ATRACTOR

Given two paths (with the same start and end points), we can ask if they are homotopic, that is, if one path can be continuously deformed into the other. The answer to this question depends on the paths considered, but also on the surface we are working on. Taking the model with graded rubber strings, the idea is to keep fixed the start and end points while moving one rubber string into the other always on the surface. The condition "always on the surface" is important! For instance, if the surface has holes, we cannot pass over them, we have to go around them in some way possible!

A simple case where this "detail" makes the difference is to compare the possible continuous deformations of paths in the plane, or in the plane minus a point (the plane with a hole).

In the apps below we explore this situation:

(Click on the figures to access the apps.)

The mathematical definition of homotopy contains these properties through the codomain of the homotopy and its continuity. Considering \(H\) from \([0,1]\times[0,1]\) to \(S\), the image set of \(H\) is in \(S\) and, therefore, the points not in \(S\) cannot be in paths defined by \(H\) - that is, we cannot ignore the holes going through them! Furthermore, as \(H\) is continuous, all paths defined by \(H\) are continuous, so we also cannot jump over the holes!