An, Byung Hee; Drummond-Cole, Gabriel C.; Knudsen, Ben

Subdivisional Spaces and Graph Braid Groups

Doc. Math. 24, 1513-1583 (2019)
DOI: 10.25537/dm.2019v24.1513-1583


We study the problem of computing the homology of the configuration spaces of a finite cell complex \(X\). We proceed by viewing \(X\), together with its subdivisions, as a \textit{subdivisional space} -- a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose \(X\) and show that the homology of the configuration spaces of \(X\) is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. \par Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to Swiatkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.

Mathematics Subject Classification

20F36, 55R80, 55U05, 05C10


configuration spaces, cell complexes, subdivisional spaces, graphs, braid groups


  • [Abr00]. A. Abrams, Configuration spaces and braid groups of graphs, Ph.D. thesis, UC Berkeley, 2000. MR2701024.
  • [ADCK]. B. H. An, G. C. Drummond-Cole, and B. Knudsen, Edge stabilization in the homology of graph braid groups, 2018. arxiv 1806.05585.
  • [AF15]. D. Ayala and J. Francis, Factorization homology of topological manifolds, J. Topology 8 (2015), no. 4, 1045-1084. DOI 10.1112/jtopol/jtv028; zbl 1350.55009; MR3431668; arxiv 1206.5522.
  • [AFT17]. D. Ayala, J. Francis, and H. Tanaka, Factorization homology of stratified spaces, Selecta Math. (N.S.) 23 (2017), no. 1, 293-362. DOI 10.1007/s00029-016-0242-1; zbl 1365.57037; MR3595895; arxiv 1409.0848.
  • [AP17]. B. H. An and H. W. Park, On the structure of braid groups on complexes, Topology Appl. 226 (2017), no. 1, 86-119. DOI 10.1016/j.topol.2017.05.001; zbl 06728657; MR3660266; arxiv 1508.03699.
  • [Arn69]. V. Arnold, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227-231. DOI 10.1007/BF01098313; zbl 0277.55002; MR0242196.
  • [BC88]. C.-F. Bödigheimer and F. Cohen, Rational cohomology of configuration spaces of surfaces, Algebraic Topology and Transformation Groups, Lecture Notes in Math., vol. 1361, Springer, 1988, pp. 7-13. zbl 0666.55008; MR0979504.
  • [BCT89]. C.-F. Bödigheimer, F. Cohen, and L. Taylor, On the homology of configuration spaces, Topology 28 (1989), 111-123. DOI 10.1016/0040-9383(89)90035-9; zbl 0689.55012; MR0991102.
  • [Bel04]. P. Bellingeri, On presentations of surface braid groups, J. Algebra 274 (2004), no. 2, 543-563. DOI 10.1016/j.jalgebra.2003.12.009; zbl 1081.20045; MR2043362; arxiv math/0110129.
  • [BM08]. A. Bondy and M. R. Murty, Graph theory, Grad. Texts in Math., vol. 244, Springer-Verlag, 2008. zbl 1134.05001; MR2368647.
  • [Böd87]. C.-F. Bödigheimer, Stable splittings of mapping spaces, Algebraic topology (H. Miller and D. Ravenel, eds.), Lecture Notes in Math., vol. 1286, Springer, 1987, pp. 174-187. zbl 0641.55006; MR0922926.
  • [CE80]. W. H. Cunningham and J. Edmonds, A combinatorial decomposition theory, Canad. J. Math. 32 (1980), 734-765. DOI 10.4153/CJM-1980-057-7; zbl 0442.05054; MR0586989.
  • [CLM76]. F. Cohen, T. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., no. 533, Springer, 1976. zbl 0334.55009; MR0436146.
  • [DI04]. D. Dugger and D. Isaksen, Topological hypercovers and \(\mathbb{A}^1\)-realizations, Math. Z. 246 (2004), no. 4, 667-689. DOI 10.1007/s00209-003-0607-y; zbl 1055.55016; MR2045835; arxiv math/0111287.
  • [Dug17]. D. Dugger, A primer on homotopy colimits, preprint, 2008-2017.
  • [Far03]. M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211-221. DOI 10.1007/s00454-002-0760-9; zbl 1038.68130; MR1957228; arxiv math/0111197.
  • [Far05]. M. Farber, Collision free motion planning on graphs, Algorithmic Foundations of Robotics VI (M. Erdmann, M. Overmars, D. Hsu, and F. van der Stappen, eds.), Springer Tracts Adv. Robot., vol. 17, Springer, 2005, pp. 123-138. arxiv math/0406361.
  • [Far06]. D. Farley, Homology of tree braid groups, preprint, 2006.
  • [FN62]. E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. DOI 10.7146/math.scand.a-10517; zbl 0136.44104; MR0141126.
  • [For98]. R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90-145. DOI 10.1006/aima.1997.1650; zbl 0896.57023; MR1612391.
  • [FS05]. D. Farley and L. Sabalka, Discrete Morse theory and graph braid groups, Alg. Geom. Topol. 5 (2005), 1075-1109. DOI 10.2140/agt.2005.5.1075; zbl 1134.20050; MR2171804; arxiv math/0410539.
  • [FS08]. D. Farley and L. Sabalka, On the cohomology rings of tree braid groups, J. Pure Appl. Algebra 212 (2008), no. 1, 53-71. DOI 10.1016/j.jpaa.2007.04.011; zbl 1137.20027; MR2355034; arxiv math/0602444.
  • [FS12]. D. Farley and L. Sabalka, Presentations of graph braid groups, Forum Math. 24 (2012), 827-860. DOI 10.1515/form.2011.086; zbl 1262.20041; MR2949126; arxiv 0907.2730.
  • [FT00]. Y. Félix and J.-C. Thomas, Rational Betti numbers of configuration spaces, Topology Appl. 102 (2000), 139-149. DOI 10.1016/S0166-8641(98)00148-5; zbl 0948.55014; MR1741482.
  • [Gal01]. S. Gal, Euler characteristic of the configuration space of a complex, Colloq. Math. 89 (2001), no. 1, 61-67. DOI 10.4064/cm89-1-4; zbl 0982.55012; MR1853415; arxiv math/0202143.
  • [Ghr02]. R. Ghrist, Configuration spaces and braid groups on graphs in robotics, Knots, braids, and mapping class groups - papers dedicated to Joan S. Birman (New York, 1998), AMS/IP Stud. Adv. Math., vol. 24, Amer. Math. Soc., 2002, pp. 29-40. zbl 1010.55012; MR1873106; arxiv math/9905023.
  • [Gin13]. G. Ginot, Notes on factorization algebras, factorization homology and applications. Preprint , 2013. arxiv 1307.5213.
  • [GJ09]. P. Goerss and J. F. Jardine, Simplicial homotopy theory, Modern Birkhäuser Classics, Birkhäuser, 2009. DOI 10.1007/978-3-0346-0189-4; zbl 1195.55001; MR2840650.
  • [GK98]. R. Ghrist and D. Koditschek, Safe cooperative robot patterns via dynamics on graphs, Robotics Research: The Eighth International Symposium (Y. Shirai and S. Hirose, eds.), Springer, 1998, pp. 81-92. MR1882808.
  • [Hir02]. P. Hirschhorn, Model categories and their localizations, Math. Surveys Monogr., vol. 99, Amer. Math. Soc., Providence, Rhode Island, 2002. zbl 1017.55001; MR1944041.
  • [HKRS14]. J. M. Harrison, J. P. Keating, J. M. Robbins, and A. Sawicki, n-particle quantum statistics on graphs, Comm. Math. Phys. 330 (2014), 1293-1326. DOI 10.1007/s00220-014-2091-0; zbl 1294.81049; MR3227513; arxiv 1304.5781.
  • [IMW16]. S. Ivanov, R. Mikhailov, and J. Wu, On nontriviality of certain homotopy groups of spheres, Homology Homotopy Appl. 18 (2016), no. 2, 337-344. DOI 10.4310/HHA.2016.v18.n2.a18; zbl 1366.55008; MR3576002; arxiv 1506.00952.
  • [KKP12]. J. H. Kim, H. K Ko, and H. W. Park, Graph braid groups and right-angled Artin groups, Trans. Amer. Math. Soc. 364 (2012), 309-360. DOI 10.1090/S0002-9947-2011-05399-7; zbl 1281.20040; MR2833585; arxiv 0805.0082.
  • [Knu17]. B. Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Alg. Geom. Topol. 17 (2017), no. 5, 3137-3187. DOI 10.2140/agt.2017.17.3137; zbl 1377.57025; MR3704255; arxiv 1405.6696.
  • [Kon92]. M. Kontsevich, Intersection theory on the moduli space of curves and matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1-23. DOI 10.1007/BF02099526; zbl 0756.35081; MR1171758.
  • [KP12]. K. H. Ko and H. W. Park, Characteristics of graph braid groups, Discrete Comput. Geom. 48 (2012), no. 4, 915-963. DOI 10.1007/s00454-012-9459-8; zbl 1256.05100; MR3000570; arxiv 1101.2648.
  • [KS06]. M. Kashiwara and P. Schapira, Categories and sheaves, Grundlehren Math. Wiss., vol. 332, Springer, 2006. DOI 10.1007/3-540-27950-4; zbl 1118.18001; MR2182076.
  • [LS05]. R. Longoni and P. Salvatore, Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375-380. DOI 10.1016/; zbl 1063.55015; MR2114713; arxiv math/0401075.
  • [Lur09]. J. Lurie, Higher topos theory, Ann. of Math. Stud., Princeton University Press, 2009. DOI 10.1515/9781400830558; zbl 1175.18001; MR2522659; arxiv math/0608040.
  • [Lüt14]. Daniel Lütgehetmann, Configuration spaces of graphs, Master's thesis, Freie Universität Berlin, 2014.
  • [LW69]. A. Lundell and S. Weingram, The topology of CW complexes, The University Series in Higher Mathematics, vol. 4, Springer, 1969. zbl 0207.21704; MR3822092.
  • [McD75]. D. McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91-107. DOI 10.1016/0040-9383(75)90038-5; zbl 0296.57001; MR0358766.
  • [Men27]. K. Menger, Zur allgemeinen Kurventheorie, Fund. Math. 10 (1927), no. 1, 96-115. DOI 10.4064/fm-10-1-96-115; JFM 53.0561.01.
  • [MS17]. T. Maciazek and A. Sawicki, Homology groups for particles on one-connected graphs, J. Math. Phys. 58 (2017), no. 6. DOI 10.1063/1.4984309; zbl 1368.81078; MR3659699; arxiv 1606.03414.
  • [Ram18]. E. Ramos, Stability phenomena in the homology of tree braid groups, Alg. Geom. Topol. 18 (2018), no. 4, 2305-2337. DOI 10.2140/agt.2018.18.2305; zbl 1387.20041; MR3797068; arxiv 1609.05611.
  • [Rie14]. E. Riehl, Categorical homotopy theory, New Mathematical Monographs, vol. 24, Cambridge University Press, 2014. DOI 10.1017/CBO9781107261457; zbl 1317.18001; MR3221774.
  • [Sab09]. L. Sabalka, On rigidity and the isomorphism problem for tree braid groups, Groups Geom. Dyn. 3 (2009), no. 3, 469-523. DOI 10.4171/GGD/67; zbl 1231.20033; MR2516176; arxiv 0711.1160.
  • [Seg73]. G. Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213-221. DOI 10.1007/BF01390197; zbl 0267.55020; MR0331377.
  • [Smi67]. L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), no. 1, 58-93. DOI 10.2307/1994364; zbl 0177.51402; MR0216504.
  • [Świ01]. Jacek Swiatkowski, Estimates for homological dimension of configuration spaces of graphs, Colloq. Math. 89 (2001), no. 1, 69-79. DOI 10.4064/cm89-1-5; zbl 1007.55013; MR1853416.
  • [Tot96]. B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057-1067. DOI 10.1016/0040-9383(95)00058-5; zbl 0857.57025; MR1404924.
  • [Tut66]. W. T. Tutte, Connectivity in graphs, Mathematical Expositions, vol. 15, Toronto University Press, 1966. zbl 0146.45603; MR0210617.
  • [Wag37]. K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), no. 1, 570-590. DOI 10.1007/BF01594196; zbl 0017.19005; MR1513158.
  • [WG]. J. D. Wiltshire-Gordon, Models for configuration space in a simplicial complex, to appear in Colloq. Math. arxiv 1706.06626.


An, Byung Hee
Center for Geometry and Physics, Institute for Basic Science, Pohang 37673, Republic of Korea
Drummond-Cole, Gabriel C.
Center for Geometry and Physics, Institute for Basic Science, Pohang 37673, Republic of Korea
Knudsen, Ben
Mathematics Department, Northeastern University, Boston 02115, USA