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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.

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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References:

[1] I. Basabe, J. Gonzalez, Y. Rudyak and D. Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014), 2103-2124. · Zbl 1348.55005
[2] A. Borat, Higher dimensional simplicial complexity, New York J. Math. 26 (2020), 1130-1144. · Zbl 1456.55001
[3] O. Cornea, G. Lupton, J. Oprea and D. Tanr, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs, Vol. 103, American Mathematical Society, 2003. · Zbl 1032.55001
[4] M. Farber, Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics, EMS, 2008. · Zbl 1148.55011
[5] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211-221. · Zbl 1038.68130
[6] M. Farber and M. Grant, Symmetric motion planning, Topology and Robotics, Contemporary Mathematics Series (M. Farber, R. Ghrist, M. Burger and D. Koditschek, eds.), American Mathematical Society, Providence, RI, 438 (2007), 85-104. · Zbl 1143.70013
[7] E. Macias-Virgos and D. Mosquera-Lois, Homotopic distance between maps, Math. Proc. Cambridge Philos. Soc. (2021), 1-21, DOI: 1017/S0305004121000116. · Zbl 1462.55009
[8] E. Macias-Virgos and D. Mosquera-Lois, Homotopic distance between functors, J. Homotopy Relat. Struct. 15 (2020), 537-555. · Zbl 1462.55009
[9] J. Oprea and J. Strom, Mixing categories, Proc. Amer. Math. Soc. 139 (2011), 3383-3392. · Zbl 1227.55003
[10] Y. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), 916-920; Erratum: Topology Appl. 157 (2010), p. 1118. · Zbl 1187.55001
[11] A.S. Svarc, The genus of a fiber space, Amer. Math. Soc. Transl. 55 (1966), 49-140. · Zbl 0178.26202
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