Edge stabilization in the homology of graph braid groups

An, Byung Hee¹ ; Drummond-Cole, Gabriel² ; Knudsen, Ben³

¹ Department of Mathematics Education, Teachers College, Kyungpook National University, Daegu, South Korea
² Center for Geometry and Physics, Institute for Basic Science, Pohang, South Korea
³ Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 24 (2020) no. 1, pp. 421-469.

Résumé

We introduce a novel type of stabilization map on the configuration spaces of a graph which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains which contains strictly more information than the homology-level action. We show that the resulting differential graded module is almost never formal over the ring of edges.

Détail

DOI : 10.2140/gt.2020.24.421

Classification : 13D40, 20F36, 55R80, 05C40
Keywords: configuration spaces, braid groups, graphs, growth of Betti numbers, connectivity, homological stability

Bibliographie

[1] A D Abrams, Configuration spaces and braid groups of graphs, PhD thesis, University of California, Berkeley (2000)

[2] B H An, G C Drummond-Cole, B Knudsen, Subdivisional spaces and graph braid groups, Doc. Math. 24 (2019) 1513

[3] V I Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969) 227

[4] S Chettih, D Lütgehetmann, The homology of configuration spaces of trees with loops, Algebr. Geom. Topol. 18 (2018) 2443 | DOI

[5] T Church, J S Ellenberg, B Farb, FI–modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015) 1833 | DOI

[6] F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, 533, Springer (1976) | DOI

[7] D Eisenbud, Commutative algebra: with a view toward algebraic geometry, 150, Springer (1995) | DOI

[8] D Farley, L Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005) 1075 | DOI

[9] D Farley, L Sabalka, On the cohomology rings of tree braid groups, J. Pure Appl. Algebra 212 (2008) 53 | DOI

[10] Ś R Gal, Euler characteristic of the configuration space of a complex, Colloq. Math. 89 (2001) 61 | DOI

[11] R Ghrist, Configuration spaces and braid groups on graphs in robotics, from: "Knots, braids, and mapping class groups : papers dedicated to Joan S Birman" (editors J Gilman, W W Menasco, X S Lin), AMS/IP Stud. Adv. Math. 24, Amer. Math. Soc. (2001) 29

[12] J M Harrison, J P Keating, J M Robbins, A Sawicki, n–particle quantum statistics on graphs, Comm. Math. Phys. 330 (2014) 1293 | DOI

[13] J H Kim, K H Ko, H W Park, Graph braid groups and right-angled Artin groups, Trans. Amer. Math. Soc. 364 (2012) 309 | DOI

[14] D E Knuth, Big omicron and big omega and big theta, SIGACT News 8 (1976) 18 | DOI

[15] K H Ko, J H La, H W Park, Graph 4–braid groups and Massey products, Topology Appl. 197 (2016) 133 | DOI

[16] K H Ko, H W Park, Characteristics of graph braid groups, Discrete Comput. Geom. 48 (2012) 915 | DOI

[17] D Lütgehetmann, Configuration spaces of graphs, master’s thesis, Freie Universität Berlin (2014)

[18] D Lütgehetmann, Representation stability for configuration spaces of graphs, preprint (2017)

[19] D Lütgehetmann, D Recio-Mitter, Topological complexity of configuration spaces of fully articulated graphs and banana graphs, preprint (2018)

[20] T Maciązek, A Sawicki, Homology groups for particles on one-connected graphs, J. Math. Phys. 58 (2017) | DOI

[21] T Maciązek, A Sawicki, Non-abelian quantum statistics on graphs, Comm. Math. Phys. 371 (2019) 921 | DOI

[22] D Mcduff, Configuration spaces of positive and negative particles, Topology 14 (1975) 91 | DOI

[23] S Papadima, A I Suciu, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006) 533 | DOI

[24] E Ramos, Stability phenomena in the homology of tree braid groups, Algebr. Geom. Topol. 18 (2018) 2305 | DOI

[25] E Ramos, Configuration spaces of graphs with certain permitted collisions, Discrete Comput. Geom. 62 (2019) 912 | DOI

[26] L Sabalka, On rigidity and the isomorphism problem for tree braid groups, Groups Geom. Dyn. 3 (2009) 469 | DOI

[27] J Świątkowski, Estimates for homological dimension of configuration spaces of graphs, Colloq. Math. 89 (2001) 69 | DOI