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For a closed path avoiding zero, choose a continuous lift $\gamma (t)/|\gamma (t)|=e^{i\theta (t)}$ and set $w(\gamma ;0)=(\theta (1)-\theta (0))/(2\pi )$. Conjugation replaces $\theta$ by $-\theta$, as does reversing the path, so both winding numbers are $-w(\gamma ;0)$.

Homotopy invariance of winding number states that a homotopy $H(s,t)$ through closed paths in ${\mathbb{C}}^{\ast }$ preserves winding number. Uniform continuity divides the $s$ interval so successive paths differ pointwise by less than the minimum modulus of the earlier path; the supplied perturbation lemma makes their winding numbers equal. Chaining the subdivisions proves the result.

If no $z\in S^1$ obeyed $z^n=-g(z)$, the loops $z^n$ and $-g(z)$ could be joined without crossing zero by the straight-line homotopy, so they would have equal winding number. The first has winding $n$. Since $g{|}_{S^1}$ extends over the disc, it is null-homotopic and has winding $0$; multiplication by $-1$ does not change it. This contradicts $n\ne 0$.

The Poincare index of a planar vector field around a simple closed curve containing no zero on the curve is the winding number of the vector-field direction along that curve.

Single-valuedness of the condensate field gives total complex argument change $2\pi q$ on a closed loop avoiding zeros, where $q$ is an integer winding number. Integrating the superfluid velocity around that loop gives the displayed circulation. It is unchanged by smooth deformations of the loop that do not cross a quantum vortex.

A quantum vortex has integer winding number of a condensate's complex argument. For atomic mass $m$, its superfluid velocity gives circulation $2\pi \hslash q/m$ for integer $q$. A straight vortex has a density-depleted core and, outside it, tangential velocity $q\hslash /(mr)$. Its long-range flow energy is proportional to $q^2\log (R/a)$, while its angular momentum has the sign of $q$.

Every smooth complex line bundle on ${\mathbb{CP}}^1$ is isomorphic to exactly one twisting bundle $\mathcal{O}(k)$. Trivializing on the two standard affine charts leaves a transition function on ${\mathbb{C}}^{\ast }$; its winding number determines $k$, and a partition of unity removes the zero-winding factor.

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.