algebraic topology AlgebraicTopology 2010-09-28 02:15:06 2010-10-24 18:11:17 Topic non-Abelian fundamental group fundamental groupoid homotopy theory homology theory cohomology group simplicial complex fundamental groupoid functor covering homotopical excision diagram of spaces crossed module crossed complex Lie algebroid Lie groupoid cubical higher homotopy groupoid homology and cohomology theory fundamental functor fundamental groupoid functor groupoid category algebroid category crossed complexes complex modules homology groups and groupoids. homotopy theory groupoids categorical algebra topological categories topological groupoids Lie groupoids Lie algebroids higher-dimensional algebra higher-dimensional groupoids Van Kampen theorems approximation theorem Hurewicz theorem algebraic theories \section{Algebraic topology} \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, of course, its most recent development-- \PMlinkname{non-Abelian Algebraic Topology (NAAT)}{http://planetphysics.org/encyclopedia/NonAbelianAlgebraicTopology6.html}. 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 Non-Abelian Quantum Algebraic Topology (NAQAT) \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{Duality in algebraic topology and category theory} \begin{enumerate} \item Tanaka-Krein duality \item Grothendieck duality \item Categorical duality \item Tangled duality \item DA5 \item DA6 \item DA7 \end{enumerate} \subsection{Category theory applications} \begin{enumerate} \item Abelian categories \item Topological category \item Fundamental groupoid functor \item Categorical Galois theory \item Non-Abelian algebraic topology \item Group category \item Groupoid category \item $\mathcal{T}op$ category \item Topos and topoi axioms \item Generalized toposes \item Categorical logic and algebraic topology \item Meta-theorems \item Duality between spaces and algebras \end{enumerate} \subsection{Index of categories} The following is a listing of categories relevant to algebraic topology: \begin{enumerate} \item \PMlinkexternal{Algebraic categories}{http://www.uclouvain.be/17501.html} \item Topological category \item Category of sets, Set \item Category of topological spaces \item Category of Riemannian manifolds \item Category of CW-complexes \item Category of Hausdorff spaces \item Category of Borel spaces \item Category of CR-complexes \item Category of graphs \item Category of spin networks \item Category of groups \item Galois category \item Category of fundamental groups \item Category of Polish groups \item Groupoid category \item Category of groupoids (or groupoid category) \item Category of Borel groupoids \item Category of fundamental groupoids \item Category of functors (or functor category) \item Double groupoid category \item Double category \item Category of Hilbert spaces \item Category of quantum automata \item R-category \item Category of algebroids \item Category of double algebroids \item Category of dynamical systems \end{enumerate} \subsection{Index of functors} \emph{The following is a contributed listing of functors:} \begin{enumerate} \item Covariant functors \item Contravariant functors \item Adjoint functors \item Preadditive functors \item Additive functor \item Representable functors \item Fundamental groupoid functor \item Forgetful functors \item Grothendieck group functor \item Exact functor \item Multi-functor \item Section functors \item NT2 \item NT3 \end{enumerate} \subsection{Index of natural transformations} \emph{The following is a contributed listing of natural transformations:} \begin{enumerate} \item Natural equivalence \item Natural transformations in a 2-category \item NT3 \item NT1 \item NT2 \item NT3 \end{enumerate} \subsection{Grothendieck proposals} \begin{enumerate} \item \PMlinkname{Esquisse d'un Programme}{AlexSMathematicalHeritageEsquisseDunProgramme} \item \PMlinkexternal{Pursuing Stacks}{http://www.math.jussieu.fr/~leila/grothendieckcircle/stacks.ps} \item S2 \item S3 \item S4 \end{enumerate} \subsection{Descent theory} \begin{enumerate} \item D1 \item D2 \item D3 \item D4 \end{enumerate} \subsection{Higher dimensional algebra (HDA)} \begin{enumerate} \item Categorical groups \item Double groupoids \item Double algebroids \item Bi-algebroids \item $R$-algebroid \item $2$-category \item $n$-category \item Super-category \item weak n-categories \item Bi-dimensional Geometry \item \PMlinkname{Noncommutative geometry}{NoncommutativeGeometry} \item Higher-Homotopy theories \item \PMlinkname{Higher-Homotopy Generalized van Kampen Theorem (HGvKT)}{GeneralizedVanKampenTheoremsHigherDimensional} \item H1 \item H2 \item H3 \item H4 \end{enumerate} \subsubsection{Axioms of cohomology theory} \begin{enumerate} \item A1 \item A2 \item A3 \item A4 \item A5 \item A6 \item A7 \end{enumerate} \subsubsection{Axioms of homology theory} \begin{enumerate} \item A1 \item A2 \item A3 \item A4 \item A5 \item A6 \end{enumerate} \subsection{Non-Abelian Algebraic Topology (NAAT)} \begin{enumerate} \item \PMlinkexternal{An overview of Nonabelian Algebraic Topology}{http://arxiv.org/PS_cache/math/pdf/0407/0407275v2.pdf} \item Non-Abelian categories \item Non-commutative groupoids (including non-Abelian groups) \item \PMlinkname{Generalized van Kampen theorems}{GeneralizedVanKampenTheoremsHigherDimensional} \item \PMlinkname{Noncommutative Geometry (NCG)}{NoncommutativeGeometry} \item Non-commutative `spaces' of functions \item \PMlinkname{Non-Abelian Algebraic Topology textbook}{NonAbelianAlgebraicTopology5} \end{enumerate} \subsubsection{References for NAAT} [1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1--groups: a GAP4 package,(2004) (http://www.maths.bangor.ac.uk/chda/) [2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. second edition as Topology: a geometric account of general topology, homotopy types, and the fundamentalgroupoid, Ellis Horwood, Chichester (1988) 460 pp. [3] R. Brown, \PMlinkexternal{`Higher dimensional group theory’}{http://www.bangor.ac.uk/∼mas010/hdaweb2.htm} [4] R. Brown.`Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local--to--global problems', Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23--28, 2002, Contemp. Math. (2004). (to appear), UWB Math Preprint 02.26.pdf (30 pp.) [5] R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193--212. [6] R. Brown and R. Sivera, `Nonabelian algebraic topology', (in preparation) Part I is downloadable from (http://www.bangor.ac.uk/~mas010/nonab-a-t.html) [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. G'/eom.Diff., 17 (1976) 343--362. [8] R. Brown and C. D.Wensley, `1Computation and homotopical applications of induced crossed modules', J. Symbolic Computation, 35 (2003) 59-72. [9] The GAP Group, 2004, GAP --Groups, Algorithms, and Programming, version 4.4 , Technical report, (http://www.grp-system.org) [10] A. Grothendieck, ‘Pursuing Stacks’, 600p, 1983, distributed from Bangor. Now being edited by G. Maltsiniotis for the SMF. [11] P. J. Higgins, 1971, Categories and Groupoids, Van Nostrand, New York. Reprint Series, Theory and Appl. Categories (to appear). [12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects, number 131 in Translations of Mathematical Monographs, American Mathematical Society. \begin{enumerate} \item new1 \item new2 \item new3 \item new4 \end{enumerate} \subsection{13} \begin{enumerate} \item new1 \item new2 \item new3 \item new4 \end{enumerate} \subsection{14} \subsection{References} \PMlinkexternal{Bibliography on Category theory, AT and QAT}{http://planetmath.org/?op=getobj&from=objects&id=10746} \subsubsection{Textbooks and Expositions:} \begin{enumerate} \item A \PMlinkexternal{Textbook1}{http://planetmath.org/?op=getobj&from=books&id=172} \item A \PMlinkexternal{Textbook2}{http://planetmath.org/?op=getobj&from=books&id=156} \item A \PMlinkexternal{Textbook3}{http://planetmath.org/?op=getobj&from=books&id=159} \item A \PMlinkexternal{Textbook4}{http://planetmath.org/?op=getobj&from=books&id=160} \item A \PMlinkexternal{Textbook5}{http://planetmath.org/?op=getobj&from=books&id=153} \item A \PMlinkexternal{Textbook6}{http://planetmath.org/?op=getobj&from=lec&id=68} \item A \PMlinkexternal{Textbook7}{http://planetmath.org/?op=getobj&from=books&id=158} \item A \PMlinkexternal{Textbook8}{http://planetmath.org/?op=getobj&from=lec&id=75} \item A \PMlinkexternal{Textbook9}{http://planetmath.org/?op=getobj&from=lec&id=73} \item A \PMlinkexternal{Textbook10}{http://planetmath.org/?op=getobj&from=books&id=174} \item A \PMlinkexternal{Textbook11}{http://planetmath.org/?op=getobj&from=books&id=169} \item A \PMlinkexternal{Textbook12}{http://planetmath.org/?op=getobj&from=books&id=178} \item A \PMlinkexternal{Textbook13}{http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf} \item new1 \item new2 \item new3 \item new4 \end{enumerate} \subsection{Algebraic Topology and Groupoids} \begin{enumerate} \item Ronald Brown: Topology and Groupoids, BookSurge LLC (2006). \item Ronald Brown R, P.J. Higgins, and R. Sivera.: \emph{``Non-Abelian algebraic topology"}. http://www. bangor.ac.uk/mas010/nonab-a-t.html; http://www.bangor.ac.uk/mas010/nonab-t/partI010604.pdf , Springer: in press (2010). \item R. Brown and J.-L. Loday: Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc., 54:(3), 176--192, (1987). \item R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, Topology, 26: 311-337 (1987). \item R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986. \item R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. G\'eom. Diff. 17 (1976), 343--362. \item Madalina (Ruxi) Buneci.: \emph{Groupoid Representations}., Ed. Mirton: Timisoara (2003). \item Allain Connes: \emph{Noncommutative Geometry}, Academic Press 1994. \end{enumerate} \subsection{Non--Abelian Algebraic Topology and Higher Dimensional Algebra} \begin{enumerate} \item Ronald Brown: Non--Abelian Algebraic Topology, vols. I and II. 2010. (in press: Springer): \PMlinkexternal{Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical higher homotopy groupoids}{http://www.bangor.ac.uk/~mas010/rbrsbookb-e040310.pdf} \item \PMlinkexternal{Higher Dimensional Algebra: An Introduction}{http://en.wikipedia.org/wiki/Higher_dimensional_algebra} \item \PMlinkexternal{Higher Dimensional Algebra and Algebraic Topology., 282 pages, Feb. 10, 2010}{http://en.wikipedia.org/wiki/User:Bci2/Books/Higher_Dimensional_Algebra} \end{enumerate}