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 in an oriented Euclidean plane (such as the complex plane ), and a point not in the image of , the winding number (or index) of with respect to is the net number of times surrounds . It is not altogether easy to make this notion rigorous.
Let us take for the plane. We have a continuous mapping where and are some reals with and . Denote by the angle from the positive real axis to the ray from to . As moves from to , we expect to increase or decrease by a multiple of , namely where 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 . To define the winding number in this way, we need to assume that the closed path is rectifiable (so that the path integral is defined). An equivalent condition is that the real and imaginary parts of the function are of bounded variation.
But if is any continuous mapping having , the winding number is still definable, without any integration. We can break up the domain of into a finite number of intervals such that the image of , on any of those intervals, is contained in a disc which does not contain . Then emerges as a finite sum: the sum of the angles subtended at by the sides of a polygon.
Let , , and be any three distinct rays from . The three sets
are closed in , and they determine the winding number of around . 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 be any topological space homeomorphic to a circle, and let be any continuous mapping. Intuitively we expect that if a point travels once around , the point will travel around some integral number of times, say times. The notion can be made precise. Moreover, the number is determined by the three closed sets
where , , and are any three distinct points in .
|Date of creation||2013-03-22 12:56:06|
|Last modified on||2013-03-22 12:56:06|
|Last modified by||CWoo (3771)|