PlanetPhysics/Non Abelian Algebraic Topology 5

Non--Abelian algebraic topology

A relatively recent development in Algebraic Topology that began in 1960s which considers [[../TrivialGroupoid/|algebraic structures]] in dimensions greater than 1 which develop the [[../AbelianCategory3/|non-Abelian]] character of the [[../HomotopyCategory/|fundamental group]] of a [[../CoIntersections/|topological]] space, or a novel approach to higher dimensional, non-Abelian [[../CoIntersections/|algebraic]] treatments of topological invariants in Algebraic Topology.

Recent reference:

Ronald Brown, Bangor University, UK, Philip J. Higgins, Durham University, UK Rafael Sivera, University of Valencia, Spain.2010. Nonabelian Algebraic Topology: Filtered Spaces, \htmladdnormallink{crossed complexes {http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html}, Cubical Homotopy [[../QuantumOperatorAlgebra5/|groupoids]].} EMS Tracts in Mathematics, Vol.15, an EMS publication: September 2010, approx. 670 pages. Template:ISBN.

Notes

1.A central topic of the book is a Higher Homotopy van Kampen Theorem;

2.The book presents ``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 [[../Cod/|composition]] than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the [[../Work/|work]] of the first two authors since the mid 1960s...(it provides) a full account of a theory which, without using singular [[../CubicalHigherHomotopyGroupoid/|homology theory]] or [[../PiecewiseLinear/|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 [[../ExtendedHurewiczFundamentalTheorem/|homotopy groups]] as a [[../RModule/|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 $X\to Y$ 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."

3. See also Nonabelian Algebraic Topology vol.1.2007-2008.free downloads and Nonabelian Algebraic Topology textbook in 2010