Systolic category is a numerical invariant

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 of a closed manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 M, introduced by Mikhail Katz

Mikhail Katz

Mikhail Katz is an Israeli mathematician, a professor of mathematics at Bar Ilan University. His main interests are differential geometry and geometric topology; he is the author of a book about systolic geometry....

 and Yuli Rudyak

Yuli Rudyak

Yuli B. Rudyak is professor of Mathematics at University of Florida at Gainsville. He obtained his doctorate from Moscow State University under the supervision of M. M. Postnikov. His main research interests aregeometry and topology and symplectic topology....

 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. Thus, in dimensions 2 and 3, the two invariants coincide. In dimension 4, the systolic category is known to be a lower bound for the Lusternik–Schnirelmann category.