Let \(X\) be a topological space and \(G\) an abelian group. For each \(i\) there are natural short exact sequences \[ 0 \to H_i(X;\ZZ)\tensor G \to H_i(X; G) \to \Tor(H_{i-1}(X;\ZZ), G) \to 0 \] [@Hat02] and \[ 0 \to \Ext(H_{i-1}(X;\ZZ), G) \to H^i(X;G) \to \Hom(H_i(X;\ZZ), G) \to 0 \] [@Hat02]. Both sequences split, though not naturally, so \[ H_i(X;G) \cong \qty{H_i(X;\ZZ)\tensor G} \oplus \Tor(H_{i-1}(X;\ZZ), G), \qquad H^i(X;G) \cong \Hom(H_i(X;\ZZ), G) \oplus \Ext(H_{i-1}(X;\ZZ), G) .\]
If \(H_i(X;\ZZ)\) and \(H_{i-1}(X;\ZZ)\) are finitely generated with torsion subgroups \(T_i\) and \(T_{i-1}\), then \[ H^i(X;\ZZ) \cong \qty{H_i(X;\ZZ)/T_i} \oplus T_{i-1} \] [@Hat02]. If \(F\) is a field, then \(H^i(X;F) \cong \Hom_F(H_i(X;F), F)\) [@Hat02].