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

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_sysM ≤ cat_LSM 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.