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For a smooth map $\varphi :M_1\to M_2$ between connected oriented closed manifolds of equal dimension and a volume form $\omega$ on $M_2$, the topological degree is defined byThe standard area form on the unit sphere isSince ${\int }_{S^2}\omega =\mathrm{vol}(S^2)$, this gives

Finite Euclidean Yang-Mills action requires $F\to 0$ sufficiently rapidly at spatial infinity. Consequently the connection approaches a pure gauge,under the convention of the question. Compactifying the asymptotic boundary identifies it with $S_{\infty }^3$, while $SU(2)$ is itself topologically $S^3$. Thus $g{|}_{S_{\infty }^3}$ has an integer topological degree.

By $C_2=dY$ and Stokes theorem, the instanton number isup to the common simultaneous choice of trace and orientation signs. Smoothness and finite action are imposed in the interior, and $F=\pm {}^{\star }F$ selects an instanton or anti-instanton representative of the topological sector.

Let $\varphi :S^{2q-1}\to S^q$ and precompose with a sphere self-map of topological degree $d$. The resulting map of mapping cones is the identity on the bottom $q$-cell and has degree $d$ on the top $2q$-cell. Pulling back the defining cup product relation for the Hopf invariant multiplies its coefficient by $d$. In particular the element $d\eta \in {\pi }_3(S^2)$ has Hopf invariant $d$. Postcomposition by a degree-$d$ map of the target sphere instead multiplies the invariant by $d^2$.

A closed target $d$-form $\omega$ pulls back under a spacetime field $\varphi$ to a closed form $\alpha ={\varphi }^{\ast }\omega$. Its dual defines an identically conserved topological current. For a compact oriented spatial $d$-manifold and an equally dimensional oriented target with $\int \omega =1$, the spatial integral is the topological degree. Conservation follows without the field equations; changing boundary flux can change the charge.

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.