Algebraic & Geometric Topology

Equivariant topological complexity

Hellen Colman and Mark Grant

Full-text: Open access

Abstract

We define and study an equivariant version of Farber’s topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik–Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the nonequivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the nonequivariant topological complexity.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 4 (2012), 2299-2316.

Dates
Received: 2 May 2012
Revised: 2 August 2012
Accepted: 4 August 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715458

Digital Object Identifier
doi:10.2140/agt.2012.12.2299

Mathematical Reviews number (MathSciNet)
MR3020208

Zentralblatt MATH identifier
1260.55007

Subjects
Primary: 55M99: None of the above, but in this section 57S10: Compact groups of homeomorphisms
Secondary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 55R91: Equivariant fiber spaces and bundles [See also 19L47]

Keywords
equivariant LS–category equivariant sectional category equivariant topological complexity

Citation

Colman, Hellen; Grant, Mark. Equivariant topological complexity. Algebr. Geom. Topol. 12 (2012), no. 4, 2299--2316. doi:10.2140/agt.2012.12.2299. https://projecteuclid.org/euclid.agt/1513715458


Export citation

References

  • I Basabe, J González, Y B Rudyak, D Tamaki, Higher topological complexity and homotopy dimension of configuration spaces on spheres
  • I Berstein, T Ganea, The category of a map and of a cohomology class, Fund. Math. 50 (1961/1962) 265–279
  • G Cicortaş, Categorical sequences and applications, Studia Univ. Babeş-Bolyai Math. 47 (2002) 31–39
  • H Colman, Equivariant LS–category for finite group actions, from: “Lusternik–Schnirelmann category and related topics”, (O Cornea, G Lupton, J Oprea, D Tanré, editors), Contemp. Math. 316, Amer. Math. Soc. (2002) 35–40
  • O Cornea, G Lupton, J Oprea, D Tanré, Lusternik–Schnirelmann category, Math. Surveys and Monographs 103, Amer. Math. Soc. (2003)
  • T tom Dieck, Transformation groups, de Gruyter Studies in Math. 8, Walter de Gruyter & Co., Berlin (1987)
  • E Fadell, The equivariant Ljusternik–Schnirelmann method for invariant functionals and relative cohomological index theories, from: “Topological methods in nonlinear analysis”, (A Granas, editor), Sém. Math. Sup. 95, Presses Univ. Montréal (1985) 41–70
  • M Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003) 211–221
  • M Farber, Instabilities of robot motion, Topology Appl. 140 (2004) 245–266
  • M Farber, Topology of robot motion planning, from: “Morse theoretic methods in nonlinear analysis and in symplectic topology”, (P Biran, O Cornea, F Lalonde, editors), NATO Sci. Ser. II Math. Phys. Chem. 217, Springer (2006) 185–230
  • R H Fox, On the Lusternik–Schnirelmann category, Ann. of Math. 42 (1941) 333–370
  • J González, P Landweber, Symmetric topological complexity of projective and lens spaces, Algebr. Geom. Topol. 9 (2009) 473–494
  • M Grant, Topological complexity, fibrations and symmetry, Topology Appl. 159 (2012) 88–97
  • I M James, On category, in the sense of Lusternik–Schnirelmann, Topology 17 (1978) 331–348
  • W Marzantowicz, A $G$-Lusternik–Schnirelman category of space with an action of a compact Lie group, Topology 28 (1989) 403–412
  • W Singhof, On the Lusternik–Schnirelmann category of Lie groups, Math. Z. 145 (1975) 111–116
  • S Smale, On the topology of algorithms. I, J. Complexity 3 (1987) 81–89
  • E H Spanier, Algebraic topology, McGraw-Hill, New York (1966)
  • V A Vasiliev, Cohomology of braid groups and the complexity of algorithms, Funktsional. Anal. i Prilozhen. 22 (1988) 15–24, 96
  • A S Švarc, The genus of a fiber space. I, II., Amer. Math. Soc. Transl. 55 (1966) 49–140