Universal coefficient theorem for cohomology

This is the cohomological form of the universal coefficient theorem, with Ext and Hom terms rather than the homological tensor and Tor terms.

For an abelian coefficient group

G

, cohomology fits into a split short exact sequence

0 \to Ext (H_{n - 1} (X; Z), G) \to H^{n} (X; G) \to Hom (H_{n} (X; Z), G) \to 0.

Table of contents
- First cohomology of the countably punctured plane Universal coefficient theorem for cohomology

First cohomology of the countably punctured plane ( $H^{1} (R^{2} ∖ {(n, 0) : n \geq 1}; Z) ≅ \prod_{n \geq 1} Z$ )

The connected open set

X = R^{2} ∖ {(n, 0) : n \geq 1}

retracts onto a locally finite graph having one independent loop around each puncture. Hence

H_{1} (X; Z) ≅ ⨁_{n \geq 1} Z

, while the universal coefficient theorem for cohomology gives

H^{1} (X; Z) ≅ Hom (⨁_{n \geq 1} Z, Z) ≅ \prod_{n \geq 1} Z,

an uncountable group.

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