Mod-two degree of a map between closed manifolds (${\mathrm{deg}}_2(f)\in {\mathbb{F}}_2$)

For a map $f:M\to N$ between closed connected $n$-manifolds, the mod-two degree is defined by $f_{\ast }[M{]}_2={\mathrm{deg}}_2(f)[N{]}_2$. Every manifold has a mod-two orientation class, so no orientability assumption is needed. It is invariant under homotopy. A nonsurjective map has mod-two degree zero: mapping the fundamental classes into the local relative homology at a missed target point makes the source contribution zero but leaves the target local generator nonzero.