= Local degree of a continuous map
{title2=$\deg_x f$}
= Local degree
{synonym}
Let $f:M\to N$ be a <continuous map> between oriented $d$-dimensional <manifolds>, with $d\geq1$. Suppose $x$ is an isolated point of the fiber over $y=f(x)$. Choose an open neighborhood $U$ with $U\cap f^{-1}(y)=\{x\}$. The map of pairs
$$
(U,U\setminus\{x\})\longrightarrow(N,N\setminus\{y\})
$$
induces a map between rank-one degree-$d$ <local homology> groups. Relative to the <local orientation of a manifold>, this map is multiplication by an integer, the local degree. <Excision> makes it independent of the neighborhood. A <smooth map between manifolds> with invertible derivative at $x$ has local degree $\operatorname{sign}\det Df_x$, by the <inverse function theorem> and the chosen orientations.
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