Universal coefficient theorem for homology

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For an algebraic version of the theorem, see Groupprops:Universal coefficient theorem for group homology

Statement

For coefficients in an abelian group

Suppose M is an abelian group and X is a topological space. The universal coefficients theorem relates the homology groups for X with integral coefficients (i.e., with coefficients in \mathbb{Z}) to the homology groups with coefficients in M.

The theorem comes in two parts:

First, it states that there is a natural short exact sequence:

\! 0 \to H_n(X; \mathbb{Z}) \otimes M \to H_n(X;M) \to \operatorname{Tor}(H_{n-1}(X;\mathbb{Z}),M) \to 0

Second, it states that this short exact sequence splits, so we obtain:

H_n(X;M) \cong (H_n(X;\mathbb{Z}) \otimes M) \oplus \operatorname{Tor}(H_{n-1}(X;\mathbb{Z}),M)

For coefficients in a module over a principal ideal domain

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Related facts

Particular cases

Case of free abelian groups

If H_{n-1}(X;\mathbb{Z}) is a free abelian group, then we get:

H_n(X;M) \cong H_n(X;\mathbb{Z}) \otimes M

As a corollary, if all the homology groups are free abelian, then the above holds for all n.