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winding number (Definition)

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

Let us take $ \mathbb{C}$ for the plane. We have a continuous mapping $ S:[a,b]\to \mathbb{C}$ where $ a$ and $ b$ are some reals with $ a<b$ and $ S(a)=S(b)$. Denote by $ \theta(t)$ the angle from the positive 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 thinks of using integration. And indeed, in the theory of functions of a complex variable, it is proved that the value

$\displaystyle \frac{1}{2\pi i} \int_S \frac{dz}{z-z_0}$
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 assume that the closed path $ S$ is rectifiable (so that the path integral is defined). An equivalent condition is that the real and imaginary parts of the function $ S$ are of bounded variation.

But if $ S$ is any continuous mapping $ [a,b]\to \mathbb{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 that the image of $ S$, on any of those intervals, is contained in a disc which does not contain $ z_0$. Then $ 2\omega\pi$ emerges as a finite sum: the sum of 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

$\displaystyle S^{-1}(A)\qquad S^{-1}(B)\qquad S^{-1}(C)$
are closed in $ [a,b]$, and they determine the winding number of $ S$ around $ z_0$. This result can provide an alternative definition of winding numbers in $ \mathbb{C}$, and a definition in some other spaces also, but the details are rather subtle.

For one more variation on the theme, let $ S$ be any topological space homeomorphic 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$ some integral number of times, say $ n$ times. The notion can be made precise. Moreover, the number $ n$ is determined by the three closed sets

$\displaystyle f^{-1}(a)\qquad f^{-1}(b)\qquad f^{-1}(c)$
where $ a$, $ b$, and $ c$ are any three distinct points in $ S$.



"winding number" is owned by CWoo. [ full author list (2) | owner history (1) ]
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winding number and fundamental group (Result) by Dr_Absentius
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Cross-references: closed sets, circle, homeomorphic, variation, closed, polygon, sides, sum, contain, disc, contained, intervals, finite, domain, bounded variation, imaginary parts, equivalent, path integral, rectifiable, closed path, properties, integer, functions, multiple, ray, real axis, positive, angle, reals, continuous mapping, plane, number, image, point, complex plane, Euclidean plane, oriented, closed curve, variable, complex, analytic functions, connection, topology, algebraic
There are 13 references to this entry.

This is version 5 of winding number, born on 2002-08-14, modified 2003-06-13.
Object id is 3291, canonical name is WindingNumber.
Accessed 11657 times total.

Classification:
AMS MSC: 55M25 (Algebraic topology :: Classical topics :: Degree, winding number)
 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
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