TopicOnAxiomsAndFoundationsOfHomologyAndCohomologyTheories 2009-04-19 07:59:57 2009-04-19 08:00:40 Topic axioms and foundations of homology and cohomology theories % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Axioms for Homology and Cohomology theories} \begin{enumerate} \item Axioms for homology theory and uniqueness theorems \item Cech types \item $K$-theory \item Generalized cohomology \item Generalized (extraordinary) homology and cohomology theories \item Galois Cohomology and Categorical Galois theories \item (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) \item Bordism and cobordism theories, formal group laws \item Homology with local coefficients, equivariant cohomology \item Sheaf cohomology \item Cohomology in Noncommutative algebraic geometry \item Classifying spaces for foliations; Gelfand-Fuks cohomology \item Intersection homology and cohomology \item Elliptic cohomology \item Equivariant homology and cohomology \item Homology and homotopy of topological groups and related structures \item Homotopy Quantum Field Theories and Axiomatic Quantum Field Theories \item Non-Abelian Homological Algebra \item Grothendieck's `Anabelian Geometry' \item Other homology theories--Your new additions \end{enumerate} \begin{thebibliography} {99} \bibitem{AllenHatcher2k1} Hatcher, A. 2001. \PMlinkexternal{\emph{Algebraic Topology} (textbook on line).}{http://www.math.cornell.edu/~hatcher/AT/AT.pdf}, Cambridge University Press; Cambridge, UK., 405 pages. \bibitem{RB2k6} Ronald Brown: Topology and Groupoids, BookSurge LLC (2006). \bibitem{RBHS2k7} Ronald Brown R, P.J. Higgins, and R. Sivera.: \emph{``Non-Abelian algebraic topology''}, (2008). \bibitem{RBJL87} 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). \bibitem{RBJL86} R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, {\em Topology}, 26: 311-337 (1987). \bibitem{RBCS76} R. Brown and C.B. Spencer: Double groupoids and crossed modules, {\em Cahiers Top. G\'eom. Diff.}, 17 (1976), 343-362. \bibitem{AC94} Allain Connes: \emph{Noncommutative Geometry}, Academic Press 1994. \bibitem{May1999} May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago \end{thebibliography}