Brouwer degree

Suppose that $M$ and $N$ are two oriented differentiable manifoldsof dimension $n$ (without boundary) with $M$ compact and $N$ connected and suppose that$f:M\to N$ is a differentiable mapping. Let $Df(x)$ denote thedifferential mapping at the point $x\in M$,that is the linear mapping $Df(x):T_x(M)\to T_{f(x)}(N)$. Let $\mathrm{sign}Df(x)$ denote the signof the determinant of $Df(x)$. That is the sign is positive if $f$ preservesorientation and negative if $f$ reverses orientation.

Definition.

Let $y\in N$ be a regular value, then we define the Brower degree (or justdegree) of $f$ by

$$\mathrm{deg}f:=\sum _{x\in f^{-1}(y)}\mathrm{sign}Df(x).$$

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

Note that this degree coincides with the degree (http://planetmath.org/Degree5) as defined for maps of spheres.