PlanetPhysics/Derivation of Cohomology Group Theorem

Introduction
Let $$X_g$$ be a general CW-complex and consider the set $$\left\langle{X_g, K(G,n)}\right\rangle$$ of basepoint preserving homotopy classes of maps from $$X_g$$ to Eilenberg-MacLane spaces $$K(G, n)$$ for $$n \geq 0 $$, with $$G$$ being an Abelian group.

\begin{theorem}(Fundamental, [or reduced] cohomology \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}, }).

There exists a natural group isomorphism: $$ \iota : \left\langle(X_g, K(G,n))\right\rangle \cong \overline{H}^n (X_g;G) $$ for all CW-complexes $$X_g$$, with $$G$$ any Abelian group and all $$n \geq 0$$. Such a group isomorphism has the form $$\iota ([f]) = f^*(\Phi)$$ for a certain distinguished class in the cohomology group $$\Phi \in \overline{H}^n (X_g;G)$$, (called a ""fundamental class ).

\end{theorem}

Derivation of the cohomology group theorem for connected CW-complexes.
For connected CW-complexes, $$X$$, the set $$\left\langle X_g, K(G,n))\right\rangle$$ of basepoint preserving homotopy classes maps from $$X_g$$ to Eilenberg-MacLane spaces $$K(G, n)$$ is replaced by the set of non-basepointed homotopy classes $$[X, K(\pi,n)]$$, for an Abelian group $$G = \pi$$ and all $$n \geq 1$$, because every map $$X \to K(\pi,n)$$ can be homotoped to take basepoint to basepoint, and also every homotopy between basepoint -preserving maps can be homotoped to be basepoint-preserving when the image space $$K(\pi,n)$$ is simply-connected.

Therefore, the natural group isomorphism in {\mathbf Eq. (0.1)} becomes: $$ \iota : [X, K(\pi,n)] \cong \overline{H}^n (X;\pi) $$

When $$n =1$$ the above group isomorphism results immediately from the condition that $$\pi = G$$ is an Abelian group. QED {\mathbf Remarks.}


 * 1) A direct but very tedious proof of the (reduced) cohomology theorem can be obtained by constructing maps and homotopies cell-by-cell.

This also raises the interesting question of the propositions that hold for non-Abelian groups G, and generalized cohomology theories.
 * 1) An alternative, categorical derivation via duality and generalization of the proof of the cohomology group theorem  is possible by employing the categorical definitions of a limit, colimit/cocone, the definition of Eilenberg-MacLane spaces (as specified under related), and by verification of the axioms for reduced cohomology groups (pp. 142-143 in Ch.19 and p. 172 of ref. ).

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