Homology groups determine cohomology groups

Statement
Suppose $$X$$ is a topological space. Then, knowledge of each of the (singular) homology groups $$H_n(X;\mathbb{Z})$$ up to isomorphism determines each of the cohomology groups $$H^n(X;\mathbb{Z})$$ up to isomorphism.

Facts used

 * 1) uses::Dual universal coefficients theorem

Proof
The proof follows directly from Fact (1), which provides an explicit computational procedure.