Degrees of maps of the zero-sphere
= Degrees of maps of the zero-sphere
{title2=$\deg f\in\{-1,0,1\}$}
Define <mapping degree> for $S^0$ using its rank-one <reduced homology> group $\widetilde H_0(S^0;\mathbb Z)$. Its generator is the difference of its two points. The four maps of that two-point set consist of the identity, the interchange, and two constant maps; their degree multipliers are $1,-1,0,0$. Thus <sphere maps of arbitrary integer degree> require positive <sphere> dimension.