Representation stability for pure braid group Milnor fibers

Miller, Jeremy; Tosteson, Philip

doi:10.1090/tran/8466

Representation stability for pure braid group Milnor fibers
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by Jeremy Miller and Philip Tosteson PDF

Trans. Amer. Math. Soc. 374 (2021), 8177-8199 Request permission

Abstract:

We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Settepanella to representation stability in the sense of Church–Ellenberg–Farb, answering a question of Denham.

References

V. I. Arnol′d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231 (Russian). MR 242196
Thomas Church and Jordan S. Ellenberg, Homology of FI-modules, Geom. Topol. 21 (2017), no. 4, 2373–2418. MR 3654111, DOI 10.2140/gt.2017.21.2373
Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
Thomas Church and Benson Farb, Representation theory and homological stability, Adv. Math. 245 (2013), 250–314. MR 3084430, DOI 10.1016/j.aim.2013.06.016
Thomas Church, Jeremy Miller, Rohit Nagpal, and Jens Reinhold, Linear and quadratic ranges in representation stability, Adv. Math. 333 (2018), 1–40. MR 3818071, DOI 10.1016/j.aim.2018.05.025
Graham Denham, Problem 10, Mathematisches Forschungsinstitut, Oberwolfach Report No. 2 (2018), 74–75, https://www.mfo.de/occasion/1803/www_view.
E. Getzler, Operads and moduli spaces of genus $0$ Riemann surfaces, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 199–230. MR 1363058, DOI 10.1007/978-1-4612-4264-2_{8}
E. Getzler, Resolving mixed Hodge modules on configuration spaces, Duke Math. J. 96 (1999), no. 1, 175–203. MR 1663927, DOI 10.1215/S0012-7094-99-09605-9
Wee Liang Gan and Liping Li, Linear stable range for homology of congruence subgroups via FI-modules, Selecta Math. (N.S.) 25 (2019), no. 4, Paper No. 55, 11. MR 3997138, DOI 10.1007/s00029-019-0500-0
Trevor Hyde and Jeffrey C. Lagarias, Polynomial splitting measures and cohomology of the pure braid group, Arnold Math. J. 3 (2017), no. 2, 219–249. MR 3664267, DOI 10.1007/s40598-017-0064-z
Patricia Hersh and Victor Reiner, Representation stability for cohomology of configuration spaces in $\Bbb R^d$, Int. Math. Res. Not. IMRN 5 (2017), 1433–1486. With an appendix written jointly with Steven Sam MR 3658170, DOI 10.1093/imrn/rnw060
Allen Hatcher and Nathalie Wahl, Stabilization for mapping class groups of 3-manifolds, Duke Math. J. 155 (2010), no. 2, 205–269. MR 2736166, DOI 10.1215/00127094-2010-055
André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI 10.1006/aima.1993.1055
Alexander Kupers and Jeremy Miller, $E_n$-cell attachments and a local-to-global principle for homological stability, Math. Ann. 370 (2018), no. 1-2, 209–269. MR 3747486, DOI 10.1007/s00208-017-1533-3
John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
Jeremy Miller and Jennifer C. H. Wilson, Higher-order representation stability and ordered configuration spaces of manifolds, Geom. Topol. 23 (2019), no. 5, 2519–2591. MR 4019898, DOI 10.2140/gt.2019.23.2519
Peter Patzt, Central stability homology, Math. Z. 295 (2020), no. 3-4, 877–916. MR 4125676, DOI 10.1007/s00209-019-02365-y
Andrew Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), no. 3, 987–1027. MR 3425385, DOI 10.1007/s00222-015-0581-0
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
Simona Settepanella, A stability-like theorem for cohomology of pure braid groups of the series $A$, $B$ and $D$, Topology Appl. 139 (2004), no. 1-3, 37–47. MR 2051096, DOI 10.1016/j.topol.2003.09.003
Richard P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), no. 2, 132–161. MR 654618, DOI 10.1016/0097-3165(82)90017-6
Sheila Sundaram and Volkmar Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1389–1420. MR 1340186, DOI 10.1090/S0002-9947-97-01565-1