homotopy class

In mathematics, a homotopy class is a way to categorize different paths within a geometric region. Two paths are considered homotopic if one can be continuously deformed into the other without leaving the region and keeping the endpoints fixed.1

Imagine drawing a path on a piece of paper with a hole in it. If you can slide one path into another without lifting your pen and without crossing the hole, they are homotopic. If a path goes around the hole, it cannot be deformed into a path that doesn't go around the hole without passing through the hole itself.1

The set of all paths that are homotopic to each other forms a homotopy class. These classes can be organized into algebraic structures called groups, such as the fundamental group of a region. The structure of these groups reveals important information about the shape of the region, such as the presence and number of holes.1