overview of algebraic topology OverviewOfAlgebraicTopology 2010-10-07 01:35:34 2010-10-10 22:10:26 Topic omega-groupoid higher homotopy algebraic topology nonabelian algebraic topology manifold homology group cohomology theory nonabelian theory quantum geometry quantum algebra quantum algebraic topology category thyeory noncommutative geometry overview algebraic topology nonabelian algebraic topology invarinats groupoid topological space higher homotopy theory higher homotopy van Kampen theorem crossed complex module omega-groupoid double groupoid \section{An Overview of Algebraic Topology topics} \subsection{Introduction} \emph{Algebraic topology} (AT) utilizes algebraic approaches to solve topological problems, such as the classification of surfaces, proving duality theorems for manifolds and approximation theorems for topological spaces. A central problem in algebraic topology is to find algebraic invariants of topological spaces, which is usually carried out by means of homotopy, homology and cohomology groups. There are close connections between algebraic topology, \PMlinkname{Algebraic Geometry (AG)}{AlgebraicGeometry), Non-commutative Geometry and NAAT. On the other hand, there are also close ties between algebraic geometry and number theory. \subsection{Outline} \begin{enumerate} \item Homotopy theory and fundamental groups \item Topology and groupoids; \PMlinkname{van Kampen theorem}{VanKampensTheorem} \item Homology and cohomology theories \item Duality \item Category theory applications in algebraic topology \item Index of categories, functors and natural transformations \item \PMlinkexternal{Grothendieck's Descent theory}{http://www.uclouvain.be/17501.html} \item `Anabelian geometry' \item Categorical Galois theory \item Higher dimensional algebra (HDA) \item Quantum algebraic topology (QAT) \item \PMlinkexternal{Non-Abelian Quantum Algebraic Topology}{http://aux.planetphysics.org/files/lec/61/ANAQAT20c.pdf} \item Quantum Geometry \item \PMlinkexternal{Non-Abelian algebraic topology (NAAT)}{http://planetphysics.org/encyclopedia/NonAbelianAlgebraicTopology6.html} \end{enumerate} \subsection{Homotopy theory and fundamental groups} \begin{enumerate} \item Homotopy \item Fundamental group of a space \item Fundamental theorems \item van Kampen theorem \item Whitehead groups, torsion and towers \item Postnikov towers \end{enumerate} \subsection{Topology and Groupoids} \begin{enumerate} \item Topology definition, axioms and basic concepts \item Fundamental groupoid \item Topological groupoid \item van Kampen theorem for groupoids \item Groupoid pushout theorem \item Double groupoids and crossed modules \item new4 \end{enumerate} \subsection{Homology theory} \begin{enumerate} \item Homology group \item Homology sequence \item Homology complex \item new4 \end{enumerate} \subsection{Cohomology theory} \begin{enumerate} \item Cohomology group \item Cohomology sequence \item DeRham cohomology \item new4 \end{enumerate} \subsection{Non-Abelian Algebraic Topology} \begin{enumerate} \item Crossed Complexes \item Modules \item omega-groupoids \item double groupoids \item Higher Homotopy van Kampen Theorems \end{enumerate}