Degree of a continuous map

Summary

Let $f$ be a continuous map between closed connected oriented manifolds of the same dimension. Its degree is an integer that counts, with signs, how many times the source wraps around the target. For maps from the $n$-sphere to itself it is defined by the action on the top homology group $H_n(S^n; \Z) \cong \Z$. The degree is a homotopy invariant, and for maps to a sphere it is a complete invariant by the Hopf degree theorem. It underlies the proofs of the Brouwer fixed point theorem, the hairy ball theorem and the fundamental theorem of algebra.