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

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