Lusternik–Schnirelmann category

Last updated • 1 min readFrom Wikipedia, The Free Encyclopedia

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopy invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic. For example, if is a sphere, this takes the value two.

Sometimes a different normalization of the invariant is adopted, which is one less than the definition above. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).

In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for the LS-category.

It was, as originally defined for the case of a manifold, the lower bound for the number of critical points that a real-valued function on could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).

The invariant has been generalized in several different directions (group actions, foliations, simplicial complexes, etc.).

See also

Related Research Articles

References

Ralph Fox

Ralph Hartzler Fox was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.

The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.

Floris Takens Dutch mathematician

Floris Takens was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.