DerivationOfCohomologyGroupTheorem 2009-01-26 22:17:37 2009-01-27 11:59:37 Derivation cohomology group theorem fundamental class cohomology group fundamental cohomology theorem derivation of cohomology group theorem for connected CW-complexes % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym,color,enumerate} \usepackage{xypic} \xyoption{curve} \usepackage[mathscr]{eucal} %the next gives two direction arrows at the top of a 2 x 2 matrix \newcommand{\directs}[2]{\def\objectstyle{\scriptstyle} \objectmargin={0pt} \xy (0,4)*+{}="a",(0,-2)*+{\rule{0em}{1.5ex}#2}="b",(7,4)*+{\;#1}="c" \ar@{->} "a";"b" \ar @{->}"a";"c" \endxy } \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{conjecture}{Conjecture}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \theoremstyle{plain} \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{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\Ce}{\mathsf{C}} \newcommand{\Q}{\mathsf{Q}} \newcommand{\grp}{\mathsf{G}} \newcommand{\dgrp}{\mathsf{D}} \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{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{\operatorname{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}} \newcommand{\hdgb}{\boldsymbol{\rho}^\square} \newcommand{\hdg}{\rho^\square_2} \renewcommand{\leq}{{\leqslant}} \renewcommand{\geq}{{\geqslant}} \def\red{\textcolor{red}} \def\magenta{\textcolor{magenta}} \def\blue{\textcolor{blue}} \def\<{\langle} \def\>{\rangle} \subsection{Introduction} Let $X_g$ be a general CW-complex and consider the set $\left\langle{X_g, K(G,n)}\right\rangle$ of basepoint preserving homotopy classes of maps from $X_g$ to Eilenberg-MacLane spaces $K(G, n)$ for $n \geq 0 $, with $G$ being an Abelian group. \begin{theorem}({\em Fundamental, [or reduced] cohomology theorem, \cite{AllenHatcher2k1}}). There exists a natural group isomorphism: \begin{equation} \iota : \left\langle(X_g, K(G,n))\right\rangle \cong \overline{H}^n (X_g;G) \end{equation} for all CW-complexes $X_g$ , with $G$ any Abelian group and all $n \geq 0$. Such a group isomorphism has the form $\iota ([f]) = f^*(\Phi)$ for a certain distinguished class in the cohomology group $\Phi \in \overline{H}^n (X_g;G)$, (called a ``\emph{fundamental class}''). \end{theorem} \subsection{Derivation of the cohomology group theorem for connected CW-complexes.} For connected CW-complexes, $X$, the set $\left\langle X_g, K(G,n))\right\rangle$ of basepoint preserving homotopy classes maps from $X_g$ to Eilenberg-MacLane spaces $K(G, n)$ is replaced by the set of non-basepointed homotopy classes $[X, K(\pi,n)]$, for an Abelian group $G = \pi$ and all $n \geq 1$, because every map $X \to K(\pi,n)$ can be homotoped to take basepoint to basepoint, and also every homotopy between basepoint -preserving maps can be homotoped to be basepoint-preserving when the image space $K(\pi,n)$ is simply-connected. Therefore, the {\em natural group isomorphism} in {\bf Eq. (0.1)} becomes: \begin{equation} \iota : [X, K(\pi,n)] \cong \overline{H}^n (X;\pi) \end{equation} When $n =1$ the above group isomorphism results immediately from the condition that $\pi = G$ is an Abelian group. QED {\bf Remarks.} \begin{enumerate} \item A direct but very tedious proof of the (reduced) cohomology theorem can be obtained by constructing maps and homotopies cell-by-cell. \item An alternative, categorical derivation {\em via} duality and generalization of the proof of the cohomology group theorem (\cite{May1999}) is possible by employing the categorical definitions of a limit, colimit/cocone, the definition of Eilenberg-MacLane spaces (as specified under related), and by verification of the axioms for reduced cohomology groups (pp. 142-143 in Ch.19 and p. 172 of ref. \cite{May1999}). This also raises the interesting question of the propositions that hold for non-Abelian groups G, and generalized cohomology theories. \end{enumerate} \begin{thebibliography} {9} \bibitem{AllenHatcher2k1} Hatcher, A. 2001. \PMlinkexternal{Algebraic Topology.}{http://www.math.cornell.edu/~hatcher/AT/AT.pdf}, Cambridge University Press; Cambridge, UK., (Theorem 4.57, pp.393-405). \bibitem{May1999} May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago \end{thebibliography}