systolic category

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

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Systolic geometry