homotopic paths

In mathematics, homotopy is a concept used to classify geometric regions by examining the different types of paths within them. Two paths with the same endpoints are considered homotopic if one can be continuously deformed into the other without leaving the region and keeping the endpoints fixed. This deformation is visualized as a continuous map that gradually transforms one path into another.

A key aspect of homotopy involves paths that start and end at the same point. The collection of all paths homotopic to each other forms a "homotopy class." These classes can be organized into an algebraic structure called a group, known as the fundamental group of the region. The structure of this group reveals information about the region's topology, such as the presence of holes. For instance, a region without holes has a fundamental group with only one element, as all closed paths are homotopic. In contrast, a region with a hole will have different homotopy classes for paths that wind around the hole a different number of times.1