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.

Sequential Collision-Free Optimal Motion Planning Algorithms in Punctured Euclidean spaces

“…The notion of the nth sequential or higher topological complexity was introduced by Rudyak in [9] and further developed in [1]. We follow [10] to recall the basic definitions and properties.…”

The topological complexity of pure graph braid groups is stably maximal

“…The "higher" or "sequential" topological complexity TC r is a numerical homotopy invariant defined in [Rud10] and developed in [BGRT14], recovering Farber's topological complexity [Far03] for r = 2 and the Lusternik-Schnirelmann category for r = 1. The reader is referred to these references for definitions; we recall only that 1 r TC r is bounded by the homotopy dimension in non-pathological settings.…”

Độ Phức Tạp Tôpô Bậc Cao Của Tích Các Không Gian Tôpô

“…Do đó, thông thường ta chỉ có thể đưa ra các chặn trên và chặn dưới cho bất biến này. Trong [3], [4], [5] các tác giả đã đưa ra một số kết quả về độ phức tạp tôpô bậc cao cho một số không gian cụ thể.…”