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:
Eilenberg-MacLane spaces $$K(\pi,n)$$; (source: ref );
 * 1) In order to determine all cohomology operations one needs only to compute the cohomology of all


 * 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.