cohomology group theorem

The following theorem involves Eilenberg-MacLane spaces in relation to cohomology groups for connected 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 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 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 all Eilenberg-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.

Title	cohomology group theorem
Canonical name	CohomologyGroupTheorem
Date of creation	2013-03-22 18:14:43
Last modified on	2013-03-22 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