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