PlanetPhysics/Derivation of Cohomology Group Theorem

Introduction

Let $X_g$ be a general CW-complex and consider the set $⟨X_g,K(G,n)⟩$ of basepoint preserving [[../HomotopyCategory/|homotopy classes of maps]] from $X_g$ to Eilenberg-MacLane spaces $K(G,n)$ for $n\ge 0$, with $G$ being an [[../TrivialGroupoid/|Abelian group]].

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

There exists a natural [[../TrivialGroupoid/|group]] [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]]:

$\iota :⟨(X_g,K(G,n))⟩\cong \stackrel{n}{\stackrel{‾}{H}}(X_g;G)$

for all CW-complexes $X_g$ , with $G$ any Abelian group and all $n\ge 0$. Such a group isomorphism has the form $\iota ([f])=f^{*}(\mathrm{\Phi })$ for a certain distinguished class in the cohomology group $\mathrm{\Phi }\in \stackrel{n}{\stackrel{‾}{H}}(X_g;G)$, (called a ""fundamental class ).

\end{theorem}

Derivation of the cohomology group theorem for connected CW-complexes.

For connected CW-complexes, $X$, the set $⟨X_g,K(G,n))⟩$ of basepoint preserving [[../ThinEquivalence/|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 $n\ge 1$, 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:

$\iota :[X,K(\pi ,n)]\cong \stackrel{n}{\stackrel{‾}{H}}(X;\pi )$

When $n=1$ the above group isomorphism results immediately from the condition that $\pi =G$ is an Abelian group. [[../LQG2/|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.

1. An alternative, categorical derivation via [[../GroupoidSymmetries/|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 [[../CohomologyTheoryOnCWComplexes/|cohomology groups]] (pp. 142-143 in Ch.19 and p. 172 of ref. ).

This also raises the interesting question of the [[../Predicate/|propositions]] that hold for [[../AbelianCategory3/|non-Abelian]] groups G, and generalized [[../NoncommutativeGeometry4/|cohomology theories]].

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