PlanetPhysics/Cohomology Group Theorem

The following theorem involves Eilenberg-MacLane spaces in relation to 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 $X$, Abelian groups $\pi$ and integers $n\geq 0$. Let us also consider the set of non-basepointed homotopy 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 natural isomorphisms:

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

\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 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 non-Abelian, one still has that $[X,K(\pi ,1)]\cong Hom(\pi _{1}(X),\pi )/\pi$, that is, the conjugacy class or representation of $\pi _{1}$ into $\pi$;

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