Preliminary Data.

Let be a general CW complex and consider the set of basepoint preserving homotopy classes of maps from to Eilenberg-MacLane spaces for , with being an Abelian group.

Fundamental, (reduced) Cohomology Theorem, [1]. There exists a natural group isomorphism:

(0.1)

for all CW complexes , with any Abelian group and all . Such a group isomorphism has the form for a certain distinguished class in the cohomology group , (called a “fundamental class”).

Derivation of the Cohomology Group Theorem for Connected CW complexes. For connected CW complexes, , the set of basepoint preserving homotopy classes maps from to Eilenberg-MacLane spaces is replaced by the set of non-basepointed homotopy classes , for an Abelian group and all , because every map 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 is simply-connected.

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

(0.2)

When the above group isomorphism results immediately from the condition that 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.

Bibliography

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