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 $f:M\to N$ is a differentiable mapping. If $y\in N$ is a regular value 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 ${\mathrm{deg}}_{2}f=\mathrm{deg}f(mod2)$.

Title	degree mod 2 of a mapping
Canonical name	DegreeMod2OfAMapping
Date of creation	2013-03-22 14:52:39
Last modified on	2013-03-22 14:52:39
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 57R35
Synonym	degree mod 2
Synonym	degree modulo 2
Related topic	BrouwerDegree