Comptes Rendus
Géométrie et Topologie, Théorie des groupes
Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 161-167.

We provide the first known example of a finite group action on an oriented surface T that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free) orientation-preserving action on any compact oriented 3-manifold N with boundary N=T. This implies a negative solution to a conjecture of Domínguez and Segovia, as well as Uribe’s evenness conjecture for equivariant unitary bordism groups. We more generally provide sufficient conditions implying that infinitely many such group actions on surfaces exist. Intriguingly, any group with such a non-extending action is also a counterexample to the Noether problem over the complex numbers . In forthcoming work with Segovia we give a complete homological characterization of those finite groups admitting such a non-extending action, as well as more examples and non-examples. We do not address here the analogous question for non-orientation-preserving actions.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.277
Classification : 57M60, 57M10

Eric G. Samperton 1

1 Department of Mathematics, University of Illinois, 1409 West Green Street (MC-382), Urbana, Illinois 61801, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2022__360_G2_161_0,
     author = {Eric G. Samperton},
     title = {Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {161--167},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.277},
     language = {en},
}
TY  - JOUR
AU  - Eric G. Samperton
TI  - Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 161
EP  - 167
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.277
LA  - en
ID  - CRMATH_2022__360_G2_161_0
ER  - 
%0 Journal Article
%A Eric G. Samperton
%T Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds
%J Comptes Rendus. Mathématique
%D 2022
%P 161-167
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.277
%G en
%F CRMATH_2022__360_G2_161_0
Eric G. Samperton. Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 161-167. doi : 10.5802/crmath.277. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.277/

[1] Fedor A. Bogomolov The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 51 (1987) no. 3, 688, pp. 485-516 | DOI

[2] Neal Brand Classifying spaces for branched coverings, Indiana Univ. Math. J., Volume 29 (1980) no. 2, pp. 229-248 | DOI | MR | Zbl

[3] Daryl Cooper; Craig D. Hodgson; Steven P. Kerckhoff Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, 5, Mathematical Society of Japan, 2000 (with a postface by Sadayoshi Kojima) | Zbl

[4] Jesús Emilio Domínguez; Carlos Segovia Extending free actions of finite groups on surfaces, Topology Appl., Volume 305 (2022), 107898 | DOI | MR | Zbl

[5] Jordan S. Ellenberg; Akshay Venkatesh; Craig Westerland Homological stability for Hurwitz spaces and the Cohen–Lenstra conjecture over function fields, II (2013) (https://arxiv.org/abs/1212.0923v2)

[6] Monique Gradolato; Bruno Zimmermann Extending finite group actions on surfaces to hyperbolic 3-manifolds, Math. Proc. Camb. Philos. Soc., Volume 117 (1995) no. 1, pp. 137-151 | DOI | MR | Zbl

[7] Rubén A. Hidalgo On Schottky groups with automorphism, Ann. Acad. Sci. Fenn., Math., Volume 19 (1994) no. 2, pp. 259-289 | MR | Zbl

[8] Akinari Hoshi; Ming-Chang Kang; Boris E. Kunyavskii Noether’s problem and unramified Brauer groups, Asian J. Math., Volume 17 (2013) no. 4, pp. 689-713 | DOI | MR | Zbl

[9] Urban Jezernik; Primož Moravec Bogomolov multipliers of groups of order 128, Exp. Math., Volume 23 (2014) no. 2, pp. 174-180 | DOI | MR | Zbl

[10] Boris Kunyavskiĭ The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems. New Perspectives (Fedor Bogomolov, ed.) (Progress in Mathematics), Volume 282, Birkhäuser, 2010, pp. 209-217 | DOI | MR | Zbl

[11] Primož Moravec Unramified Brauer groups of finite and infinite groups, Am. J. Math., Volume 134 (2012) no. 6, pp. 1679-1704 | DOI | MR | Zbl

[12] Erich Ossa Unitary bordism of abelian groups, Proc. Am. Math. Soc., Volume 33 (1972), pp. 568-571 | DOI | MR | Zbl

[13] Marco Reni; Bruno Zimmermann Extending finite group actions from surfaces to handlebodies, Proc. Am. Math. Soc., Volume 124 (1996) no. 9, pp. 2877-2887 | DOI | MR | Zbl

[14] Russell J. Rowlett Bordism of metacyclic group actions, Mich. Math. J., Volume 27 (1980) no. 2, pp. 223-233 | MR | Zbl

[15] David J. Saltman Noether’s problem over an algebraically closed field, Invent. Math., Volume 77 (1984) no. 1, pp. 71-84 | DOI | MR | Zbl

[16] Eric Samperton Schur-type invariants of branched G-covers of surfaces, Topological phases of matter and quantum computation. AMS special session, Bowdoin College, Brunswick, ME, USA, September 24–25, 2016 (Paul Bruillard, ed.) (Contemporary Mathematics), Volume 747, American Mathematical Society, 2020, pp. 173-197 | DOI | MR | Zbl

[17] Robert E. Stong Unoriented bordism and actions of finite groups, Memoirs of the American Mathematical Society, 103, American Mathematical Society, 1970 | Zbl

[18] The GAP Group GAP – Groups, Algorithms, and Programming, 2021 (Version 4.11.1, https://www.gap-system.org)

[19] Bernardo Uribe The evenness conjecture in equivariant unitary bordism, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures (Boyan Sirakov, ed.), World Scientific; Sociedade Brasileira de Matemática, 2018, pp. 1217-1239 | Zbl

[20] Bernardo Uribe Private communication, 2021

Cité par Sources :

Commentaires - Politique