Brouwer degree

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

Definition.

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

 $\operatorname{deg}f:=\sum_{x\in f^{-1}(y)}\operatorname{sign}Df(x).$

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

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

References

• 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
Title Brouwer degree BrouwerDegree 2013-03-22 14:52:37 2013-03-22 14:52:37 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 57R35 degree DegreeMod2OfAMapping