Universal coefficient theorem for homology
For an algebraic version of the theorem, see Groupprops:Universal coefficient theorem for group homology
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
- Universal coefficient theorem for cohomology
- Dual universal coefficient theorem
- Kunneth formula for homology
- Kunneth formula for cohomology
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 .