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derivation of cohomology group theorem for connected CW-complexes
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(Derivation)
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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.
Theorem 0.1 (Fundamental, [or reduced] Cohomology Theorem, [ 1]) .
There exists a natural group isomorphism: \begin{equation} \iota : \left\langle{X_g, K(G,n)}\right\rangle \cong \overline{H}^n (X_g;G) \end{equation}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).
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 Eq. (0.1) becomes: \begin{equation} \iota : [X, K(\pi,n)] \cong \overline{H}^n (X;\pi) \end{equation} When $n =1$ the above group isomorphism results immediately from the condition that $\pi = G$ is an Abelian group. QED
- A direct but very tedious proof of the (reduced) cohomology theorem can be obtained by constructing maps and homotopies cell-by-cell.
- An alternative, categorical derivation via duality and generalization of the proof of the cohomology group theorem ([2]) 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. [2]). This also raises the interesting question of the propositions that hold for non-Abelian groups G, and generalized cohomology theories.
- 1
- Hatcher, A. 2001. Algebraic Topology., Cambridge University Press; Cambridge, UK., (Theorem 4.57, pp.393-405).
- 2
- May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
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Cross-references: theories, non-abelian groups, propositions, CH, axioms, colimit, limit, definitions, cohomology group theorem, duality, derivation, categorical, theorem, cohomology, reduced, proof, QED, image, connected, group isomorphism, abelian group, Eilenberg-MacLane spaces, maps, classes, homotopy, basepoint, CW-complex
There are 24 references to this entry.
This is version 48 of derivation of cohomology group theorem for connected CW-complexes, born on 2008-08-01, modified 2009-02-03.
Object id is 10899, canonical name is ProofOfCohomologyGroupTheorem.
Accessed 1930 times total.
Classification:
| AMS MSC: | 55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories) | | | 55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology) | | | 55P20 (Algebraic topology :: Homotopy theory :: Eilenberg-Mac Lane spaces) | | | 18-00 (Category theory; homological algebra :: General reference works ) |
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Pending Errata and Addenda
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