No abstract

2016-11-15T19:40:59

Abstract. We develop the properties of the n-th sequential topological complexity TC n , a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of TC n (X) to the Lusternik-Schnirelmann category of cartesian powers of X, to the cup-length of the diagonal embedding X → X n , and to the ratio between homotopy dimension and connectivity of X. We fully compute the numerical value of TC n for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces. Our study includes two symmetrized versions of TC n (X). The first one, unlike Farber-Grant's symmetric topological complexity, turns out to be a homotopy invariant of X; the second one is closely tied to the homotopical properties of the configuration space of cardinality-n subsets of X. Special attention is given to the case of spheres.

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 .

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 .