Cubical Homotopy TheoryGraduate 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 Blakers–Massey 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. |
Contents
| 3 | |
Homotopy fibers and cofibers | 28 |
Homotopy pullbacks and pushouts | 102 |
4 | 112 |
5 | 137 |
Pushouts | 145 |
8 | 154 |
9 | 179 |
The BlakersMassey Theorems for ncubes | 288 |
Some category theory | 339 |
Homotopy limits and colimits of diagrams of spaces | 379 |
Cosimplicial spaces | 443 |
Applications | 502 |
Appendix | 570 |
References | 600 |
| 613 | |
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Common terms and phrases
1)-connected based spaces basepoint Blakers-Massey Theorem canonical map cartesian codegeneracies cofaces cofibration colim commutative diagram Corollary cosimplicial space cubical diagrams CW complex define Definition denote dimension dual Example fibration finite functor F given hence hocolim hofiber hofiber(X holim F holim(X homeomorphism homology spectral sequence homotopy cocartesian square homotopy cofiber homotopy equivalence homotopy groups homotopy invariant homotopy limit homotopy pullback homotopy pushout homotopy resp inclusion induced map inverse isomorphism k-connected knots Lemma Let f lifting problem limits and colimits long exact sequence map f Map(Z mapping cylinder morphism n-cube natural transformation object operad proof of Theorem punctured cube quasifibration quotient reader Remark result Section simplicial sets smooth manifold spectral sequence square is homotopy statement strongly homotopy cocartesian submanifold subset subspace Suppose Taylor tower tfiber(X topological spaces topology topy total homotopy fiber vertical maps weak equivalence X₁