Consider any two oriented connected closed smooth manifolds
$M, N$ of same dimension $n \ge 1$. Let $f : M \to N$ be a smooth map. Recall that a regular value of $f$ is a point $y \in Y$ such that the differential $D_xf^*$ of $f$ is invertible at each point of
$x \in f^{−1}(y)$. It is a basic result due to Sard that the complement in $N$ of the set of regular values has Lebesgue measure zero; in particular, the set of regular values of $f$ is dense , and so non-empty.
If $y \in N$ is a regular value of $f$, then $f$ is a local diffeomorphism around every $x \in f^{−1}(y)$; it follows that $f^{−1}(y)$ is discrete in $M$; since this fiber is also compact, it is
therefore finite. For $x \in f^{−1}(y)$ define $\epsilon_x(f)$ to be $1$ if $f$ is orientation preserving at $x$ and $−1$ if f is orientation reversing at $x$.
Definition: The local degree of $f$ at a regular value $y$ is the integer
$$\text{deg}(y)(f) = \sum_{x \in f^{−1}(y)} \epsilon_x(f) $$
In other words it's the difference of the number of points in the fiber at $y$ where $f$ in a neighbourhood of $x$ is orientation preserving minus the number of points where $f$ is locally orientation reversing.
Non assume that $f: M \to N$ is in addition a finite covering in topological sense. Then one can also define purely topologically the degree of $f$ to be the number of elements in the fiber: $ \vert f^{−1}(y) \vert $
Question: Do these two definition coincide (up to $\pm 1 $ sign)? Or in equivalent terms, is a honest covering map (ie without any ramificational defects) always locally orientation preserving?
Best Answer
After having fixed orientations on $M$ and $N$ the correct statement becomes for every $y \in N$: $$ \vert \text{deg}(y)(f) \vert = \vert f^{−1}(y) \vert $$
That's because the function $M \to \{\pm1 \}$ which associates to every $x \in M$ the sign of the determinant of derivative $D_x f$ (-which is nowhere zero, because a covering is a local diffeomeorphism) is a locally constant function and because $M$ was assumed to be connected, this function is even constant.