# Kunneth formula for homology

## Contents

## Statement

Suppose and are topological spaces. We then have the following relation for the homology groups of , , and the product space .

For any and any module over a principal ideal domain for coefficients, we have:

Here, is torsion of modules over the ring .

## Related facts

- Kunneth formula for cohomology
- Universal coefficients theorem for homology
- Universal coefficients theorem for cohomology
- Dual universal coefficients theorem (computes cohomology in terms of homology)

## Particular cases

### Case of free modules

If all the homology groups are free (or more generally torsion-free) modules over , *and/or* all the homology groups , are free (or more generally torsion-free) modules over , then all the torsion part vanishes and we get:

In particular, if *all* and are free modules over and and denote the respective free ranks, and all these are finite, we obtain that:

Note that if is a field, then the above holds.

### Impact for ranks even in case of torsion

When is a principal ideal domain and all the homologies are finitely generated modules over , we can consider the *rank* as the rank of the torsion-free part of the homology modules. If denotes the free rank of the torsion-free part of , we get:

Note that this applies *even if* the homology modules have torsion.

In the special case that , the numbers are called Betti numbers, and we get:

In particular, this yields that Poincare polynomial of product is product of Poincare polynomials.

## Facts used

The Kunneth formula combines the Kunneth theorem and the Eilenberg-Zilber theorem.

## Proof

*Fill this in later*