Show that a \(p\)-sheeted covering space projection \(M \to N\) has degree \(\pm p\), when \(M\) and \(N\) are connected closed orientable manifolds.
Solution
Let \(p:M\to N\) be a \(p\)-sheeted covering between connected closed orientable \(n\)-manifolds. Choose an evenly covered ball \(B\subset N\). Its inverse image is a disjoint union of \(p\) balls \[ p^{-1}(B)=B_1\amalg\cdots\amalg B_p, \] each mapped homeomorphically to \(B\).
For each \(i\), let \(\varepsilon_i=\pm1\) be the local orientation sign of \(p|_{B_i}\). Since \(M\) and \(N\) are connected and a covering map is locally a homeomorphism, the local sign is locally constant on \(M\) and therefore constant. Thus all \(\varepsilon_i\) equal a common sign \(\varepsilon\).
By Exercise 8’s local-degree formula, \[ \deg p=\sum_{i=1}^p\varepsilon_i=p\varepsilon. \] Hence \[ \boxed{\deg p=\pm p.} \]