2004/10/21 by Mikhail G. Katz, Katz, Mikhail G., Yuli B. Rudyak +1 · 2 citations
Mathematics · #53C23 #55M30 #57N65 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #math.AT #math.DG #math.GT #math.MG #msc:53C23 #msc:55M30 #msc:57N65
paper · pdf · doi:10.48550/arxiv.math/0410456
23 pages. Communications on Pure and Applied Mathematics, to appear
openalex publication_date 2004/10/21 · arxiv created 2004/12/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M. It is denoted catsys(M), and defined in terms of the existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality catsys(M) ≤ catLS(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(M) leads us to prove or conjecture new systolic inequalities on M.