Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or Lusternik–Schnirelmann category, LS-category) is a way to measure how complex a topological space is. It tells us the smallest number of open sets we need to cover the space so that each set can be continuously shrunk to a point within the space. This idea helps mathematicians understand the shape and structure of spaces.

For example, if we look at an n-sphere, which is like an n-dimensional ball’s surface, the LS-category is 2. This means we need at least two open sets to cover the sphere with the special property mentioned above. The Lusternik–Schnirelmann theorem shows that for real projective space R P n, the LS-category is exactly n + 1.

This concept was first introduced by Lazar Lyusternik and Lev Schnirelmann and connects closely with algebraic topology. It provides a lower bound for the number of critical points a function can have on a space, similar to ideas in Morse theory. The LS-category has been expanded to study different kinds of spaces and structures, such as those with group actions, foliations, and simplicial complexes.