Degree modulo two (${\mathrm{deg}}_2$)

For a smooth map $f:X\to Y$ between compact connected $n$-manifolds and a regular value $y$, defineThe regular-value theorem makes the preimage discrete and compact, hence finite. A transverse path between regular values produces a compact one-manifold whose boundary is the two fibres, proving independence of $y$. The degree is invariant under smooth homotopy.