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**Higher homotopic distance.**
*(English)*
Zbl 1479.55006

The homotopic distance of two maps with the same domain and codomain is a concept introduced in [E. Macías-Virgós and D. Mosquera-Lois, Math. Proc. Camb. Philos. Soc. 172, No. 1, 73–93 (2022; Zbl 1485.55005)]. This notion allows a study of various categorical invariants like topological complexity and Lusternik-Schnirelmann category of maps in a unified setting. In the present article, the authors discuss the notion of higher homotopic distance of a finite family of maps, which generalizes for example the notion of higher topological complexities as introduced by Y. Rudyak [Topology Appl. 157, No. 5, 916–920 (2010; Zbl 1187.55001)].

The authors follow the line of argument of [Marcias-Virgos and Mosquera-Lois, loc. cit.] and discuss foundational results about higher homotopic distances like their homotopy invariance and their behaviour under taking Cartesian products. In particular, in the same way as the homotopic distance of two maps is bounded from above by the topological complexity of their common codomain, it is proven that higher topological complexities provide upper bounds for the higher homotopic distances as well. The authors further show how several known results about higher topological complexities, e.g. their monotonicity property, are derived from more general results about higher homotopic distances.

The authors follow the line of argument of [Marcias-Virgos and Mosquera-Lois, loc. cit.] and discuss foundational results about higher homotopic distances like their homotopy invariance and their behaviour under taking Cartesian products. In particular, in the same way as the homotopic distance of two maps is bounded from above by the topological complexity of their common codomain, it is proven that higher topological complexities provide upper bounds for the higher homotopic distances as well. The authors further show how several known results about higher topological complexities, e.g. their monotonicity property, are derived from more general results about higher homotopic distances.

Reviewer: Stephan Mescher (Halle)

### MSC:

55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |

55M99 | Classical topics in algebraic topology |

68T40 | Artificial intelligence for robotics |

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\textit{A. Borat} and \textit{T. Vergili}, Topol. Methods Nonlinear Anal. 57, No. 2, 525--534 (2021; Zbl 1479.55006)

### References:

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