A closed $n$-manifold has no homology or integral cohomology above degree $n$.Suggest a slogan
Let \(M\) be a closed connected \(n\)-manifold. Then \(H_i(M;R) = 0\) for \(i > n\) and every coefficient ring \(R\) [@Hat02], and \(H^i(M;\ZZ) = 0\) for \(i>n\) by the universal coefficient theorem [@Hat02], since \(H_n(M;\ZZ)\) is free.
In degree \(n\), \(H^n(M;\ZZ)\cong\ZZ\) if \(M\) is orientable and \(H^n(M;\ZZ)\cong\ZZ/2\) otherwise, by the universal coefficient theorem applied to \(H_n(M;\ZZ)\) and the torsion subgroup of \(H_{n-1}(M;\ZZ)\) [@Hat02].