 Degree of a continuous mapping

For $m\ge 1$, a continuous map $f:S^m\to S^m$ induces multiplication by one integer on $H_m(S^m;\mathbb{Z})\cong \mathbb{Z}$. That integer is its mapping degree:It records the signed number of times the domain wraps around the target, made precise by the degree as a sum of local degrees. The degree is invariant under homotopy, and multiplicativity of mapping degree follows by composing the induced homomorphisms.

The circle map $z\mapsto z^r$ has degree $r$ for every integer $r$. The antipodal map on $S^m$ has degree $(-1{)}^{m+1}$. A constant map has degree zero. The same fundamental-class definition gives the degree of a map between oriented manifolds for arbitrary connected closed manifolds of equal dimension with chosen orientations. For $S^0$, use reduced homology instead of its rank-two ordinary $H_0$; the degrees of maps of the zero-sphere are only $-1,0,1$.

The degree of a continuous mapping refers to a topological invariant that describes the number of times a continuous function covers its target space. This concept is most commonly applied in the context of mappings between spheres or between manifolds.