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- W4387164805 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 $1leq sleq rgeq2$, 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 $text{TC}_{r,s}(fcolon Xto Y)$ 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." @default.
- W4387164805 created "2023-09-30" @default.
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- W4387164805 date "2023-09-25" @default.
- W4387164805 modified "2023-09-30" @default.
- W4387164805 title "Higher topological complexity of a map" @default.
- W4387164805 doi "https://doi.org/10.55730/1300-0098.3453" @default.
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