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. Dual universal coefficients theorem

Proof

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