cohomology group theorem
The following theorem involves EilenbergMacLane spaces in relation^{} to cohomology groups^{} for connected CWcomplexes^{}.
Theorem 0.1.
Cohomology group theorem for connected CWcomplexes ([1]):
Let $K\mathit{}\mathrm{(}\pi \mathrm{,}n\mathrm{)}$ be EilenbergMacLane spaces for connected CW complexes (http://planetmath.org/CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams) $X$, Abelian groups^{} $\pi $ and integers $n\mathrm{\ge}\mathrm{0}$. Let us also consider the set of nonbasepointed homotopy classes $\mathrm{[}X\mathrm{,}K\mathit{}\mathrm{(}\pi \mathrm{,}n\mathrm{)}\mathrm{]}$ of nonbasepointed 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 all EilenbergMacLane spaces $K(\pi ,n)$; (source: ref [1]);

2.
When $n=1$, and $\pi $ is nonAbelian^{}, 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.
References
 1 May, J.P. 1999. A Concise Course in Algebraic Topology, The University of Chicago Press: Chicago.,p.173.
Title  cohomology group theorem 
Canonical name  CohomologyGroupTheorem 
Date of creation  20130322 18:14:43 
Last modified on  20130322 18:14:43 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  48 
Author  bci1 (20947) 
Entry type  Theorem 
Classification  msc 55N33 
Classification  msc 55N20 
Synonym  fundamental cohomology theorem 
Related topic  GroupCohomology 
Related topic  EilenbergMacLaneSpace 
Related topic  HomotopyGroups 
Related topic  HomotopyCategory 
Related topic  GroupCohomologyTopologicalDefinition 
Related topic  TangentialCauchyRiemannComplexOfCinftySmoothForms 
Related topic  HomologyTopologicalSpace 
Related topic  ProofOfCohomologyGroupTheorem 
Related topic  OmegaSpectrum 
Related topic  ACRcomplex 
Defines  conjugacy class or representation of ${\pi}_{1}$ into $\pi $ 
Defines  set of based homotopy classes of based maps 