Cubical Homotopy Theory

A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.

Brian A. Munson (Author), Ismar Voli? (Author)

9781107030251, Cambridge University Press

Hardback, published 6 October 2015

644 pages, 20 b/w illus.
23.5 x 16 x 3.8 cm, 1.01 kg

'… 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

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 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.

Preface
Part I. Cubical Diagrams: 1. Preliminaries
2. 1-cubes: homotopy fibers and cofibers
3. 2-cubes: homotopy pullbacks and pushouts
4. 2-cubes: the Blakers-Massey Theorems
5. n-cubes: generalized homotopy pullbacks and pushouts
6. The Blakers–Massey Theorems for n-cubes
Part II. Generalizations, Related Topics, and Applications: 7. Some category theory
8. Homotopy limits and colimits of diagrams of spaces
9. Cosimplicial spaces
10. Applications
Appendix
References
Index.

Subject Areas: Topology [PBP], Geometry [PBM], Mathematics [PB]