Let \(n\geq 1\) and let \(f\colon S^n \to S^n\) be a continuous map. Since \(H^n(S^n;\ZZ) \cong \ZZ\), the induced homomorphism \(f^*\colon H^n(S^n;\ZZ) \to H^n(S^n;\ZZ)\) is multiplication by a unique integer \(d\), that is, \(f^*(u) = d\,u\) for every \(u\in H^n(S^n;\ZZ)\). The degree of \(f\) is \(\deg f \coloneqq d\).
Let \(n\geq 1\) and \(f, g\colon S^n\to S^n\) be continuous.
If \(f\) and \(g\) are homotopic, then \(\deg f = \deg g\).
\(\deg f\) equals the integer \(d'\) with \(f_*(v) = d'\,v\) for every \(v\in H_n(S^n;\ZZ)\).
[@Hat02].