An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes

• Emilio J. González ^[1] ; Jesús González ^[1]
  1. [1] Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del I.P.N., México
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 3, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01043-8
• Enlaces
  • Texto completo
• Resumen

  • For a positive integer n and a finite simplicial complex K, we describe an algorithmic procedure constructing a maximal discrete gradient field W(K, n) on Abrams’ discretized configuration space DConf(K, n). Computer experimentation suggests that the field is optimal for n = 2 and complexes K such as triangulations of surfaces. We study the field W(K, n) for n = 2 and K = m,d , the d-dimensional skeleton of the m-dimensional simplex. In particular, we prove that DConf(m,d , 2) is (min{d, m − 1} − 1)-connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of DConf(m,d , 2) and, for certain values of d, we prove that DConf(m,d , 2) breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.

• Referencias bibliográficas
  • 1. https://mathoverflow.net/questions/425679/
  • 2. Abrams, A: Configuration spaces and braid groups of graphs. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–University of California, Berkeley...
  • 3. Abrams, Aaron: Gay, David, Hower, Valerie: Discretized configurations and partial partitions. Proc. Amer. Math. Soc. 141(3), 1093–1104...
  • 4. Aguilar-Guzmán, Jorge: González, Jesús, Hoekstra-Mendoza, Teresa: Farley-Sabalka’s Morse-theory model and the higher topological complexity...
  • 5. Bauer, Ulrich: Ripser: efficient computation of Vietoris-Rips persistence barcodes. J. Appl. Comput. Topol. 5(3), 391–423 (2021)
  • 6. Farber, Michael: Hanbury, Elizabeth: Topology of configuration space of two particles on a graph. II. Algebr. Geom. Topol. 10(4), 2203–2227...
  • 7. Farley, Daniel: Sabalka, Lucas: Discrete Morse theory and graph braid groups. Algebr. Geom. Topol. 5, 1075–1109 (2005)
  • 8. Farley, Daniel: Sabalka, Lucas: On the cohomology rings of tree braid groups. J. Pure Appl. Algebra 212(1), 53–71 (2008)
  • 9. Farley, Daniel: Sabalka, Lucas: Presentations of graph braid groups. Forum Math. 24(4), 827–859 (2012)
  • 10. Forman, Robin: A discrete Morse theory for cell complexes. In Geometry, topology, & physics, Conference of Proceedings Lecture Notes...
  • 11. Forman, Robin: Discrete Morse theory and the cohomology ring. Trans. Amer. Math. Soc. 354(12), 5063–5085 (2002)
  • 12. Ghrist, Robert: Configuration spaces and braid groups on graphs in robotics. In Knots, braids, and mapping class groups—papers dedicated...
  • 13. González, Emilio J., González, Jesús: An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two...
  • 14. González, Jesús: Hoekstra-Mendoza, Teresa: Cohomology ring of tree braid groups and exterior face rings. Trans. Amer. Math. Soc. Ser....
  • 15. Hatcher, Allen: Algebraic Topology. Cambridge University Press, Cambridge (2002)
  • 16. Sze-tsen, Hu.: Isotopy invariants of topological spaces. Proc. Roy. Soc. London Ser. A 255, 331–366 (1960)
  • 17. Kahle, Matthew: Random geometric complexes. Discrete Comput. Geom. 45(3), 553–573 (2011)
  • 18. Kim, Jee Hyoun, Ko, Ki Hyoung, Park, Hyo Won: Graph braid groups and right-angled Artin groups. Trans. Amer. Math. Soc. 364(1), 309–360...
  • 19. Ko, Ki Hyoung, La, Joon Hyun, Park, Hyo Won: Graph 4-braid groups and Massey products. Topol. Appl. 197, 133–153 (2016)
  • 20. Ko, Ki Hyoung, Park, Hyo Won: Characteristics of graph braid groups. Discrete Comput. Geom. 48(4), 915–963 (2012)
  • 21. Prue, Paul: Scrimshaw, Travis: Abrams’s stable equivalence for graph braid groups. Topol. Appl. 178, 136–145 (2014)
  • 22. Shapiro, Arnold: Obstructions to the imbedding of a complex in a Euclidean space. I. The first obstruction. Ann. Math. 2(66), 256–269...
  • 23. Toda, Hirosi: Composition Methods in Homotopy Groups of Spheres, vol. 49. Princeton University Press, Princeton (1962)
  • 24. Wang, Guozhen, Zhouli, Xu.: The triviality of the 61-stem in the stable homotopy groups of spheres. Ann. Math. 186(2), 501–580 (2017)