Hostname: page-component-669899f699-cf6xr Total loading time: 0 Render date: 2025-04-27T09:56:52.814Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopic distance between maps

Part of: Artificial intelligence (68Txx) Homotopy theory Fiber spaces and bundles Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  22 February 2021

E. MACÍAS–VIRGÓS  and
D. MOSQUERA–LOIS
E. MACÍAS–VIRGÓS
Affiliation:
Institute of Mathematics, University of Santiago de Compostela, Avda. Lope Gómez de Marzoa s/n. Campus Sur, Santiago de Compostela, 15782Spain. e-mails: [email protected], [email protected]
D. MOSQUERA–LOIS
Affiliation:
Institute of Mathematics, University of Santiago de Compostela, Avda. Lope Gómez de Marzoa s/n. Campus Sur, Santiago de Compostela, 15782Spain. e-mails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

MSC classification

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Secondary: 55M99: None of the above, but in this section 55P99: None of the above, but in this section 55R10: Fiber bundles 55P45: $H$-spaces and duals 68T40: Robotics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 73 - 93
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

Partially supported by MINECO Spain research project MTM2016-78647-P and by Xunta de Galicia ED431C 2019/10 with FEDER funds.

‡

Partially supported by Ministerio de Ciencia, Innovación y Universidades, grant FPU17/03443.

References

Barmak, J. A.. Algebraic Topology of Finite Topological Spaces and Applications. Lecture Notes in Mathematics 2032 (Springer, Heidelberg, 2011).CrossRefGoogle Scholar
Cornea, O., Lupton, G., Oprea, J. and Tanré, D.. Lusternik–Schnirelmann category. Math. Surveys Monogr. 103 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
Farber, M.. Topological complexity of motion planning. Discrete Comput. Geom. 29(2) (2003), 21–221.CrossRefGoogle Scholar
Farber, M.. Instabilities of robot motion. Topology Appl. 140(2-3) (2004), 245–266.CrossRefGoogle Scholar
Farber, M. and Grant, M.. Symmetric motion planning. In Topology and Robotics. Contemp. Math. 432 (American Mathematical Society, Providence, RI, 2007).Google Scholar
Farber, M. and Grant, M.. Robot motion planning, weights of cohomology classes, and cohomology operations. Proc. Amer. Math. Soc. 136(9) (2008), 3339–3349.CrossRefGoogle Scholar
Fernández–Ternero, D., Macías–Virgós, E., Minuz, E. and Vilches, J. A.. Discrete topological complexity. Proc. Amer. Math. Soc. 146(10) (2018), 4535–4548.CrossRefGoogle Scholar
Fernández–Ternero, D., Macías–Virgós, E. and Vilches, J. A.. Simplicial Lusternik–Schnirelmann category. Publicacions Matemàtiques 63 (2019), 265–293.CrossRefGoogle Scholar
Fernández–Ternero, D., Macías–Virgós, E. and Vilches, J. A.. Lusternik–Schnirelmann category of simplicial complexes and finite spaces. Topology Appl. 194 (2015), 37–50.CrossRefGoogle Scholar
Frankel, T.. Critical submanifolds of the classical groups and Stiefel manifolds. In Differential and Combinatorial Topology. (A Symposium in Honor of Marston Morse), pages 37–53, (Princeton University Press, Princeton, N.J., 1965).Google Scholar
Gómez–Tato, A., Macías–Virgós, E. and Pereira–Sáez, M. J.. Trace map, Cayley transform and LS category of Lie groups. Ann. Global Anal. Geom. 39(3) (2011), 325–335.CrossRefGoogle Scholar
Hatcher, A.. Algebraic Topology. (Cambridge University Press, Cambridge, 2002).Google Scholar
James, I. M.. On H-spaces and their homotopy groups, The Quarterly Journal of Mathematics, 11(1) (1960), 161–179.CrossRefGoogle Scholar
Kozlov, D.. Combinatorial Algebraic Topology. Algorithms and Computation in Mathematics 21 (Springer, Berlin, 2008).CrossRefGoogle Scholar
Lupton, G. and Scherer, J.. Topological complexity of H-spaces. Proc. Amer. Math. Soc. 141(5) (2013), 1827–1838.CrossRefGoogle Scholar
Mosher, R. E. and Tangora, M. C.. Cohomology Operations and Applications in Homotopy Theory. (Harper & Row Publishers, New York-London, 1968).Google Scholar
Oprea, J. and Strom, J.. Mixing categories. Proc. Amer. Math. Soc. 139(9) (2011), 3383–3392.CrossRefGoogle Scholar
Rudyak, Y. B.. On category weight and its applications. Topology 38(1) (1999), 37–55.CrossRefGoogle Scholar
Rudyak, Y. B.. On higher analogs of topological complexity. Topology Appl. 157(5) (2010), 916–920.CrossRefGoogle Scholar
Schweitzer, P. A.. Secondary cohomology operations induced by the diagonal mapping. Topology 3 (1965), 337–355.CrossRefGoogle Scholar
Singhof, W.. On the Lusternik–Schnirelmann category of Lie groups. Math. Z. 145(2) (1975), 111–116.CrossRefGoogle Scholar
Tanaka, K.. A combinatorial description of topological complexity for finite spaces. Algebr. Geom. Topol. 18(2) (2018), 779–796.CrossRefGoogle Scholar
Spanier, E. H.. Algebraic Topology (McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966).Google Scholar
Strom, J. A.. Category weight and essential category weight. Ph.D. thesis. The University of Wisconsin–Madison (1997).Google Scholar
Varadarajan, K.. On fibrations and category. Math. Z. 88 (1965), 267–273.CrossRefGoogle Scholar