The Universal Coefficient Theorem

The Universal Coefficient Theorem relates cohomology with arbitrary coefficients to homology with integer coefficients, providing a systematic way to compute cohomology from known homology groups.

The UCT for Cohomology

Definition

For an abelian group $G$ and a non-negative integer $n$ , the Ext group $\operatorname{Ext}^1_\mathbb{Z}(A, G)$ is the first derived functor of $\operatorname{Hom}(-, G)$ . Concretely, given a free resolution $0 \to F_1 \to F_0 \to A \to 0$ , we have $\operatorname{Ext}^1(A, G) = \operatorname{coker}(\operatorname{Hom}(F_0, G) \to \operatorname{Hom}(F_1, G))$ For finitely generated abelian groups, $\operatorname{Ext}^1(\mathbb{Z}, G) = 0$ and $\operatorname{Ext}^1(\mathbb{Z}/n\mathbb{Z}, G) \cong G/nG$ .

Theorem7.5Universal Coefficient Theorem for Cohomology

For any space $X$ and abelian group $G$ , there is a natural short exact sequence $0 \to \operatorname{Ext}^1(H_{n-1}(X), G) \to H^n(X; G) \to \operatorname{Hom}(H_n(X), G) \to 0$ This sequence splits (though not naturally), giving $H^n(X; G) \cong \operatorname{Hom}(H_n(X), G) \oplus \operatorname{Ext}^1(H_{n-1}(X), G)$

Computational Examples

ExampleCohomology of the Klein bottle

The Klein bottle $K$ has $H_0(K) \cong \mathbb{Z}$ , $H_1(K) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ , $H_2(K) = 0$ . With $G = \mathbb{Z}$ :

$H^0(K) \cong \operatorname{Hom}(\mathbb{Z}, \mathbb{Z}) = \mathbb{Z}$
$H^1(K) \cong \operatorname{Hom}(\mathbb{Z} \oplus \mathbb{Z}/2, \mathbb{Z}) \oplus \operatorname{Ext}^1(\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z} \oplus 0 = \mathbb{Z}$
$H^2(K) \cong \operatorname{Hom}(0, \mathbb{Z}) \oplus \operatorname{Ext}^1(\mathbb{Z} \oplus \mathbb{Z}/2, \mathbb{Z}) \cong 0 \oplus \mathbb{Z}/2 = \mathbb{Z}/2\mathbb{Z}$

Note that $H^2(K; \mathbb{Z}) \neq 0$ even though $H_2(K) = 0$ ; the torsion in $H_1$ contributes via the Ext term.

The UCT for Homology

Definition

The Tor group $\operatorname{Tor}_1^\mathbb{Z}(A, G)$ is the first derived functor of $- \otimes G$ . For finitely generated abelian groups, $\operatorname{Tor}_1(\mathbb{Z}, G) = 0$ and $\operatorname{Tor}_1(\mathbb{Z}/n\mathbb{Z}, G) \cong \{g \in G : ng = 0\}$ .

Theorem7.6Universal Coefficient Theorem for Homology

For any space $X$ and abelian group $G$ , there is a natural split short exact sequence $0 \to H_n(X) \otimes G \to H_n(X; G) \to \operatorname{Tor}_1(H_{n-1}(X), G) \to 0$

RemarkField coefficients

When $G = k$ is a field, both $\operatorname{Ext}^1(-, k)$ and $\operatorname{Tor}_1(-, k)$ vanish for torsion-free groups, and the universal coefficient theorems simplify dramatically: $H^n(X; k) \cong \operatorname{Hom}_k(H_n(X; k), k)$ is just the dual vector space. This explains why cohomology with field coefficients is often the cleanest setting for computations.