On the growth of topological complexity
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On the growth of topological complexity Daisuke Kishimoto1
· Atsushi Yamaguchi2
Received: 2 July 2020 / Accepted: 30 August 2020 © Springer Nature Switzerland AG 2020
Abstract . In many cases, the Let TCr (X ) denote the r th topological complexity of a space XP(x) generating function r ≥1 TCr +1 (X )x r is a rational function (1−x) 2 where P(x) is a polynomial with P(1) = cat(X ), that is, the asymptotic growth of TCr (X ) with respect to r is cat(X ). In this paper, we introduce a lower bound MTCr (X ) of TCr (X ) for a rational space X , and estimate the growth of MTCr (X ). Keywords Topological complexity · Module topological complexity · LS-category Mathematics Subject Classification 55M30 · 55P62
1 Introduction The topological complexity TC(X ) was introduced by Farber (2003) for the motion planning problem (Latombe 1991; Sharir 1997), which measures discontinuity of the process of robot motion planning in the configuration space X . The higher topological complexity TCr (X ) was introduced by Rudyak (2010) as a next step towards capturing the complexity of tasks that can be given to robots besides the motion problem, so that TCr (X ) measures discontinuity of the process of robot motion planning of a series of places to visit, in a specific order. We recall the precise definition of TCr (X ). For a space X , let X r denote the r th Cartesian product of X , and let r : X → X r denote the diagonal map r (x) = (x, x, . . . , x) for x ∈ X . The r th topological complexity TCr (X ) of a space X is defined to be the least integer n such that there is an open cover X r = U0 ∪ U1 ∪ · · · ∪ Un having the property that each Ui has a homotopy section si : Ui → X
B
Daisuke Kishimoto [email protected] Atsushi Yamaguchi [email protected]
1
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
2
Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan
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D. Kishimoto, A. Yamaguchi
of r , that is, r ◦ si is homotopic to the inclusion Ui → X r . Then TC2 (X ) is the topological complexity of Farber (2003), and TCr (X ) for r > 2 is the higher topological complexity of Rudyak (2010). It is known that TCr (X ) is a homotopy invariant of X . For a space X , we can define a formal power series F X (x) =
∞
TCr +1 (X )x r .
r =1
Farber and Oprea (2019) asked the following question. Let cat(X ) denote the LScategory of a space X . Question 1.1 For which finite CW-complex X is F X (x) a rational function P(x) (1 − x)2 such that P(x) is a polynomial with P(1) = cat(X )? As is observed in Farber and Oprea (2019) [and proved in Farber et al. (2020)], Question 1.1 is asking whether or not TCr +1 (X ) = TCr (X ) + cat(X ) for all r large enough. Farber et al. (2020) proved that if TCr (X ) = zclr (X ; k) and cat(X ) = cup(X ; k) for some field k and all r large enough, then F X (x) satisfies the condition in Question 1.1, where zclr (X ; k) and cup(X ; k) denote the r th zerodivisors cup-length and the cup-length of X over k, respectively. They also
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