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Cubical Homotopy Theory

Part of New Mathematical Monographs

  • Date Published: October 2015
  • availability: Available
  • format: Hardback
  • isbn: 9781107030251


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About the Authors
  • 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.

    • Accessible graduate-level text with 300 examples
    • Contemporary treatment of classical topics will make it a valuable reference for researchers
    • Explores applications that may inspire readers to pursue further research in the area
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    Reviews & endorsements

    '… 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

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    Product details

    • Date Published: October 2015
    • format: Hardback
    • isbn: 9781107030251
    • length: 644 pages
    • dimensions: 235 x 160 x 38 mm
    • weight: 1.01kg
    • contains: 20 b/w illus.
    • availability: Available
  • Table of Contents

    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

  • Authors

    Brian A. Munson, United States Naval Academy, Maryland
    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 Volić, Wellesley College, Massachusetts
    Ismar Volić 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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