Degree of an orientation-preserving or reversing diffeomorphism

The inverse $F^{-1}:N\to M$ is continuous, so for each compact $K\subset N$, the set $F^{-1}(K)$ is the continuous image of $K$ under that inverse and is compact. Thus $F$ is proper and [F1] defines its degree.

If $F$ preserves orientation, [F2] gives ${\int }_M{F}^{\ast }\omega ={\int }_N\omega$ for every compactly supported top form $\omega$; if it reverses orientation, it gives ${\int }_M{F}^{\ast }\omega =-{\int }_N\omega$. Uniqueness in [F1] yields respectively $\mathrm{deg}(F)=1$ and $\mathrm{deg}(F)=-1$. For dimension zero this compares the two supplied point-orientation signs; connectedness makes the sign constant. Empty manifolds are excluded, the zero form is harmless because an integral-one class supplies uniqueness, and no representatives or families are chosen.