The systole (or systolic category) is a numerical
invariant 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
Mikhail "Mischa" Gershevich Katz (, ; born 1958)[Curriculum vitae](_blank)
retrieved ...
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. 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.
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