derivation of cohomology group theorem for connected CW-complexesProofOfCohomologyGroupTheorem2008-08-01 11:34:352009-02-03 04:05:35Derivationcohomology group theoremfundamental classcohomology groupfundamental cohomology theoremderivation of the cohomology group theorem for connected CW complexesfundamental cohomology group theorem % this is the default PlanetMath preamble. as your knowledge
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\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}(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
\subsection{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}