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Lusternik‐Schnirelmann category and systolic category of low‐dimensional manifolds
Author(s) -
Katz Mikhail G.,
Rudyak Yuli B.
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20146
Subject(s) - mathematics , conjecture , homotopy , inequality , invariant (physics) , dimension (graph theory) , pure mathematics , metric (unit) , combinatorics , mathematical analysis , operations management , economics , mathematical physics
We show that the geometry of a Riemannian manifold ( M , ) is sensitive to the apparently purely homotopy‐theoretic invariant of M known as the Lusternik‐Schnirelmann category, denoted cat LS ( M ). Here we introduce a Riemannian analogue of cat LS ( M ), called the systolic category of M . It is denoted cat sys ( M ) and defined in terms of the existence of systolic inequalities satisfied by every metric , as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat sys M ≤ cat LS M is satisfied, which typically turns out to be an equality, e.g., in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality and that both categories are sensitive to Massey products. The comparison with the value of cat LS ( M ) leads us to prove or conjecture new systolic inequalities on M . © 2006 Wiley Periodicals, Inc.