Lusternik-Schnirelmann category and systolic category of low-dimensional manifolds

Abstract

Abstract. 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 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 G, 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 .

Small values of the Lusternik–Schnirelmann category for manifolds

“…If a closed manifold M n is n-essential then cat LS M D n; see eg the paper by the second and third authors [24] and the book by the second author [22, Theorem 12.5.2].…”

“…Since H 2n .K.Z p ; 1// D 0, it follows from the obstruction theory and Poincaré duality that f can be deformed into the .2n 1/-skeleton of K.Z p ; 1/, cf [1,Section 8]. Hence, M is not 2n-essential, and thus cat LS M < 2n [24].…”

Systolic inequalities and Massey products in simply-connected manifolds

“…Namely, the two categories (i.e. the two integers) coincide for 2-complexes [KRS06], as well as for 3-manifolds, orientable or not [KR06,KR07], attain their maximal value simultaneously, both admit a lower bound in terms of real cup-length, both are sensitive to Massey products, etc. Definition 1.1.…”

“…see [KR06] for details. The definitions of the systolic invariants involved may also be found in [Gr83,CrK03,KL05].…”

“…This line of investigation is inspired by M. Gromov's remarks [Gr83,7.4.C ′ , p. 96] and [Gr83, 7.5.C, p. 102], outlining a program for obtaining geometric inequalities associated to nontrivial Massey products of any length. The first step in the program was carried out in [KR06] in the presence of a nontrivial triple Massey product in a manifold with unit Betti numbers.…”