Up: Winding Numbers

Homotopy

A homotopy is a very general kind of mathematical eqivalence between functions. In the case of winding numbers, we are interested in equivalent curves, so we want to think of a curve as a function .

If and are two curves, we say they are homotopic if there exists a continuous function with the following properties:

H(s,0) = H(s,1) for all s in [0,1].

Note for fixed s₀, is a curve in the plane. Because of the third condition, its ends must meet up. The other two conditions say the when s=0, the curve you get is and when s=1, you get .

If you think of s as a time parameter, we begin deforming at s=0, and when we reach time s=1, the curve has evolved into . The curve is the curve it has evolved into at time s₀.

Up: Winding Numbers

The Geometry Center Home Page

Comments to: rminer@geom.umn.edu
Created: Fri Sep 8 11:39:00 1995 --- Last modified: Wed Apr 17 17:01:45 1996