= Degree as a sum of local degrees
{title2=$\deg f=\sum_{x\in f^{-1}(y)}\deg_x f$}
For a <continuous map> $f:M\to N$ of closed connected oriented $d$-dimensional <manifolds>, suppose $f^{-1}(y)=\{x_1,\ldots,x_s\}$ is finite. Then
$$
\deg f=\sum_{j=1}^s\deg_{x_j}f.
$$
By <excision>, the degree-$d$ <relative homology> group $H_d(M,M\setminus f^{-1}(y);\mathbb Z)$ is the direct sum of the local groups at $x_j$. The <fundamental class> maps to the tuple of local orientation generators. Mapping this tuple to $H_d(N,N\setminus\{y\};\mathbb Z)$ adds their local degrees. Naturality identifies the result with the image of $(\deg f)[N]$. If the fiber is empty, the same argument gives degree zero.
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