derivation of cohomology group theorem

0.1 Introduction

Let X_g be a general CW-complex and consider the set ⟨Xg,K (G, n)⟩ of basepoint preserving homotopy classes of maps from X_g to Eilenberg-MacLane spaces K(G,n) for n≥0, with G being an Abelian group.

Theorem 0.1. (Fundamental, [or reduced] cohomology theorem, [1]).

There exists a natural group isomorphism:

∼ --n ι : ⟨(Xg, K (G, n))⟩ = H (Xg; G)

(0.1)

for all CW-complexes X_g , with G any Abelian group and all n≥0. Such a group isomorphism has the form ι([f]) = f^∗(Φ) for a certain distinguished class in the cohomology group Φ ∈Hⁿ(X_g; G), (called a “ fundamental class”).

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

For connected CW-complexes, X, the set ⟨Xg, K (G, n))⟩ 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(π,n)], for an Abelian group G = π and all n≥1, because every map X → K(π,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(π,n) is simply-connected.

Therefore, the natural group isomorphism in Eq. (0.1) becomes:

--n ι : [X, K (π, n)] ∼= H (X; π)

(0.2)

When n = 1 the above group isomorphism results immediately from the condition that π = G is an Abelian group. QED Remarks.

1.: A direct but very tedious proof of the (reduced) cohomology theorem can be obtained by constructing maps and homotopies cell-by-cell.
2.: 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.

References

[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

"derivation of cohomology group theorem" is owned by bci1.
(view preamble)