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
- {{cite journal
| last1=Dranishnikov | first1=Alexander N.
| last2=Rudyak | first2=Yuli B.
| title=Stable systolic category of manifolds and the cup-length
| journal=Journal of Fixed Point Theory and Applications
| volume=6
| issue=1
| pages=165–177
| date=2009
| doi=10.1007/s11784-009-0118-5
| arxiv=0812.4637}}
- {{cite journal
| last1=Katz | first1=Mikhail G. | authorlink1=Mikhail Katz
| last2=Rudyak | first2=Yuli B.
| title=Bounding volume by systoles of 3-manifolds
| journal=Journal of the London Mathematical Society
| volume=78
| issue=2
| date=2008
| pages=407–417
| doi=10.1112/jlms/jdm105
| arxiv=math/0504008}}
- {{cite journal
| last1=Dranishnikov | first1=Alexander N.
| last2=Katz | first2=Mikhail G. | authorlink2=Mikhail Katz
| last3=Rudyak | first3=Yuli B.
| title=Cohomological dimension, self-linking, and systolic geometry
| journal=Israel Journal of Mathematics
| volume=184
| issue=1
| pages=437–453
| date=2011
| doi=10.1007/s11856-011-0075-8
| arxiv=0807.5040}}
- {{cite journal
| last1=Brunnbauer | first1=Michael
| title=On manifolds satisfying stable systolic inequalities
| journal=Mathematische Annalen
| volume=342
| issue=4
| date=2008
| pages=951–968
| doi=10.1007/s00208-008-0263-y
| arxiv=0708.2589}}
- {{cite journal
| last1=Katz | first1=Mikhail G. | authorlink1=Mikhail Katz
| last2=Rudyak | first2=Yuli B.
| date=2006
| title=Lusternik–Schnirelmann category and systolic category of low dimensional manifolds
| journal=Communications on Pure and Applied Mathematics
| volume=59
| issue=10
| pages=1433–1456
| doi=10.1002/cpa.20146
| arxiv=math/0410456}}
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