winding number

Winding numbers are a basic notion in algebraic topology, and play animportant role in connection with analytic functions of a complex variable.Intuitively, given a closed curve $t\mapsto S(t)$ in an orientedEuclidean plane (such as the complex plane $ℂ$), and a point$p$ not in the image of $S$, the winding number (or index) of $S$ with respectto $p$ is the net number of times $S$ surrounds $p$. It is not altogethereasy to make this notion rigorous.

Let us take $ℂ$ for the plane. We havea continuous mapping $S:[a,b]\to ℂ$ where $a$ and $b$ are somereals with and $S(a)=S(b)$. Denote by $\theta (t)$ the angle from thepositive real axis to the ray from $z_0$ to $S(t)$. As $t$ moves from $a$ to$b$, we expect $\theta$ to increase or decrease by a multiple of $2\pi$,namely $2\omega \pi$ where $\omega$ is the winding number. One therefore thinksof using integration. And indeed, in the theory offunctions of a complex variable, it is proved that the value

$$\frac{1}{2\pi i}{\int }_S\frac{dz}{z-z_0}$$

is an integer and has the expected properties of a winding number around$z_0$. To define the winding number in this way, we need to assumethat the closed path $S$ is rectifiable (so that the pathintegral is defined). An equivalent condition is that the real and imaginaryparts of the function $S$ are of bounded variation.

But if $S$ is any continuous mapping $[a,b]\to ℂ$ having$S(a)=S(b)$, the winding number is still definable, without any integration.We can break up the domain of $S$ into a finite number of intervals such thatthe image of $S$, on any of those intervals, is contained in a disc whichdoes not contain $z_0$. Then $2\omega \pi$ emerges as a finite sum: the sumof the angles subtended at $z_0$ by the sides of a polygon.

Let $A$, $B$, and $C$ be any three distinct rays from $z_0$.The three sets

$$S^{-1}(A)\mathit{\hspace{1em}\hspace{1em}}{S}^{-1}(B)\mathit{\hspace{1em}\hspace{1em}}{S}^{-1}(C)$$

are closed in $[a,b]$, and they determinethe winding number of $S$ around $z_0$. This result can provide an alternativedefinition of winding numbers in $ℂ$, and a definition in some otherspaces also, but the details are rather subtle.

For one more variation on the theme, let $S$ be any topological spacehomeomorphic to a circle, and let$f:S\to S$ be any continuous mapping. Intuitively we expect that if a point$x$ travels once around $S$, the point $f(x)$ will travel around $S$ someintegral number of times, say $n$ times. The notion can be made precise.Moreover, the number $n$ is determined by the three closed sets

$$f^{-1}(a)\mathit{\hspace{1em}\hspace{1em}}{f}^{-1}(b)\mathit{\hspace{1em}\hspace{1em}}{f}^{-1}(c)$$

where $a$, $b$, and $c$ are any three distinct points in $S$.