Retracting compact 3-manifolds onto their boundary components

Abstract

Let $X$ be a compact 3-manifold and $A$ a boundary component of $X$ to which $X$ retracts. We propose the problem of classifying such $(X,A)$ up to PL homeomorphisms in a suitable class of manifolds. We give a complete solution in a special case. We show in this case there are exactly twenty-four basic pairs $(X,A)$ and all other cases are obtained from these by some obvious modifications. If $A$ is assumed to be different from a 2-sphere, there are six basic pairs.