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# 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 $X$,
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*:

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

# References

- 1
May, J.P. 1999.
*A Concise Course in Algebraic Topology*, The University of Chicago Press: Chicago.,p.173.

## Mathematics Subject Classification

55N33*no label found*55N20

*no label found*

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