Suppose $f: X \to Y$ is a homeomorphism and $X,Y$ are orientable manifolds. For $U \subset X$ and $V \subset Y$ both open balls homeomorphic to $\mathbb{R}^n,$ we can define the local orientation at a point $p \in U$ as the relative homology group $H_n(U, U -p),$ and similarly for $q \in V$.
Since $X$ and $Y$ are orientable, their fundamental class induces a choice of generator for each group $H_n(U, U-p),$ and similarly again for $q \in V$.
Suppose now that $f(x)=y$ for fixed $x \in U, y \in V.$ Then $f$ induces a map $f_*: H_n(U, U-x) \to H_n(V, V-y).$ Since these groups are both isomorphic to $\mathbb{Z}$ with a choice of generator inherited from the orientation of $X$ and $Y,$ it follows that $f_*$ is just $\pm 1.$ We call this induced map into $\{\pm 1\}$ the local degree at $x.$
I want to prove the following: the local degree is constant on $U.$