Talk:PlanetPhysics/Cohomology Group Theorem

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\begin{document}

 The following \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} involves Eilenberg-MacLane spaces in \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} to \htmladdnormallink{cohomology groups}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} for connected CW-complexes.

\begin{theorem}
Cohomology group theorem for connected CW-complexes (\cite{MJP1999}):

Let $K(\pi,n)$ be Eilenberg-MacLane spaces for connected
CW complexes $X$,
\htmladdnormallink{Abelian groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\pi$ and integers $n \geq 0$. Let us also consider the set of non-basepointed \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} classes $[X, K(\pi,n)]$ of non-basepointed maps $\eta :X \to K(\pi,n)$ and the cohomolgy groups $\overline{H}^n(X;\pi)$. Then, there exist the following \emph{\htmladdnormallink{natural isomorphisms}{http://planetphysics.us/encyclopedia/NaturalIsomorphism.html}}:

\begin{equation}
[X, K(\pi,n)] \cong \overline{H}^n(X;\pi),
\end{equation}

\end{theorem}

\begin{proof}
For a complete proof of this theorem the reader is referred to ref.
\cite{MJP1999}
\end{proof}
\subsection{Related remarks:}

\begin{enumerate}
\item In order to determine all cohomology \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} one needs only to compute the cohomology of all
Eilenberg-MacLane spaces $K(\pi,n)$; (source: ref \cite{MJP1999});

\item When $n = 1$, and $\pi$ is \emph{\htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}}, one still has that $[X,K(\pi ,1)] \cong Hom(\pi_1(X),\pi)/\pi$, that is, the conjugacy class or \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of $\pi_1$ into $\pi$;

\item A derivation of this result based on the fundamental cohomology theorem is also attached.
\end{enumerate}


\begin{thebibliography}{9}

\bibitem{MJP1999}
May, J.P. 1999. \emph{A Concise Course in Algebraic Topology}, The University of Chicago Press: Chicago.,p.173.
\end{thebibliography} 

\end{document}