Degree of a map

Definition

Let $M$ and $N$ be compact connected orientable manifolds. Suppose we choose fundamental classes for both $M$ and $N$. Then given a continuous map $f:M\to N$, the degree of $f$ is defined as the unique integer $d$ such that the fundamental class of $M$ gets mapped to $d$ times the fundamental class of $N$.

We can talk unambiguously about the degree of a self-map of a compact connected orientable manifold, without needing to chose a fundamental class for it. For different manifolds, making different choices of orientation may change the value of the degree upto sign; in magnitude it remains the same.