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.

Facts

Self-maps

• For a sphere of dimension at least 1, the degree classifies the homotopy type of self-maps. This is essentially because spheres have only the highest homology and no other homology, and their cohomology rings need to satisfy no other constraints
• For a torus, every degree can be realized as the degree of a self-map, but the degree does not classify the homotopy type of the map.

Maps between manifolds

• There exists a degree one map from any compact connected orientable manifold to the sphere.
• In general, given two compact connected orientable manifolds, there may or may not exist degree one maps between them. In fact, existence of degree one maps arranges the compact connected orientable manifolds (upto homotopy) in a partial order.
• The degree of a covering map is (upto sign) its usual degree in the sense of a covering map.

For compact connected Lie groups

Further information: degree homomorphism of a compact connected Lie group