Lusternik-Schnirelmann category

Recently, an intriguing link has emerged with the Lusternik-Schnirelmann category.

In dimension 4, the systolic category is known to be a lower bound for the Lusternik-Schnirelmann category.

One of these problems, regarding the Lusternik-Schnirelmann category, came to be known as Ganea's conjecture.

A Whitehead-Ganea approach for proper Lusternik-Schnirelmann category.

Recently (see the '06 paper by Katz and Rudyak below), the link with the Lusternik-Schnirelmann category has emerged.

Katz, M.; Rudyak, Y. (2006) Lusternik-Schnirelmann category and systolic category of low dimensional manifolds.

Thus, a notion of systolic category of a manifold has been defined and investigated, exhibiting a connection to the Lusternik-Schnirelmann category (LS category).

F. Takens, The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category, Inventiones Mathematicae 6 (1968), 197-244.

Octav Cornea, Gregory Lupton, John Oprea, Daniel Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, 103.

Systolic category is a numerical invariant of a closed manifold M, introduced by Mikhail Katz and Yuli Rudyak in 2006, by analogy with the Lusternik-Schnirelmann category.

The invariant is defined in terms of the systoles of M and its covers, as the largest number of systoles in a product yielding a curvature-free lower bound for the total volume of M. The invariant is intimately related to the Lusternik-Schnirelmann category.

In mathematics, the Lyusternik-Schnirelmann category (or, Lusternik-Schnirelmann category, LS-category, or simply, category) of a topological space is the homotopical 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.