Higher topological complexity and its symmetrization

Abstract

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.

Symmetrized topological complexity

“…The upper bound TC Σ (S n ) ≤ 2 follows immediately from Theorem 4.2. When n is even, the lower bound TC Σ (S n ) ≥ 2 follows from TC Σ (S n ) ≥ TC(S n ), as noted already in [1].…”

“…Many of these difficulties are overcome if one uses a modified definition due to Basabe, González, Rudyak and Tamaki [1]. Namely, consider π : P X → X × X as a Z/2-equivariant map, and let TC Σ (X) denote the minimal integer k such that X × X admits a cover U 0 , U 1 , .…”

“…, U k by invariant open sets, on each of which π admits an equivariant local section. (The authors of [1] in fact define a higher version of this invariant, associated with the iterated diagonal ∆ n : X → X n ; here we concern ourselves only with the n = 2 case of their definition.) Following a suggestion of J. González, we refer to this invariant as symmetrized topological complexity, reserving the name symmetric topological complexity for the FarberGrant version.…”

“…The authors of [1] left open the question of whether the first inequality can be sharp for non-contractible spaces, or the determination of TC Σ (X) for well-known manifolds, notably odd spheres.…”

Cellular Stratified Spaces

“…Recently the author renewed his interest in configuration spaces during a joint project with Basabe, González, and Rudyak [BGRT14] on higher symmetric topological complexities. The notion of cellular stratified spaces was discovered during the discussion with them.…”