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EMS Tracts in Mathematics, Vol.15, an EMS publication: September 2010, approx. 650 pages. ISBN 978-3-13719-083-8.

Abstract:

The book presents, according to the authors: ``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...(it provides) a full account of a theory which, without using singular homology theory or simplicial approximation, but employing filtered spaces and methods analogous to those used originally for the fundamental group or groupoid, obtains for example: --the Brouwer degree theorem; --the Relative Hurewicz theorem, seen as a special case of a homotopical excision theorem giving information on relative homotopy groups as a module over the fundamental group; --non-Abelian information on second relative homotopy groups of mapping cones, and of unions; -homotopy information on the space of pointed maps X ==> Y when X is a CW--complex of dimension n and Y is connected and has no homotopy between 1 and n; this result again involves the fundamental groups... 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." ============================ AMS MSC: 55-01 (Algebraic topology :: Instructional exposition ) 18B40 (Category theory; homological algebra :: Special categories :: Groupoids, semigroupoids, semigroups, groups ) 20-XX (Group theory and generalizations) 55Nxx (Homology and cohomology theories) 55Pxx (Homotopy theory) 20L05 (Group theory and generalizations :: Groupoids ) 20Kxx (Abelian groups) 20K45 (Group theory and generalizations :: Abelian groups :: Topological methods) MSC CLASSIFICATION: msc:55-01, msc:55N20, msc:55N40, msc:55N99, msc:55N15, msc:55N30, msc:18-00, msc:11E72, msc:11F23, msc:57N65, msc:57R19 3. See also Nonabelian Algebraic Topology vol.1.2007-2008.free downloads at: http://planetphysics.org/?op=getobj&from=books\&id=167

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Copyright@2010 EMS by Ronald Brown et al. http://www.bangor.ac.uk/~mas010/nonab-a-t.html

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