cohomology group theorem

The following theorem involves Eilenberg-MacLane spaces in relation to cohomology groups forconnected CW-complexes.

Theorem 0.1.

Cohomology group theorem for connected CW-complexes ([1]):

Let $K\mathit{}\mathrm{(}\pi \mathrm{,}n\mathrm{)}$ be Eilenberg-MacLane spaces for connectedCW complexes (http://planetmath.org/CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams) $X$,Abelian groups $\pi$ and integers $n\mathrm{\ge }\mathrm{0}$. Let us also consider the set of non-basepointed homotopy classes $\mathrm{[}X\mathrm{,}K\mathit{}\mathrm{(}\pi \mathrm{,}n\mathrm{)}\mathrm{]}$ of non-basepointed maps $\eta \mathrm{:}X\mathrm{\to }K\mathit{}\mathrm{(}\pi \mathrm{,}n\mathrm{)}$ and the cohomolgy groups (http://planetmath.org/GroupCohomology) ${\overline{H}}^n\mathit{}\mathrm{(}X\mathrm{;}\pi \mathrm{)}$. Then, there exist the following natural isomorphisms:

$$[X,K(\pi ,n)]\cong {\overline{H}}^n(X;\pi ),$$(0.1)

0.1 Related remarks:

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

2. 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 $\pi$;

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