A decomposition into homeomorphic handlebodies with naturally equivalent involutions (Q912436)

Summary: \textit{J. S. Downing} [Proc. Am. Math. Soc. 24, 241-244 (1970; Zbl 0189.547)] extended the well-known result that any closed 3-manifold X contains a handlebody H such that cl(X-H) is homeomorphic to H in the case where X is a compact 3-manifold with nonvoid boundary. We show that if X is a compact 3-manifold with involution h having 2-dimensional fixed point set, then X contains an h-invariant handlebody H such that the involutions induced on H and cl(X-H) are naturally equivalent.