Skip to main content
SpringerLink
Log in

Farley–Sabalka’s Morse-Theory Model and the Higher Topological Complexity of Ordered Configuration Spaces on Trees

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

Using the ordered analogue of Farley–Sabalka’s discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber’s equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological complexities of ordered configuration spaces on trees on any number of particles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. All cohomology groups in this paper are taken with \({\mathbb {Z}}\)-coefficients.

  2. It suffices to totally order the non-deleted edges adjacent to v.

  3. See [9], particularly Theorem 4.2, for the \({\mathscr {S}}_n\)-equivariant ingredient in the third item of Corollary 3.5.

  4. The spliced graph G(3, T) has vertices given by the six permutations ijk, and edges given by all the edges (with the indicated repetitions) shown in Figs. 2 and 3.

  5. Here and below, piling up is done according to (9), without leaving empty spots in the tree (as in (11)).

  6. Stacks like these are uniquely determined, as the tree has been assumed to be n-sufficiently subdivided.

  7. In this kind of expressions we extend, in the obvious way, the abuse of terminology in (10).

  8. An \(a_0\)-local T-branch is a sequence of edges \((a_0,b), (b,c), (c,d),\ldots \) (it implies \(a_0<b<c<d<\ldots \)).

References

  1. Abrams, A.D.: Configuration Spaces and Braid Groups of Graphs. PhD thesis, University of California, Berkeley (2000)

  2. Basabe, I., González, J., Rudyak, Y.B., Tamaki, D.: Higher topological complexity and its symmetrization. Algebr. Geom. Topol. 14(4), 2103–2124 (2014)

    Article  MathSciNet  Google Scholar 

  3. Chettih, S.: Dancing in the Stars: Topology of Non-\(k\)-Equal Configuration Spaces of Graphs. PhD thesis, University of Oregon (2016)

  4. Cornea, O., Lupton, G., Oprea, J., Tanré, D.: Lusternik–Schnirelmann Category. Mathematical Surveys and Monographs, vol. 103. American Mathematical Society, Providence (2003)

  5. Farber, M.: Configuration spaces and robot motion planning algorithms. In: Combinatorial and Toric Homotopy. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 35, pp. 263–303. World Scientific, Hackensack (2018)

  6. Farley, D., Sabalka, L.: Discrete Morse theory and graph braid groups. Algebr. Geom. Topol. 5, 1075–1109 (2005)

    Article  MathSciNet  Google Scholar 

  7. Farley, D., Sabalka, L.: Presentations of graph braid groups. Forum Math. 24(4), 827–859 (2012)

    Article  MathSciNet  Google Scholar 

  8. Forman, R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)

    Article  MathSciNet  Google Scholar 

  9. Freij, R.: Equivariant discrete Morse theory. Discrete Math. 309(12), 3821–3829 (2009)

    Article  MathSciNet  Google Scholar 

  10. Grant, M., Lupton, G., Oprea, J.: Spaces of topological complexity one. Homol. Homotopy Appl. 15(2), 73–81 (2013)

    Article  MathSciNet  Google Scholar 

  11. Kim, J.H., Ko, K.H., Park, H.W.: Graph braid groups and right-angled Artin groups. Trans. Am. Math. Soc. 364(1), 309–360 (2012)

    Article  MathSciNet  Google Scholar 

  12. Lütgehetmann, D., Recio-Mitter, D.: Topological complexity of configuration spaces of fully articulated graphs and banana graphs. Discrete Comput. Geom. 65(3), 693–712 (2021)

    Article  MathSciNet  Google Scholar 

  13. Scheirer, S.: Topological complexity of \(n\) points on a tree. Algebr. Geom. Topol. 18(2), 839–876 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del I.P.N., Av. Instituto Politécnico Nacional 2508, San Pedro Zacatenco, Ciudad de México, 07000, Mexico

    Jorge Aguilar-Guzmán, Jesús González & Teresa Hoekstra-Mendoza

Authors
  1. Jorge Aguilar-Guzmán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jesús González
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Teresa Hoekstra-Mendoza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesús González.

Additional information

Editor in Charge: János Pach

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aguilar-Guzmán, J., González, J. & Hoekstra-Mendoza, T. Farley–Sabalka’s Morse-Theory Model and the Higher Topological Complexity of Ordered Configuration Spaces on Trees. Discrete Comput Geom 67, 258–286 (2022). https://doi.org/10.1007/s00454-021-00306-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-021-00306-3

Keywords

Mathematics Subject Classification

Search

Navigation