New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
 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 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. Readership 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