Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds

Abstract

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 $\partial N=T$. 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 that imply 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 $\mathbb{C}$. 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.

7 pages. Moved appendix into main body, various other minor changes. To appear in Comptes Rendus - Mathématique

References in corpus (2)

• Noether's problem and unramified Brauer groups
• Extending free group action on surfaces

Cited by in corpus (1)

• Oriented and unitary equivariant bordism of surfaces