If $X$ and $Y$ are compact, boundaryless smooth manifolds, $f\colon X \to \mathbb R^n$ and $g\colon Y \to \mathbb R^n$ with $f(X) \cap g(Y) = \emptyset$, and with $\dim (X) +\dim (Y) =n-1$. Define the $$L_2(f,g)=\deg_2(u_{f,g}) \text{ mod }2$$ where $u_{f,g}(x,y)=\dfrac{f(x)-g(x)}{|f(x)-g(x)|}$.
Show that $L_2(f,g)$ is well defined.
I could use some help starting this one. Just a direction to head. Thanks!