Sequential parametrized topological complexity of group epimorphisms
DOI:
https://doi.org/10.12775/TMNA.2025.009Keywords
Parametrized topological complexity, sectional category, group epimorphismsAbstract
We introduce and study the sequential analogue of Grant's parametrized topological complexity of group epimorphisms, which generalizes the sequential topological complexity of groups. We derive bounds for sequential parametrized topological complexity based on the cohomological dimension of certain subgroups, thereby extending the corresponding bounds for sequential topological complexity of groups. We also obtain sequential analogs of (new) lower bounds on parametrized topological complexity of epimorphisms which are recently obtained by Espinosa Baro, Farber, Mescher and Oprea. Finally, we utilize these results to provide alternative computations for the sequential parametrized topological complexity of planar Fadell-Neuwirth fibrations.References
M. Arkowitz and J. Strom, The sectional category of a map, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 639–652.
A. Baro, M. Farber, S. Mescher and J. Oprea, Sequential topological complexity of aspherical spaces and sectional categories of subgroup inclusions, Math. Ann. 391 (2025), 4555–4605.
I. Basabe, J. González, Y. Rudyak and D. Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014), 2103–2124.
R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College, Department of Pure Mathematics, London, 1981.
R. Bieri and B. Eckmann, Groups with homological duality generalizing Poincaré duality., Invent. Math. 20 (1973), 103–124.
K. Brown, Cohomology of Groups, Springer–Verlag, New York, Berlin, 1982.
D. Cohen, M. Farber and S. Weinberger, Topology of parametrized motion planning algorithms, SIAM J. Appl. Algebra Geom. 5 (2021), 229–249.
D. Cohen, M. Farber and S. Weinberger, Parametrized topological complexity of collision-free motion planning in the plane, Ann. Math. Artif. Intell. 90 (2022), 999–1015.
D. Cohen and G. Pruidze, Motion planning in tori, Bull. Lond. Math. Soc. 40 (2008), 249–262.
D. Cohen and A. Suciu, Homology of iterated semidirect products of free groups., J. Pure Appl. Algebra 126 (1998), 87–120.
D. Cohen and L. Vandembroucq, Topological complexity of the Klein bottle, J. Appl. Comput. Topol. 1 (2017), 199–213 .
A. Dranishnikov, On topological complexity of non-orientable surfaces, Topology Appl. 232 (2017), 61–69.
A. Dranishnikov, On dimension of product of groups, Algebra Discrete Math. 28 (2019), 203–212.
A. Dranishnikov, On topological complexity of hyperbolic groups, Proc. Amer. Math. Soc. 148 (2020), 4547–4556.
S. Eilenberg and T. Ganea, On the Lusternik–Schnirelmann category of abstract groups, Ann. Of Math. (2) 65 (1957), 517–518.
E. Fadell L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118.
M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221.
M. Farber, M. Grant, G. Lupton and J. Oprea, Bredon cohomology and robot motion planning, Algebr. Geom. Topol. 19 (2019), 2023–2059.
M. Farber and J. Oprea, Higher topological complexity of aspherical spaces, Topology Appl. 258 (2019), 142–160.
M. Farber and A. Paul, Sequential parametrized motion planning and its complexity, Topology Appl. 321 (2022), paper no. 108256, 23 pp.
R. Fox, On the Lusternik–Schnirelmann category, Ann. Of Math. (2) 42 (1941), 333–370.
M. Grant, Topological complexity, fibrations and symmetry, Topology Appl. 159 (2012), 88–97.
M. Grant, Parametrised topological complexity of group epimorphisms, Topol. Methods Nonlinear Anal. 60 (2022), 287–303.
M. Grant, G. Lupton and J. Oprea, New lower bounds for the topological complexity of aspherical spaces, Topology Appl. 189 (2015), 78–91.
M. Grant and D. Recio-Mitter, Topological complexity of subgroups of Artin’s braid groups, Contemporary Mathematics, vol. 702, Topological Complexity and Related Topics, Amer. math. Soc., 2018, pp. 165–176.
S. Hughes and K. Li, Higher topological complexity of hyperbolic groups, J. Appl. Comput. Topol. 6 (2022), 323–329.
K. Li, On the topological complexity of toral relatively hyperbolic groups, Proc. Amer. Math. Soc. 150 (2022), 967–974.
L.Lusternik and L. Schnirelmann, Méthodes Topologiques Dans les Problèmes Variationnels, Actualités Scientifiques Et Industrielles, vol. 188; Exposés Sur L’analyse Mathématique Et Ses Applications, vol. 3, Hermann, Paris, 1934.
J.P. May, A Concise Course in Algebraic Topology, University of Chicago Press, Chicago, IL, 1999.
J.P. May and K. Ponto, More Concise Algebraic Topology: Localization, Completion, and Model Categories, University of Chicago Press, Chicago, IL, 2012.
Y. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), 916–920.
A.S. Schwarz, The genus of a fibered space, Tr. Mosk. Mat. Obs. 10 (1961), 217–272.
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