 Solution

For closed connected oriented $d$-manifolds, the degree of a map between oriented manifolds is the integer defined by $f_{\ast }[M]=(\mathrm{deg}f)[N]$ in $H_d(N;\mathbb{Z})$. If $y\in N$ has finite fiber $\{x_1,\dots ,x_r\}$, choose disjoint small neighbourhoods $U_i$ containing no other points of that fiber. The map of local relative homology groupssends the local orientation generator at $x_i$ to $d_i$ times the one at $y$. This integer is its local degree of a continuous map, independent of the small neighbourhood by excision. Excision also identifies $H_d(M,M\setminus f^{-1}(y))$ with the direct sum of the local groups, and the global fundamental class restricts to their orientation generators. Naturality of the maps to relative homology therefore proves the degree as a sum of local degrees:The empty fiber gives degree zero. For smooth maps at a regular value, a local degree is the sign of the derivative determinant, but general continuous maps can have larger local multiplicities.