Talk:PlanetPhysics/Non Abelian Algebraic Topology 5

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\section{Non--Abelian algebraic topology}

A relatively recent development in Algebraic Topology that began in 1960s which considers \htmladdnormallink{algebraic structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in dimensions greater than 1 which develop the \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} character of the \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} of a \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space, or a novel approach to higher dimensional, non-Abelian \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} treatments of topological invariants in Algebraic Topology.

\subsection{Recent reference:} Ronald Brown, Bangor University, UK, Philip J. Higgins, Durham University, UK Rafael Sivera, University of Valencia, Spain.2010. \emph{Nonabelian Algebraic Topology: Filtered Spaces, \htmladdnormallink{crossed complexes}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html}, Cubical Homotopy \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}.} EMS Tracts in Mathematics, Vol.15, an EMS publication: September 2010, approx. 670 pages. ISBN 978-3-13719-083-8.

\subsection{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 \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} than those commonly in use. Exploration of these uses of higher dimensional versions of groupoids has been largely the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} of the first two authors since the mid 1960s...(it provides) a full account of a theory which, without using singular \htmladdnormallink{homology theory}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} or \htmladdnormallink{simplicial}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} 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 \htmladdnormallink{homotopy groups}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} as a \htmladdnormallink{module}{http://planetphysics.us/encyclopedia/RModule.html} 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 $X \rightarrow Y$ 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 \htmladdnormallink{Nonabelian Algebraic Topology vol.1.2007-2008.free downloads}{http://planetphysics.org/?op=getobj&from=books&id=167} and \htmladdnormallink{Nonabelian Algebraic Topology textbook in 2010}{http://planetphysics.org/?op=getobj&from=books&id=374}

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