singular homological degree

For \(n \in \mathbb{N}_0\) and a continuous map \(f: S^n \rightarrow S^n\), the singular homological degree is defined as

\begin{align*} \varphi(f) \in \mathbb{Z} \end{align*}

where \(H_n\) is the singulary homology (topology)

\begin{align*} \varphi: \mathrm{Hom}_{\mathrm{Ab}}(\mathrm{H}_n(S^n), \mathrm{H}_n(S^n)) \rightarrow \mathbb{Z} \end{align*}

is the canonical group-isomorphism (cf. endomorphism ring of the integers as integers)