[Le groupe de difféotopie d’un corps en anses est virtuellement un groupe de dualité]
We show that the mapping class group of a handlebody is a virtual duality group, in the sense of Bieri and Eckmann. In positive genus we give a description of the dualising module of any torsion-free, finite-index subgroup of the handlebody mapping class group as the homology of the complex of non-simple disc systems.
Nous montrons que le groupe de difféotopie d’un corps en anses est virtuellement un groupe de dualité, au sens de Bieri–Eckmann. Pour tout genre positif, nous donnons une description du module dualisant de tout sous-groupe sans torsion et d’indice fini du groupe de difféotopie du corps en anses comme l’homologie du complexe des systèmes de disques non simples.
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Keywords: Bieri–Eckmann duality, mapping class groups, handlebodies, Steinberg module, disk complex
Mots-clés : Dualité de Bieri-Eckmann, groupes de difféotopie, corps en anses, module de Steinberg, complexe des disques
Dan Petersen 1 ; Richard D. Wade 2
CC-BY 4.0
Dan Petersen; Richard D. Wade. The handlebody group is a virtual duality group. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1007-1028. doi: 10.5802/jep.341
@article{JEP_2026__13__1007_0,
author = {Dan Petersen and Richard D. Wade},
title = {The handlebody group is a~virtual~duality~group},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1007--1028},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.341},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.341/}
}
TY - JOUR AU - Dan Petersen AU - Richard D. Wade TI - The handlebody group is a virtual duality group JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1007 EP - 1028 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.341/ DO - 10.5802/jep.341 LA - en ID - JEP_2026__13__1007_0 ER -
%0 Journal Article %A Dan Petersen %A Richard D. Wade %T The handlebody group is a virtual duality group %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 1007-1028 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.341/ %R 10.5802/jep.341 %G en %F JEP_2026__13__1007_0
[BE73] - “Groups with homological duality generalizing Poincaré duality”, Invent. Math. 20 (1973), p. 103-124 | DOI | Zbl | MR
[BF00] - “The topology at infinity of ”, Invent. Math. 140 (2000) no. 3, p. 651-692 | MR | DOI | Zbl
[BG20] - “Homotopy type of the complex of free factors of a free group”, Proc. London Math. Soc. (3) 121 (2020) no. 6, p. 1737-1765 | DOI | Zbl | MR
[Bro75] - “Homological criteria for finiteness”, Comment. Math. Helv. 50 (1975), p. 129-135 | DOI | Zbl | MR
[Bro94] - Cohomology of groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York, 1994 | MR
[Brü24] - “(Non-)Vanishing of high-dimensional group cohomology”, 2024 | arXiv | Zbl
[BS73] - “Corners and arithmetic groups”, Comment. Math. Helv. 48 (1973), p. 436-491 | MR | DOI | Zbl
[BSV18] - “On the bordification of outer space”, J. London Math. Soc. (2) 98 (2018) no. 1, p. 12-34 | MR | DOI | Zbl
[Cho08] - “Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two”, Proc. Amer. Math. Soc. 136 (2008) no. 3, p. 1113-1123 | MR | DOI | Zbl
[CV86] - “Moduli of graphs and automorphisms of free groups”, Invent. Math. 84 (1986) no. 1, p. 91-119 | MR | DOI | Zbl
[FM12] - A primer on mapping class groups, Princeton Math. Series, vol. 49, Princeton University Press, Princeton, NJ, 2012 | Zbl | MR
[Gia11] - “The framed little 2-discs operad and diffeomorphisms of handlebodies”, J. Topology 4 (2011) no. 4, p. 919-941 | DOI | Zbl | MR
[Har86] - “The virtual cohomological dimension of the mapping class group of an orientable surface”, Invent. Math. 84 (1986) no. 1, p. 157-176 | DOI | Zbl | MR
[Hen20] - “A primer on handlebody groups”, in Handbook of group actions. V, Adv. Lect. Math. (ALM), vol. 48, International Press, Somerville, MA, 2020, p. 143-177 | Zbl
[Hir03] - “Some homological invariants of the mapping class group of a three-dimensional handlebody”, Tohoku Math. J. (2) 55 (2003) no. 4, p. 543-549 | DOI | Zbl | MR
[HMNP23] - “The free factor complex and the dualizing module for the automorphism group of a free group”, Internat. Math. Res. Notices (2023) no. 22, p. 19020-19068 | DOI | Zbl | MR
[Hov99] - Model categories, Math. Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999 | Zbl | MR
[HP25] - “Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies”, J. Eur. Math. Soc. (JEMS) (2025), published online first | DOI | Zbl
[HV17] - “Tethers and homology stability for surfaces”, Algebraic Geom. Topol. 17 (2017) no. 3, p. 1871-1916 | DOI | Zbl | MR
[HW10] - “Stabilization for mapping class groups of 3-manifolds”, Duke Math. J. 155 (2010) no. 2, p. 205-269 | MR | DOI | Zbl
[Iva87] - “Complexes of curves and Teichmüller modular groups”, Uspehi Mat. Nauk 42 (1987) no. 3(255), p. 49-91 | MR | Zbl
[JW10] - “A cofinite universal space for proper actions for mapping class groups”, in In the tradition of Ahlfors-Bers. V, Contemp. Math., vol. 510, American Mathematical Society, Providence, RI, 2010, p. 151-163 | DOI | Zbl
[Loo13] - “Connectivity of complexes of separating curves”, Groups Geom. Dyn. 7 (2013) no. 2, p. 443-450 | MR | DOI | Zbl
[McC91] - “Virtually geometrically finite mapping class groups of -manifolds”, J. Differential Geom. 33 (1991) no. 1, p. 1-65 | MR | DOI | Zbl
[MV15] - Cubical homotopy theory, New Math. Monographs, vol. 25, Cambridge University Press, Cambridge, 2015 | MR | DOI | Zbl
[Qui78] - “Homotopy properties of the poset of nontrivial -subgroups of a group”, Adv. Math. 28 (1978) no. 2, p. 101-128 | DOI | Zbl
[Vog24] - “The boundary of bordified Outer space”, 2024 | arXiv | Zbl
[WW25] - “Cohen-Macaulay complexes, duality groups, and the dualizing module of ”, Internat. Math. Res. Notices (2025) no. 6, article ID rnaf059, 20 pages | Zbl | MR | DOI
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