Proper topological complexity
DOI:
https://doi.org/10.12775/TMNA.2025.010Keywords
Proper homotopy, exterior homotopy, proper Lusternik-Schnirelmann category, proper topological complexityAbstract
We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe'' area. To make it a homotopy functorial invariant we characterize it as a particular instance of the exterior sectional category of an exterior map, an invariant of the exterior homotopy category which is also deeply analyzed.References
R. Ayala, E. Domı́nguez, A. Márquez and A. Quintero, Lusternik–Schnirelmann invariants in proper homotopy theory, Pacific Journal of Math. 153 (1992), 201–215.
Z. Blaszczyk and J. Carrasquel-Vera, Topological complexity and efficiency of motion planning algorithms, Rev. Mat. Iberoam. 34 (2018), 1679–1684.
M. Cárdenas, F.F. Lasheras, F. Muro and A. Quintero, Proper L-S category, fundamental pro-groups and 2-dimensional proper co-H-spaces, Topology Appl. 153 (2005), 580–604.
M. Cárdenas, F.F. Lasheras and A. Quintero, Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces, Bull. Belgian Math. Soc. 9 (2002), 419–431.
H. Colman and M. Grant. Equivariant topological complexity, Algebr. Geom. Topol. 12 (2012), 2299–2316.
O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, Math. Surveys Monogr., vol. 103, AMS, 2003.
A. Dranishnikov, On topological complexity of twisted products, Topology Appl. 179 (2015), 74–80.
A. Dranishnikov, Topological complexity of wedges and covering maps, Proc. Amer. Math. Soc. 142 (2014), 4365–4376.
M. Farber, Topological complexity of motion planning, Discret. Comput. Geom. 29 (2003), 211–221.
M. Farber, Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2008.
M. Farber and M. Grant, Symmetric motion planning, Topology and Robotics, Contemp. Math. 438 (2007), 85–104.
J.M. Garcı́a-Calcines, A remark on proper partitions of unity, Topology Appl. 159 (2012), 3363–3371.
J.M. Garcı́a-Calcines, P.R. Garcı́a-Dı́az and A. Murillo, A Whitehead–Ganea approach for proper Lusternik–Schnirelmann category, Math. Proc. Camb. Phil. Soc. 142 (2007), 439–457.
J.M. Garcı́a-Calcines, P.R. Garcı́a-Dı́az and A. Murillo, The Ganea conjecture in proper homotopy via exterior homotopy theory, Math. Proc. Cambridge Phil. Soc. 149 (2010) 75–91.
J.M. Garcı́a-Calcines, P.R. Garcı́a-Dı́az and A. Murillo, Brown representability for exterior cohomology and cohomology with compact supports, J. London Math. Soc. 90 (2014), 184–196.
J.M. Garcı́a-Calcines, M. Garcı́a-Pinillos and L.J. Hernández-Paricio, A closed model category for proper homotopy and shape theories, Bull. Austral. Math. Soc. 57 (1998), 221–242.
W. Lubawski and W. Marzantowicz, Invariant topological complexity, Bull. London Math. Soc. 47 (2014), 101–117.
M. Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. Soc. 277 (1983), 307–321.
T. Napier and M. Ramachandran, Elementary construction of exhausting subsolutions of elliptic operators, Enseign. Math. 3 (2004), 367–390.
I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259–269.
R. Short, Relative topological complexity of a pair, Topology Appl. 248 (2018), 7–23.
E.H. Spanier, Cohomology with supports, Pacific J. Math. 123 (1986), 447–464.
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Copyright (c) 2025 José Manuel García-Calcines, Aniceto Murillo
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