cohomology group theorem | Physics Library

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 , Abelian groups $\pi$ and integers $n {\geqslant}0$ . 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:

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

(0.1)

Proof. For a complete proof of this theorem the reader is referred to ref. [1] $\qedsymbol$

Related remarks:

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]);
When , 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 $\pi$ ;
A derivation of this result based on the fundamental cohomology theorem is also attached.

Bibliography

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

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