets
570(8)
A.2 Smooth manifolds and transversality
578(8)
A.3 Spectra
586(9)
A.4 Operads and cosimplicial spaces
595(5)
References 600(13)
Index 613
Brian A. Munson is an Assistant 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. Ismar Voli is an 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.
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