PlanetPhysics/Cohomology Group Theorem

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

\begin{theorem}

Cohomology group theorem for connected CW-complexes ([1]):

Let be Eilenberg-MacLane spaces for connected CW complexes , Abelian groups and integers . Let us also consider the set of non-basepointed homotopy classes of non-basepointed maps and the cohomolgy groups . Then, there exist the following natural isomorphisms:

\end{theorem}

\begin{proof} For a complete proof of this theorem the reader is referred to ref. [1] \end{proof}

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  1. In order to determine all cohomology operations one needs only to compute the cohomology of all

Eilenberg-MacLane spaces ; (source: ref [1]);

  1. When , and is non-Abelian, one still has that , that is, the conjugacy class or representation of into ;
  1. A derivation of this result based on the fundamental cohomology theorem is also attached.

All Sources

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[1]

References

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  1. 1.0 1.1 1.2 1.3 May, J.P. 1999. A Concise Course in Algebraic Topology , The University of Chicago Press: Chicago.,p.173.