Polynomiality of surface braid and mapping class group representations

Palmer, Martin; Soulié, Arthur

doi:10.1090/tran/9409

Polynomiality of surface braid and mapping class group representations
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by Martin Palmer and Arthur Soulié;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9409

Published electronically: May 1, 2025

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Abstract:

We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally come in families, defining homological representation functors on categories associated to surface braid groups or all mapping class groups. We prove that many of these homological representation functors are polynomial. This has applications to twisted homological stability and to understanding the structure of the representation theory of the associated families of groups. Our polynomiality results are consequences of more fundamental results establishing relations amongst the coherent representations that we consider via short exact sequences of functors. As well as polynomiality, these short exact sequences also have applications to understanding the kernels of the homological representations under consideration.

References

Byung Hee An and Ki Hyoung Ko, A family of representations of braid groups on surfaces, Pacific J. Math. 247 (2010), no. 2, 257–282. MR 2734148, DOI 10.2140/pjm.2010.247.257
Cristina Ana-Maria Anghel and Martin Palmer, Lawrence-Bigelow representations, bases and duality, arXiv:2011.02388, 2020.
Reinhold Baer, Kurventypen auf Flächen, J. Reine Angew. Math. 156 (1927), 231–246 (German). MR 1581099, DOI 10.1515/crll.1927.156.231
Reinhold Baer, Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen, J. Reine Angew. Math. 159 (1928), 101–116 (German). MR 1581156, DOI 10.1515/crll.1928.159.101
Paolo Bellingeri, On presentations of surface braid groups, J. Algebra 274 (2004), no. 2, 543–563. MR 2043362, DOI 10.1016/j.jalgebra.2003.12.009
Paolo Bellingeri, Eddy Godelle, and John Guaschi, Abelian and metabelian quotient groups of surface braid groups, Glasg. Math. J. 59 (2017), no. 1, 119–142. MR 3576330, DOI 10.1017/S0017089516000070
Stephen Bigelow, Homological representations of the Iwahori-Hecke algebra, Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, 2004, pp. 493–507. MR 2172492, DOI 10.2140/gtm.2004.7.493
Stephen J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471–486. MR 1815219, DOI 10.1090/S0894-0347-00-00361-1
Joan S. Birman and Tara E. Brendle, Braids: a survey, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 19–103. MR 2179260, DOI 10.1016/B978-044451452-3/50003-4
Christian Blanchet, Martin Palmer, and Awais Shaukat, Heisenberg homology on surface configurations, Mathematische Annalen, To appear, arXiv:2109.00515, 2021.
Søren K. Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients, Math. Z. 270 (2012), no. 1-2, 297–329. MR 2875835, DOI 10.1007/s00209-010-0798-y
Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
Werner Burau, Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 179–186 (German). MR 3069652, DOI 10.1007/BF02940722
Jacques Darné, Martin Palmer, and Arthur Soulié, When the lower central series stops: a comprehensive study for braid groups and their relatives, Mem. Amer. Math. Soc., To appear, https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo, arXiv:2201.03542, 2022.
James F. Davis and Paul Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR 1841974, DOI 10.1090/gsm/035
Aurélien Djament, Antoine Touzé, and Christine Vespa, Décompositions à la Steinberg sur une catégorie additive, Ann. Sci. Éc. Norm. Supér. (4) 56 (2023), no. 2, 427–516 (French, with English and French summaries). MR 4598727, DOI 10.24033/asens.2538
Aurélien Djament and Christine Vespa, Foncteurs faiblement polynomiaux, Int. Math. Res. Not. IMRN 2 (2019), 321–391 (French, with English and French summaries). MR 3903561, DOI 10.1093/imrn/rnx099
Samuel Eilenberg and Saunders Mac Lane, On the groups $H(\Pi ,n)$. II. Methods of computation, Ann. of Math. (2) 60 (1954), 49–139. MR 65162, DOI 10.2307/1969702
D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 214087, DOI 10.1007/BF02392203
Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Suslin, General linear and functor cohomology over finite fields, Ann. of Math. (2) 150 (1999), no. 2, 663–728. MR 1726705, DOI 10.2307/121092
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
Daniel Grayson, Higher algebraic $K$-theory. II (after Daniel Quillen), Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin-New York, 1976, pp. 217–240. MR 574096
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Reinhard Häring-Oldenburg and Sofia Lambropoulou, Knot theory in handlebodies, J. Knot Theory Ramifications 11 (2002), no. 6, 921–943. Knots 2000 Korea, Vol. 3 (Yongpyong). MR 1936243, DOI 10.1142/S0218216502002050
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Nikolai V. Ivanov, On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 149–194. MR 1234264, DOI 10.1090/conm/150/01290
Craig Jackson and Thomas Kerler, The Lawrence-Krammer-Bigelow representations of the braid groups via $U_q(\mathfrak {sl}_2)$, Adv. Math. 228 (2011), no. 3, 1689–1717. MR 2824566, DOI 10.1016/j.aim.2011.06.027
Toshitake Kohno, Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 4, 144–147. MR 846350
Toshitake Kohno, One-parameter family of linear representations of Artin’s braid groups, Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986) Adv. Stud. Pure Math., vol. 12, North-Holland, Amsterdam, 1987, pp. 189–200. MR 948243, DOI 10.2969/aspm/01210189
Daan Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131–156. MR 1888796, DOI 10.2307/3062152
R. J. Lawrence, Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1990), no. 1, 141–191. MR 1086755
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Dan Margalit, Problems, questions, and conjectures about mapping class groups, Breadth in contemporary topology, Proc. Sympos. Pure Math., vol. 102, Amer. Math. Soc., Providence, RI, 2019, pp. 157–186. MR 3967367, DOI 10.1090/pspum/102/12
Jules Martel, A homological model for $U_q\mathfrak {sl}2$ Verma modules and their braid representations, Geom. Topol. 26 (2022), no. 3, 1225–1289. MR 4466648, DOI 10.2140/gt.2022.26.1225
Tetsuhiro Moriyama, The mapping class group action on the homology of the configuration spaces of surfaces, J. Lond. Math. Soc. (2) 76 (2007), no. 2, 451–466. MR 2363426, DOI 10.1112/jlms/jdm077
Martin Palmer, A comparison of twisted coefficient systems, arXiv:1712.06310, 2017.
Martin Palmer and Arthur Soulié, Irreducibility of surface braid and mapping class group representations, In preparation.
Martin Palmer and Arthur Soulié, The pro-nilpotent Lawrence-Krammer-Bigelow representation, J. Pure Appl. Algebra 229 (2025), no. 6, Paper No. 107952. MR 4885633, DOI 10.1016/j.jpaa.2025.107952
Martin Palmer and Arthur Soulié, Annex file to the present article, https://mdp.ac/papers/polynomiality/annex-file.pdf, arXiv:2302.08827v3/anc, 2024.
Martin Palmer and Arthur Soulié, Topological representations of motion groups and mapping class groups—a unified functorial construction, Ann. H. Lebesgue 7 (2024), 409–519 (English, with English and French summaries). MR 4799903
Oscar Randal-Williams and Nathalie Wahl, Homological stability for automorphism groups, Adv. Math. 318 (2017), 534–626. MR 3689750, DOI 10.1016/j.aim.2017.07.022
Takuya Sakasai, A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces, Handbook of Teichmüller theory. Volume III, IRMA Lect. Math. Theor. Phys., vol. 17, Eur. Math. Soc., Zürich, 2012, pp. 531–594. MR 2952771, DOI 10.4171/103-1/10
Arthur Soulié, The Long-Moody construction and polynomial functors, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 4, 1799–1856 (English, with English and French summaries). MR 4010870, DOI 10.5802/aif.3282
Arthur Soulié, Generalized Long-Moody functors, Algebr. Geom. Topol. 22 (2022), no. 4, 1713–1788. MR 4495668, DOI 10.2140/agt.2022.22.1713
Andreas Stavrou, Cohomology of configuration spaces of surfaces as mapping class group representations, Trans. Amer. Math. Soc. 376 (2023), no. 4, 2821–2852. MR 4557882, DOI 10.1090/tran/8804
Masaaki Suzuki, Geometric interpretation of the Magnus representation of the mapping class group, Kobe J. Math. 22 (2005), no. 1-2, 39–47. MR 2203329

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Polynomiality of surface braid and mapping class group representations
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Abstract: