Involutions of hyperbolic spatial graph exteriors whose fixed point sets are closed surfaces

Abstract

A [Formula: see text]-component non-splittable spatial graph [Formula: see text] in [Formula: see text] possibly admits a smooth involution on its exterior whose fixed point set is a closed surface [Formula: see text] of genus [Formula: see text]. It is known that [Formula: see text] holds if [Formula: see text] is a graph link in [Formula: see text]. In this paper, we consider [Formula: see text] to be a hyperbolic spatial graph in a closed orientable 3-manifold [Formula: see text]. We first study the case where [Formula: see text] is the 3-sphere, and provide a method for constructing [Formula: see text] and [Formula: see text] with [Formula: see text], where [Formula: see text] is the Euler characteristic of [Formula: see text] for [Formula: see text]. Next, we study the case where [Formula: see text] is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries.