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) orientationpreserving 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.

Extending finite free actions of surfaces

“…(ii) If 𝐵 0 (𝐺) ≠ 0, then there might be both, extendable and nonextendable, free actions of 𝐺 on surfaces. In [30], there is observed that the group 𝐺 = SmallGroup(3 5 , 28) ≅ (ℤ 9 ⋊ ℤ 9 ) ⋊ ℤ 3 satisfies (2) in Theorem 2 (𝑀(𝐺) ≅ ℤ 9 and 𝐵 0 (𝐺) ≅ ℤ 3 ), so there exists a free action of it that cannot extend. In Section 5, we consider some free actions of this group and observe that some of them extend to a handlebody.…”

“…Let us consider the group 𝐺 = SmallGroup(3 5 , 28) that, by [30], admits some free action on genus 244 that does not extend. There are many tuples (𝑎, 𝑏, 𝑟, 𝑡) of elements of 𝐺, satisfying that [𝑎, 𝑏][𝑟, 𝑡] = 1 and 𝐺 = ⟨𝑎, 𝑏, 𝑟, 𝑡⟩.…”

“…In [30], Samperton observed that $$B_{0}(G)$$ provides an obstruction for the extendability of (all possible) free actions of $$G$$.…”

“…Note that these groups have the property that their Bogomolov multiplier $$B_{0}(G)$$ is zero. Generalizing these examples, in [30], Samperton proved that if $$B_{0}(G)=0$$, then any free action of $$G$$ extends. In the same paper, it was proved that if $$B_{0}(G) \ne 0$$ and $$G$$ does not contain a dihedral subgroup, then it admits a free action that does not extend (e.g., $$G={\rm SmallGroup}(3^{5},28)$$ in the Gap Library [31]).…”

“…So, by Theorem 3, this normal subgroup $$N$$ provides us with a free action of $$G_{N}=F/N$$ as a group of orientation‐preserving homeomorphisms of a closed orientable surface $$S_{N}$$ such that $$S_{N}/G_{N}=R$$ and which does not extend to a handlebody (note that there is a normal subgroup $$H<G_{N}$$ with $$S=S_{N}/H$$ and $$G=G_{N}/H$$). Two situations may happen at this point: either (i) $$G_{N}$$ does not extend (so, providing more examples as in [30]) or (ii) $$G_{N}$$ extends but it does not extend to a handlebody (providing a negative answer to Zimmermann's question). We interpret this construction as a fiber product to see that, if we also assume that all of the Sylow subgroups of $$G$$ are abelian, then $$B_{0}(G_{N})=0$$, providing in this way examples as required.…”