Algebraic & Geometric Topology Volume 24, issue 7 (2024)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

An algorithmic discrete gradient field and the cohomology algebra of configuration spaces of two points on complete graphs

Emilio J González and Jesús González

Algebraic & Geometric Topology 24 (2024) 3719–3758

DOI: 10.2140/agt.2024.24.3719

Bibliography

1	A D Abrams, Configuration spaces and braid groups of graphs, PhD thesis, University of California, Berkeley (2000) MR2701024
2	J Aguilar-Guzmán, J González, T Hoekstra-Mendoza, Farley–Sabalka’s Morse-theory model and the higher topological complexity of ordered configuration spaces on trees, Discrete Comput. Geom. 67 (2022) 258 MR4356250
3	B H An, G C Drummond-Cole, B Knudsen, Edge stabilization in the homology of graph braid groups, Geom. Topol. 24 (2020) 421 MR4080487
4	K Barnett, M Farber, Topology of configuration space of two particles on a graph, I, Algebr. Geom. Topol. 9 (2009) 593 MR2491587
5	I Basabe, J González, Y B Rudyak, D Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014) 2103 MR3331610
6	U Bauer, C Lange, M Wardetzky, Optimal topological simplification of discrete functions on surfaces, Discrete Comput. Geom. 47 (2012) 347 MR2872542
7	S Chettih, D Lütgehetmann, The homology of configuration spaces of trees with loops, Algebr. Geom. Topol. 18 (2018) 2443 MR3797072
8	M Farber, Collision free motion planning on graphs, from: "Algorithmic Foundations of Robotics, VI" (editors M Erdmann, M Overmars, D Hsu, F van der Stappen), Springer Tracts Adv. Robot. 17, Springer (2005) 123
9	M Farber, E Hanbury, Topology of configuration space of two particles on a graph, II, Algebr. Geom. Topol. 10 (2010) 2203 MR2745669
10	D Farley, Homology of tree braid groups, from: "Topological and asymptotic aspects of group theory" (editors R Grigorchuk, M Mihalik, M Sapir, Z Šunik), Contemp. Math. 394, Amer. Math. Soc. (2006) 101 MR2216709
11	D Farley, Presentations for the cohomology rings of tree braid groups, from: "Topology and robotics" (editors M Farber, R Ghrist, M Burger, D Koditschek), Contemp. Math. 438, Amer. Math. Soc. (2007) 145 MR2359035
12	D Farley, L Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005) 1075 MR2171804
13	D Farley, L Sabalka, On the cohomology rings of tree braid groups, J. Pure Appl. Algebra 212 (2008) 53 MR2355034
14	D Farley, L Sabalka, Presentations of graph braid groups, Forum Math. 24 (2012) 827 MR2949126
15	R Forman, A discrete Morse theory for cell complexes, from: "Geometry, topology, & physics" (editor S T Yau), Conf. Proc. Lect. Notes Geom. Topol. 4, International Press (1995) 112 MR1358614
16	R Forman, Discrete Morse theory and the cohomology ring, Trans. Amer. Math. Soc. 354 (2002) 5063 MR1926850
17	R Ghrist, Configuration spaces and braid groups on graphs in robotics, from: "Knots, braids, and mapping class groups" (editors J Gilman, W W Menasco, X S Lin), AMS/IP Stud. Adv. Math. 24, Amer. Math. Soc. (2001) 29 MR1873106
18	R Ghrist, Configuration spaces, braids, and robotics, from: "Braids" (editors A J Berrick, F R Cohen, E Hanbury, Y L Wong, J Wu), Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 19, World Sci. (2010) 263 MR2605308
19	R W Ghrist, D E Koditschek, Safe cooperative robot dynamics on graphs, SIAM J. Control Optim. 40 (2002) 1556 MR1882808
20	E J Gonzalez, J Gonzalez, An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes, (2023) arXiv:2307.14454
21	J González, T Hoekstra-Mendoza, Cohomology ring of tree braid groups and exterior face rings, Trans. Amer. Math. Soc. Ser. B 9 (2022) 1065 MR4524602
22	S Harker, K Mischaikow, M Mrozek, V Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math. 14 (2014) 151 MR3160710
23	B Knudsen, The topological complexity of pure graph braid groups is stably maximal, Forum Math. Sigma 10 (2022) MR4498780
24	K H Ko, H W Park, Characteristics of graph braid groups, Discrete Comput. Geom. 48 (2012) 915 MR3000570
25	L Lampret, Chain complex reduction via fast digraph traversal, preprint (2019) arXiv:1903.00783
26	T Lewiner, H Lopes, G Tavares, Applications of Forman’s discrete Morse theory to topology visualization and mesh compression, IEEE Trans. Visualiz. Comput. Graphics 10 (2004) 499
27	T Macią.zek, A Sawicki, Homology groups for particles on one-connected graphs, J. Math. Phys. 58 (2017) 062103 MR3659699
28	F Mori, M Salvetti, (Discrete) Morse theory on configuration spaces, Math. Res. Lett. 18 (2011) 39 MR2770581
29	M Mrozek, B Batko, Coreduction homology algorithm, Discrete Comput. Geom. 41 (2009) 96 MR2470072
30	J R Munkres, Elements of algebraic topology, Addison-Wesley (1984) MR0755006
31	E Ramos, An application of the theory of FI-algebras to graph configuration spaces, Math. Z. 294 (2020) 1 MR4050061
32	M Salvetti, S Settepanella, Combinatorial Morse theory and minimality of hyperplane arrangements, Geom. Topol. 11 (2007) 1733 MR2350466
33	C Severs, J A White, On the homology of the real complement of the k–parabolic subspace arrangement, J. Combin. Theory Ser. A 119 (2012) 1336 MR2915650
34	J Shareshian, Discrete Morse theory for complexes of 2–connected graphs, Topology 40 (2001) 681 MR1851558
35	V A Vassiliev, Cohomology of knot spaces, from: "Theory of singularities and its applications" (editor V I Arnold), Adv. Soviet Math. 1, Amer. Math. Soc. (1990) 23 MR1089670
36	V A Vassiliev, Complexes of connected graphs, from: "The Gelfand Mathematical Seminars, 1990–1992" (editors L Corwin, I Gelfand, J Lepowsky), Birkhäuser (1993) 223 MR1247293
37	V A Vassiliev, Topology of two-connected graphs and homology of spaces of knots, from: "Differential and symplectic topology of knots and curves" (editor S Tabachnikov), Amer. Math. Soc. Transl. Ser. 2 190, Amer. Math. Soc. (1999) 253 MR1738399