Turkish Journal of Mathematics
Abstract
The higher topological complexity of a space $X$, $\text{TC}_r(X)$, $r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$, have been introduced by Rudyak and Pavesiç, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map $f$, $\text{TC}_{r,s}(f)$, for $1\leq s\leq r\geq2$, which simultaneously extends Rudyak's and Pavesiç notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy invariance and the behavior of $\text{TC}_{r,s}$ under products and compositions of maps, as well as the dependence of $\text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for $\text{TC}_{r,s}(f\colon X\to Y)$ in terms of categorical invariants associated to $X$, $Y$ and $f$. In particular, we describe within one the value of $\text{TC}_{r,s}$ in the case of the nontrivial double covering over real projective spaces, as well as for their complex counterparts.
DOI
10.55730/1300-0098.3453
Keywords
Higher topological complexity, sectional categor
First Page
1616
Last Page
1642
Recommended Citation
ZAPATA, CESAR AUGUSTO IPANAQUE and GONZÁLEZ, JESÚS
(2023)
"Higher topological complexity of a map,"
Turkish Journal of Mathematics: Vol. 47:
No.
6, Article 3.
https://doi.org/10.55730/1300-0098.3453
Available at:
https://journals.tubitak.gov.tr/math/vol47/iss6/3