Extending free actions of finite groups on unoriented surfaces

TL;DR

The paper extends the classical Schur and Bogomolov multipliers to the unoriented setting by defining the unoriented Schur multiplier $\mathcal{M}(G;\mathbb{Z}_2)$ and proving the isomorphism $\mathcal{M}(G;\mathbb{Z}_2) \cong H^2(G;\mathbb{Z}_2)$. It introduces the unoriented Bogomolov multiplier $B_0(G;\mathbb{Z}_2)$ as a quotient by toral/cobordism classes and shows that it is trivial for abelian, dihedral, symmetric, and alternating groups, but provides a nontrivial example at order $64$. A Hopf-type formula and a cohomological framework via square-central extensions are developed, enabling computations and linking obstructions to extending free actions on unoriented surfaces to $3$-manifolds with $G$-actions. The results yield a criterion: a free action of $G$ on an unoriented closed surface extends equivariantly to a $3$-manifold if and only if the corresponding class in $B_0(G;\mathbb{Z}_2)$ is trivial, highlighting new obstructions in the unoriented case and suggesting avenues for unoriented Noether-type rationality problems.

Abstract

We present the unoriented versions of the Schur and Bogomolov multipliers associated with a finite group $G$. We show that the unoriented Schur multiplier is isomorphic to the second cohomology group $H^2(G;\ZZ_2)$. We define the unoriented Bogomolov multiplier as the quotient of the unoriented Schur multiplier by the subgroup generated by classes over the disjoint union of tori, Klein bottles, and projective spaces. We prove that the unoriented Bogomolov multiplier is trivial for abelian, dihedral, symmetric, and alternating groups. Since $H^2(G;\ZZ_2)$ is trivial for any group of odd order, there are numerous examples where the classical Bogomolov multiplier is nontrivial while its unoriented counterpart is trivial. Nevertheless, we exhibit a group of order $64$ for which the unoriented Bogomolov multiplier is nontrivial.

Figures (7)

• Figure 1: Elementary $G$-cobordisms over the cylinder, pair of pants, and disc.
• Figure 2: A $G$-cobordism over the Möbius strip.
• Figure 3: A $G$-cobordism over $\mathbb{R} P^2$.
• Figure 4: Two equivalent representations of a $G$-cobordism over a handle of genus one.
• Figure 5: Two equivalent representations of a $G$-cobordism over a Klein bottle.

Theorems & Definitions (34)

• Theorem 1: ASSU23
• Theorem 2: Mi52
• Example 1.1: The projective space $\mathbb{R} P^2$
• Example 1.2: A handle of genus $n$
• Example 1.3: The Klein bottle with a boundary circle
• Definition 3
• Proposition 4
• proof
• Theorem 5
• proof