The Burau representations of loop braid groups

Palmer, Martin; Soulié, Arthur

Comptes Rendus. Mathématique, Tome 360 (2022) no. G7, pp. 781-797.

Nous donnons une construction topologique simple et naturelle des représentations de Burau des groupes de tresses soudées. Il en existe en fait quatre versions : ces représentations peuvent être définies pour les groupes de tresses soudées étendues ou non étendues, et dans ces deux cas, il y a une version réduite et une autre non réduite. Pour trois d’entre elles, d’un point de vue rigoureusement algébrique, on peut aisément déterminer les matrices correspondant aux générateurs des groupes considérés. En revanche, la quatrième est plus subtile et ne semble pas évidente à déterminer d’un strict point de vue combinatoire, alors qu’elle est topologiquement très naturelle à définir.

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.

Classification : 20C12, 20F36, 20J05, 57M07, 57M10

[1] Baez, John C.; Wise, Derek K.; Crans, Alissa S. Exotic statistics for strings in 4D $BF$ theory, Adv. Theor. Math. Phys., Volume 11 (2007) no. 5, pp. 707-749 | DOI | MR | Zbl

[2] Bardakov, Valerij G. Extending representations of braid groups to the automorphism groups of free groups, J. Knot Theory Ramifications, Volume 14 (2005) no. 8, pp. 1087-1098 | DOI | MR | Zbl

[3] Bellingeri, Paolo; Bodin, Arnaud The braid group of a necklace, Math. Z., Volume 283 (2016) no. 3-4, pp. 995-1010 | DOI | MR | Zbl

[4] Bellingeri, Paolo; Soulié, Arthur A note on representations of welded braid groups, J. Knot Theory Ramifications, Volume 29 (2020) no. 12, 2050082 | DOI | MR | Zbl

[5] Bigelow, Stephen The Burau representation is not faithful for $n=5$, Geom. Topol., Volume 3 (1999), pp. 397-404 | DOI | MR | Zbl

[6] Bigelow, Stephen Homological representations of the Iwahori-Hecke algebra, Proceedings of the Casson Fest (Geometry and Topology Monographs), Volume 7, Geometry and Topology Publications (2004), pp. 493-507 | DOI | MR | Zbl

[7] Binz, Ernst; Fischer, Hans R. The manifold of embeddings of a closed manifold, Differential geometric methods in mathematical physics (Proc. Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978) (Lecture Notes in Physics), Volume 139, Springer (1981), pp. 310-329 (with an appendix by P. Michor) | MR

[8] Brendle, Tara E.; Hatcher, Allen E. Configuration spaces of rings and wickets, Comment. Math. Helv., Volume 88 (2013) no. 1, pp. 131-162 | DOI | MR | Zbl

[9] Brownstein, Alan; Lee, Ronnie Cohomology of the group of motions of $n$ strings in $3$-space, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) (Contemporary Mathematics), Volume 150, American Mathematical Society, 1993, pp. 51-61 | DOI | MR | Zbl

[10] Burau, Werner Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Volume 11, Springer (1935), pp. 179-186 | DOI | Zbl

[11] Cerf, Jean Topologie de certains espaces de plongements, Bull. Soc. Math. Fr., Volume 89 (1961), pp. 227-380 | DOI | Numdam | MR | Zbl

[12] Collins, Donald J. Cohomological dimension and symmetric automorphisms of a free group, Comment. Math. Helv., Volume 64 (1989) no. 1, pp. 44-61 | DOI | MR | Zbl

[13] Dahm, David Michael A generalization of braid theory, ProQuest LLC, 1962, 59 pages Thesis (Ph.D.)–Princeton University | MR

[14] Damiani, Celeste A journey through loop braid groups, Expo. Math., Volume 35 (2017) no. 3, pp. 252-285 | DOI | MR | Zbl

[15] Damiani, Celeste; Martin, Paul; Rowell, Eric C. Generalisations of Hecke algebras from Loop Braid Groups (2020) (http://arxiv.org/abs/2008.04840)

[16] Fenn, Roger; Rimányi, Richárd; Rourke, Colin The braid-permutation group, Topology, Volume 36 (1997) no. 1, pp. 123-135 | DOI | MR | Zbl

[17] Goldsmith, Deborah L. The theory of motion groups, Mich. Math. J., Volume 28 (1981) no. 1, pp. 3-17 | MR | Zbl

[18] Hatcher, Allen E. A proof of the Smale conjecture, $\mathrm{Diff}\left(S^3\right)\simeq \mathrm{O}\left(4\right)$, Ann. Math., Volume 117 (1983) no. 3, pp. 553-607 | DOI | MR | Zbl

[19] Ibrahim, Abdoulrahim On matrix representations for generalized braid groups (2021) (http://arxiv.org/abs/2109.03583)

[20] Jensen, Craig; McCammond, Jon; Meier, John The integral cohomology of the group of loops, Geom. Topol., Volume 10 (2006), pp. 759-784 | DOI | MR | Zbl

[21] Kádár, Zoltán; Martin, Paul; Rowell, Eric C.; Wang, Zhenghan Local representations of the loop braid group, Glasg. Math. J., Volume 59 (2017) no. 2, pp. 359-378 | DOI | MR | Zbl

[22] Kassel, Christian; Turaev, Vladimir Braid groups, Graduate Texts in Mathematics, 247, Springer, 2008, xii+340 pages (with the graphical assistance of Olivier Dodane) | DOI | MR

[23] Lawrence, R. J. Homological representations of the Hecke algebra, Commun. Math. Phys., Volume 135 (1990) no. 1, pp. 141-191 | DOI | MR | Zbl

[24] Lima, Elon L. On the local triviality of the restriction map for embeddings, Comment. Math. Helv., Volume 38 (1963), pp. 163-164 | DOI | MR

[25] Long, Darren D.; Paton, Martin The Burau representation is not faithful for $n\ge 6$, Topology, Volume 32 (1993) no. 2, pp. 439-447 | DOI | MR | Zbl

[26] McCool, James On basis-conjugating automorphisms of free groups, Can. J. Math., Volume 38 (1986) no. 6, pp. 1525-1529 | DOI | MR | Zbl

[27] Moody, John Atwell The Burau representation of the braid group $B_n$ is unfaithful for large $n$, Bull. Am. Math. Soc., Volume 25 (1991) no. 2, pp. 379-384 | DOI | MR | Zbl

[28] Palais, Richard S. Local triviality of the restriction map for embeddings, Comment. Math. Helv., Volume 34 (1960), pp. 305-312 | DOI | MR | Zbl

[29] Palmer, Martin Homological stability for moduli spaces of disconnected submanifolds, I, Algebr. Geom. Topol., Volume 21-3 (2021), pp. 1371-1444 | DOI | MR | Zbl

[30] Palmer, Martin; Soulié, Arthur Topological representations of motion groups and mapping class groups – a unified functorial construction (2021) (https://arxiv.org/abs/1910.13423v4)

[31] Palmer, Martin; Soulié, Arthur Irreducibility and polynomiality of motion group and mapping class group representations (2022) (preprint in progress, to be posted on the arXiv soon)

[32] Palmer, Martin; Soulié, Arthur Verifying the extended loop braid relations for the reduced Burau representation of ${\mathbf{LB}}_4^{\prime }$ (2022) (supplementary materials for the present article; available at either https://mdp.ac/papers/loop-Burau/index.html or https://arxiv.org/src/2109.11468v2/anc)

[33] Vershinin, Vladimir V. On homology of virtual braids and Burau representation, J. Knot Theory Ramifications, Volume 10 (2001) no. 5, pp. 795-812 Knots in Hellas ’98, Vol. 3 (Delphi) | DOI | MR | Zbl