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(π,n) be Eilenberg-MacLane spaces for connected CW complexes X, Abelian groups π and integers n≥0. Let us also consider the set of non-basepointed homotopy classes [X,K(π,n)] of non-basepointed maps η : X → K(π,n) and the cohomolgy groups Hⁿ(X; π). Then, there exist the following natural isomorphisms:

--n [X, K (π,n )] ∼= H (X; π),

(0.1)

Proof. For a complete proof of this theorem the reader is referred to ref. [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(π,n); (source: ref [1]);
2.: When n = 1, and π is non-Abelian, one still has that [X,K(π, 1)]Hom(π₁(X),π)∕π, that is, the conjugacy class or representation of π₁ into π;
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.

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