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

TL;DR

This paper presents the first example of a free, orientation-preserving finite group action on a surface that cannot extend to any compact oriented 3-manifold, challenging existing conjectures and revealing new insights into group actions.

Contribution

It provides the first known example of non-extendable free actions on surfaces and establishes conditions for their existence, impacting conjectures in topology and algebra.

Findings

First example of non-extendable free surface action

Counterexample to a conjecture of Domínguez and Segovia

Implications for equivariant unitary bordism groups

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 Dom\'inguez 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.