The systole (or systolic category) is a numerical

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

of a closed

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

''M'', introduced by Mikhail Katz and

Yuli Rudyak Yuli B. Rudyak is a professor of Mathematics at the University of Florida in Gainesville, Florida. He obtained his doctorate from Moscow State University under the supervision of M. M. Postnikov. His main research interests are geometry and topol ...

in 2006, by analogy with the

Lusternik–Schnirelmann category In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X is the homotopy invariant defined to be the smallest integer number k such that there is an open covering \_ of X w ...

. 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.

Bibliography

* Dranishnikov, A.; Rudyak, Y. (2009) Stable systolic category of manifolds and the cup-length. ''Journal of Fixed Point Theory and Applications'' 6, no. 1, 165–177. * Katz, M.; Rudyak, Y. (2008) Bounding volume by systoles of 3-manifolds. '' Journal of the London Mathematical Society'' 78, no 2, 407–417. * Dranishnikov, A.; Katz, M.; Rudyak, Y. (2011) Cohomological dimension, self-linking, and systolic geometry. ''

Israel Journal of Mathematics '' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the jou ...

'' 184, no 1, 437–453. See . * Brunnbauer, M. (2008) On manifolds satisfying stable systolic inequalities. ''

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

'' 342, no. 4, 951–968. * Katz, M.; Rudyak, Y. (2006) Lusternik–Schnirelmann category and systolic category of low dimensional manifolds. '' Communications on Pure and Applied Mathematics'' 59, no. 10, 1433–1456. {{Systolic geometry navbox Systolic geometry