PlanetPhysics/Cohomology Group Theorem

The following [[../Formula/|theorem]] involves Eilenberg-MacLane spaces in [[../Bijective/|relation]] to [[../CohomologyTheoryOnCWComplexes/|cohomology groups]] for connected CW-complexes.

\begin{theorem}

Cohomology group theorem for connected CW-complexes ():

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

$[X,K(\pi ,n)]\cong \stackrel{n}{\stackrel{‾}{H}}(X;\pi ),$

\end{theorem}

\begin{proof} For a complete proof of this theorem the reader is referred to ref. \end{proof}

Related remarks:

1. In order to determine all cohomology [[../Cod/|operations]] one needs only to compute the cohomology of all

Eilenberg-MacLane spaces $K(\pi ,n)$; (source: ref );

1. When $n=1$, and $\pi$ is [[../AbelianCategory3/|non-Abelian]], one still has that $[X,K(\pi ,1)]\cong Hom({\pi }_1(X),\pi )/\pi$, that is, the conjugacy class or [[../CategoricalGroupRepresentation/|representation]] of ${\pi }_1$ into $\pi$;

1. A derivation of this result based on the fundamental cohomology theorem is also attached.