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Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids
Ronald Brown, Bangor University, Gwynedd, United Kingdom, Philip J. Higgins, Durham University, United Kingdom, and Rafael Sivera, Universitat de València, Spain
A publication of the European Mathematical Society.
EMS Tracts in Mathematics
2011; 703 pp; hardcover
Volume: 15
ISBN-10: 3-03719-083-3
ISBN-13: 978-3-03719-083-8
List Price: US$118
Member Price: US$94.40
Order Code: EMSTM/15
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The main theme of this book is that the use of filtered spaces rather than just topological spaces allows the development of basic algebraic topology in terms of higher homotopy groupoids; these algebraic structures better reflect the geometry of subdivision and composition than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the work of the first two authors since the mid 1960s.

The structure of the book is intended to make it useful to a wide class of students and researchers for learning and evaluating these methods, primarily in algebraic topology but also in higher category theory and its applications in analogous areas of mathematics, physics, and computer science.

Part I explains the intuitions and theory in dimensions 1 and 2, with many figures and diagrams, and a detailed account of the theory of crossed modules. Part II develops the applications of crossed complexes. The engine driving these applications is the work of Part III on cubical \(\omega\)-groupoids, their relations to crossed complexes, and their homotopically defined examples for filtered spaces. Part III also includes a chapter suggesting further directions and problems, and three appendices give accounts of some relevant aspects of category theory. Endnotes for each chapter give further history and references.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Graduate students and research mathematicians interested in algebraic topology.

Table of Contents

Part I. 1- and 2-dimensional results
  • Introduction to Part I
  • History
  • Homotopy theory and crossed modules
  • Basic algebra of crossed modules
  • Coproducts of crossed \(P\)-modules
  • Induced crossed modules
  • Double groupoids and the 2-dimensional Seifert-van Kampen Theorem
Part II. Crossed complexes
  • Introduction to Part II
  • The basics of crossed complexes
  • The higher homotopy Seifert-van Kampen Theorem (HHSvKT) and its applications
  • Tensor products and homotopies of crossed complexes
  • Resolutions
  • The cubical classifying space of a crossed complex
  • Nonabelian cohomology: spaces, groupoids
Part III. Cubical \(\omega\)-groupoids
  • Introduction to Part III
  • The algebra of crossed complexes and cubical \(\omega\)-groupoids
  • The cubical homotopy \(\omega\)-groupoid of a filtered space
  • Tensor products and homotopies
  • Future directions?
  • Appendices
  • Bibliography
  • Glossary of symbols
  • Index
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