Extending free actions of finite groups on surfaces (Q2052578)

Working with oriented manifolds and orientation-preserving diffeomorphisms, the basic problem considered in the present paper is the following: when does a free action of a finite group \(G\) on a closed surface \(S\) extend to an action of \(G\), not necessarily free, on a compact 3-manifold \(M\) with boundary \(\partial M = S\)? (An action is free if every nontrivial element acts without fixed points.) The obstruction for an extension to an action which is free also on the 3-manifold \(M\) is an element of the classical 2-dimensional bordism group of \(G\), isomorphic to the Schur multipier \(H_2(G, \mathbb Z)\) of \(G\) (for a low-dimensional algebraic proof of this isomorphism, see a paper by the reviewer [Monatsh. Math. 104, 247--253 (1987; Zbl 0627.57009)]); as a consequence, free actions on surfaces do not extend to free actions on 3-manifolds, in general. It is the main result of the present paper that an extension of a free action on a surface to an action on a compact 3-manifold always exists if \(G\) is a finite abelian, dihedral, symmetric or alternating group. The proofs introduce some new and interesting perspective to the problem by using the language of 2-dimensional principal \(G\)-bordisms or extendable \(G\)-cobordisms over surfaces, decomposing such \(G\)-cobordisms into some types of basic extendable pieces. For the first possible counterexamples to such an extension for some other types of groups, see a recent preprint by \textit{E. Samperton} [``Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds'', Preprint, \url{arXiv:210706982}]. The authors discuss also briefly the problem of extending arbitrary (i.e. not necessarily free) finite group-actions on surfaces to 3-manifolds; here, by results of \textit{R. A. Hidalgo} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 19, No. 2, 259--289 (1994; Zbl 0821.30029)] and \textit{M. Reni} and \textit{B. Zimmermann} [Proc. Am. Math. Soc. 124, No. 9, 2877--2887 (1996; Zbl 0868.57018)], actions of finite dihedral groups always extend to a handlebody, and one can characterize which actions of finite cyclic and abelian groups extend (and if so, again to a handlebody); an interesting question here is whether an extending action on a hyperbolic surface (e.g., to a handlebody) extends also to a hyperbolic 3-manifold with totally geodesic boundary (and then also by isometries).