## Mapping class group actions on configuration spaces and the Johnson filtration

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- by Andrea Bianchi, Jeremy Miller and Jennifer C. H. Wilson PDF
- Trans. Amer. Math. Soc.
**375**(2022), 5461-5489 Request permission

## Abstract:

Let $F_n(\Sigma _{g,1})$ denote the configuration space of $n$ ordered points on the surface $\Sigma _{g,1}$ and let $\Gamma _{g,1}$ denote the mapping class group of $\Sigma _{g,1}$. We prove that the action of $\Gamma _{g,1}$ on $H_i(F_n(\Sigma _{g,1});\mathbb {Z})$ is trivial when restricted to the $i$th stage of the Johnson filtration $\mathcal {J}(i)\subset \Gamma _{g,1}$. We give examples showing that $\mathcal {J}(2)$ acts nontrivially on $H_3(F_3(\Sigma _{g,1}))$ for $g\ge 2$, and provide two new conceptual reinterpretations of a certain group introduced by Moriyama.## References

- Andrea Bianchi,
*Splitting of the homology of the punctured mapping class group*, J. Topol.**13**(2020), no. 3, 1230–1260. MR**4125755**, DOI 10.1112/topo.12153 - Nathan Broaddus,
*Homology of the curve complex and the Steinberg module of the mapping class group*, Duke Math. J.**161**(2012), no. 10, 1943–1969. MR**2954621**, DOI 10.1215/00127094-1645634 - C. J. Earle and A. Schatz,
*Teichmüller theory for surfaces with boundary*, J. Differential Geometry**4**(1970), 169–185. MR**277000** - R. Fox and L. Neuwirth,
*The braid groups*, Math. Scand.**10**(1962), 119–126. MR**150755**, DOI 10.7146/math.scand.a-10518 - D. B. Fuks,
*Cohomology of the braid group $\textrm {mod}\ 2$*, Funkcional. Anal. i Priložen.**4**(1970), no. 2, 62–73 (Russian). MR**0274463** - André Gramain,
*Le type d’homotopie du groupe des difféomorphismes d’une surface compacte*, Ann. Sci. École Norm. Sup. (4)**6**(1973), 53–66 (French). MR**326773** - 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 - Dennis Johnson,
*An abelian quotient of the mapping class group ${\cal I}_{g}$*, Math. Ann.**249**(1980), no. 3, 225–242. MR**579103**, DOI 10.1007/BF01363897 - Dennis Johnson,
*The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves*, Topology**24**(1985), no. 2, 113–126. MR**793178**, DOI 10.1016/0040-9383(85)90049-7 - 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 - Manuel Krannich,
*Homological stability of topological moduli spaces*, Geom. Topol.**23**(2019), no. 5, 2397–2474. MR**4019896**, DOI 10.2140/gt.2019.23.2397 - Eduard Looijenga,
*Torelli group action on the configuration space of a surface*, Journal of Topology and Analysis, DOI 10.1142/S1793525322500030. - 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 - 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 - 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 - Louis Solomon,
*The Steinberg character of a finite group with $BN$-pair*, Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, 1969, pp. 213–221. MR**0246951** - Andreas Stavrou,
*Cohomology of configuration spaces of surfaces as mapping class group representations*, arXiv:2107.08462, 2021.

## Additional Information

**Andrea Bianchi**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, Copenhagen 2100, Denmark
- MR Author ID: 1324567
- ORCID: 0000-0003-0209-6416
- Email: anbi@math.ku.dk
**Jeremy Miller**- Affiliation: Department of Mathematics, Purdue University, 150 North University, West Lafayette, Indiana 47907
- MR Author ID: 1009804
- Email: jeremykmiller@purdue.edu
**Jennifer C. H. Wilson**- Affiliation: Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, Michigan 48109
- MR Author ID: 906642
- ORCID: 0000-0002-6601-7468
- Email: jchw@umich.edu
- Received by editor(s): August 30, 2021
- Received by editor(s) in revised form: November 29, 2021
- Published electronically: May 4, 2022
- Additional Notes: The first author was supported by the Danish National Research Foundation through the Centre for Geometry and Topology (DNRF151) and the European Research Council under the European Union Horizon 2020 research and innovation programme (grant agreement No. 772960).

The second author was supported in part by NSF grant DMS-1709726 and a Simons Foundation Collaboration Grants for Mathematicians.

The third author was supported in part by NSF grant DMS-1906123. - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5461-5489 - MSC (2020): Primary 55R80, 57K20
- DOI: https://doi.org/10.1090/tran/8637
- MathSciNet review: 4469226