Mod-two Poincare duality ($H_k(M;{\mathbb{F}}_2)\cong H^{n-k}(M;{\mathbb{F}}_2)$)

For a closed $n$-dimensional manifold, Poincare duality holds over ${\mathbb{F}}_2$ without an orientation assumption. The mod-two fundamental class exists because orientation signs disappear. The resulting complementary-degree intersection pairing is a perfect pairing.