# degree mod 2 of a mapping

Suppose that $M$ and $N$ are two 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. If $y\in N$ is a regular value of $f$, then we denote by $\#f^{-1}(y)$ 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

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

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

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

## References

• 1 John W. Milnor. . The University Press of Virginia, Charlottesville, Virginia, 1969.
Title degree mod 2 of a mapping DegreeMod2OfAMapping 2013-03-22 14:52:39 2013-03-22 14:52:39 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 57R35 degree mod 2 degree modulo 2 BrouwerDegree