For a map \(f: M \to N\) between connected closed orientable \(n\)-manifolds, suppose there is a ball \(B \subset N\) such that \(f^{-1}(B)\) is the disjoint union of balls \(B_i\) each mapped homeomorphically by \(f\) onto \(B\). Show the degree of \(f\) is \(\sum_i \varepsilon_i\) where \(\varepsilon_i\) is \(+1\) or \(-1\) according to whether \(f: B_i \to B\) preserves or reverses local orientations induced from given fundamental classes \([M]\) and \([N]\).
Solution
Choose \(y\in\operatorname{int}B\). Then \[ f^{-1}(y)=\{x_1,\ldots,x_r\},\qquad x_i\in\operatorname{int}B_i. \] Excision identifies \[ H_n(M,M-f^{-1}(y)) \cong \bigoplus_i H_n(B_i,B_i-\{x_i\}) \cong \bigoplus_i\mathbb Z, \] while \[ H_n(N,N-\{y\})\cong\mathbb Z. \] The image of the fundamental class \([M]\) in the direct sum of local homology groups is the tuple of local orientation generators. On the \(i\)th summand, the map induced by \(f|_{B_i}:B_i\to B\) is multiplication by \[ \varepsilon_i= \begin{cases} +1,&f|_{B_i}\text{ preserves the chosen local orientations},\\ -1,&f|_{B_i}\text{ reverses them}. \end{cases} \] Hence the image of \([M]\) in \(H_n(N,N-\{y\})\) is \[ \left(\sum_i\varepsilon_i\right)\mu_y. \] But the image of \([M]\) under \(f_*\) is also \(\deg(f)[N]\), whose local image at \(y\) is \(\deg(f)\mu_y\). Therefore \[ \boxed{\deg(f)=\sum_i\varepsilon_i.} \]