Comptes Rendus
Géométrie et Topologie, Théorie des groupes
Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds
Eric G. Samperton1  
1 Department of Mathematics, University of Illinois, 1409 West Green Street (MC-382), Urbana, Illinois 61801, USA
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 ℂ. 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.

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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 
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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/

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