Let $f:X\rightarrow Y$ be a local diffeomorphism, where $X^n$, $Y^m$ are compact and orientable manifolds. I'd like to show that $deg(f)=\text{#}(f^{-1}(z))$ or $deg(f)=-\text{#}(f^{-1}(z))$, for $z \in Y$, where # means the number of elements in the set.
It is clear to me that, in this case, $f$ must be a cover with a finite number of leaves, but I do not know how to go further on this.
It was quite a dumb question. It is enough to prove that, for any $y$ regular value of $f$, $deg(f,y)=deg(f)$.
Its the Theorem 13, of Elon Lages Lima's homology book.