Talk:PlanetPhysics/Overview of Algebraic Topology

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\section{An Overview of Algebraic Topology topics}

\subsection{Introduction} \emph{Algebraic topology} (AT) utilizes \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} approaches to solve \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} problems, such as the \htmladdnormallink{classification}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of surfaces, proving \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} 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 \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html}, homology and \htmladdnormallink{cohomology groups}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html}. There are close connections between algebraic topology, Algebraic Geometry (AG) \begin{enumerate}

\item \htmladdnormallink{homotopy theory}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} and \htmladdnormallink{fundamental groups}{http://planetphysics.us/encyclopedia/Pushout.html} \item Topology and \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}; van Kampen theorem \item Homology and cohomology theories \item Duality \item \htmladdnormallink{category theory applications}{http://planetphysics.us/encyclopedia/CategoricalOntology.html} in algebraic topology \item \htmladdnormallink{indexes of category}{http://planetphysics.us/encyclopedia/IndexOfCategories.html}, \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html} \item \htmladdnormallink{Grothendieck's Descent theory}{http://www.uclouvain.be/17501.html} \item `\htmladdnormallink{Anabelian Geometry}{http://planetphysics.us/encyclopedia/IsomorphismClass.html}' \item Categorical Galois theory \item \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}) \item Quantum algebraic topology (\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/NonNewtonian2.html}) \item \htmladdnormallink{Non-Abelian Quantum Algebraic Topology}{http://planetphysics.org/?op=getobj&from=lec&id=61}

($http://aux.planetphysics.org/files/lec/61/ANAQAT20c.pdf$) \item Quantum Geometry \item \htmladdnormallink{Non-Abelian algebraic topology (NAAT)}{http://planetphysics.org/encyclopedia/NonAbelianAlgebraicTopology6.html}

\end{enumerate}

\subsection{Homotopy theory and fundamental groups} \begin{enumerate} \item Homotopy \item \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} of a space \item Fundamental theorems \item \htmladdnormallink{Van Kampen theorem}{http://planetphysics.us/encyclopedia/VanKampenTheorems.html} \item Whitehead \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, torsion and towers \item Postnikov towers \end{enumerate}

\subsection{Topology and Groupoids} \begin{enumerate} \item Topology definition, axioms and basic \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} \item \htmladdnormallink{fundamental groupoid}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} \item \htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} \item van Kampen theorem for groupoids \item Groupoid \htmladdnormallink{pushout}{http://planetphysics.us/encyclopedia/Pushout.html} theorem \item \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} and \htmladdnormallink{crossed modules}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} \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 \htmladdnormallink{cohomology group}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} \item Cohomology sequence \item DeRham cohomology \item new4

\end{enumerate}

\subsection{Non-Abelian Algebraic Topology} \begin{enumerate}

\item \htmladdnormallink{crossed complexes}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} \item \htmladdnormallink{modules}{http://planetphysics.us/encyclopedia/RModule.html} \item \htmladdnormallink{omega-groupoids}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} \item double groupoids \item Higher Homotopy van Kampen Theorems

\end{enumerate}

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