# Universal coefficient theorem for homology

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

## Contents

## Statement

### For coefficients in an abelian group

Suppose is an abelian group and is a topological space. The **universal coefficients theorem** relates the homology groups for with integral coefficients (i.e., with coefficients in ) to the homology groups with coefficients in .

The theorem comes in two parts:

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

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

### For coefficients in a module over a principal ideal domain

*Fill this in later*

## Related facts

- Universal coefficient theorem for cohomology
- Dual universal coefficient theorem
- Kunneth formula for homology
- Kunneth formula for cohomology

## Particular cases

### Case of free abelian groups

If is a free abelian group, then we get:

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