# winding number

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^{} $\u2102$), 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 $\u2102$ for the plane. We have a continuous mapping $S:[a,b]\to \u2102$ where $a$ and $b$ are some reals with $$ 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

$$\frac{1}{2\pi i}{\int}_{S}\frac{dz}{z-{z}_{0}}$$ |

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 $[a,b]\to \u2102$ 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

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

are closed in $[a,b]$, and they *determine*
the winding number of $S$ around ${z}_{0}$. This result can provide an alternative
definition of winding numbers in $\u2102$, 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

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

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

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 |