# 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 $K(\pi,n)$ be Eilenberg-MacLane spaces for connected CW complexes (http://planetmath.org/CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams) $X$, 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 (http://planetmath.org/GroupCohomology) $\overline{H}^{n}(X;\pi)$. Then, there exist the following natural isomorphisms:

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

## 0.1 Related remarks:

1. 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. 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. 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 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