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.

If $S$ is compact and $V$ is a smooth vector field, thenis smooth and well-defined by the Hopf-Rinow theorem. The maps ${\varphi }_t(p)={\exp }_p(tV(p))$ form a smooth homotopy from the identity to $\varphi$, so ${\mathrm{deg}}_2(\varphi )=1$.