winding number

Winding numbers are a basic notion in algebraic topology, and play an important role in connection with analytic functionsMathworldPlanetmath of a complex variable. Intuitively, given a closed curve tS(t) in an oriented Euclidean plane (such as the complex planeMathworldPlanetmath ), 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 for the plane. We have a continuous mapping S:[a,b] where a and b are some reals with a<b and S(a)=S(b). Denote by θ(t) the angle from the positive real axis to the ray from z0 to S(t). As t moves from a to b, we expect θ to increase or decrease by a multiple of 2π, namely 2ωπ where ω 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


is an integer and has the expected properties of a winding number around z0. 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 equivalentPlanetmathPlanetmath condition is that the real and imaginary parts of the function S are of bounded variationMathworldPlanetmath.

But if S is any continuous mapping [a,b] 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 z0. Then 2ωπ emerges as a finite sum: the sum of the angles subtended at z0 by the sides of a polygon.

Let A, B, and C be any three distinct rays from z0. The three sets

S-1(A)  S-1(B)  S-1(C)

are closed in [a,b], and they determine the winding number of S around z0. This result can provide an alternative definition of winding numbers in , 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 spaceMathworldPlanetmath homeomorphicMathworldPlanetmath to a circle, and let f:SS 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

f-1(a)  f-1(b)  f-1(c)

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

Title winding number
Canonical name WindingNumber
Date of creation 2013-03-22 12:56:06
Last modified on 2013-03-22 12:56:06
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 55M25
Classification msc 30A99
Synonym index