degree mod 2 of a mapping

Suppose that $M$ and $N$ are two differentiable manifoldsof dimension $n$ (without boundary) with $M$ compact and $N$ connected andsuppose that$f:M\to N$ is a differentiable mapping. If $y\in N$ is a regularvalue of $f$, then we denote by $\mathrm{\#}{f}^{-1}(y)$ the number of points in $M$that map to $y$.

Definition.

Let $y\in N$ be a regular value, then we define the degree mod 2 of $f$ by

$${\mathrm{deg}}_{2}f:=\mathrm{\#}{f}^{-1}(y)(mod2).$$

It can be shown that the degree mod 2 does not depend on the regular value $y$ that we pick so that ${\mathrm{deg}}_{2}f$ is well defined.

This is similar to the Brouwer degree but does not require oriented manifolds. In fact ${\mathrm{deg}}_{2}f=\mathrm{deg}f(mod2)$.