Vacuum-boundary degree as a defect charge ($Q=\mathrm{deg}(S^{d-1}\to S^{d-1})$)

A normalized vacuum field on a sphere surrounding a core can define $S^{d-1}\to S^{d-1}$ with integer topological degree. Nonzero degree obstructs its extension through the enclosed ball while staying in the vacuum manifold. This accounts for the winding number of a vortex and the spherical degree of a magnetic monopole, with the appropriate vacuum targets.