Homotopy groups

Homotopy groups are mathematical tools used to study the shape or structure of spaces, like surfaces or shapes in higher dimensions. Imagine shrinking a loop or a sphere within a space without tearing or cutting it; the homotopy group classifies all possible ways these loops or spheres can be moved or deformed. These groups reveal information about the space’s fundamental features, such as holes or voids, and help distinguish between different shapes by analyzing how they can be continuously transformed within that space. They are essential in understanding the intrinsic geometry and topology of various mathematical and physical systems.