Mapping swirls and pseudo-spines of compact 4-manifolds

A compact subset X of the interior of a compact manifold M is a pseudo-spine of M if M − X is homeomorphic to (∂M) × [0, ∞). It is proved that a 4-manifold obtained by attaching k essential 2-handles to a B3 × S1 has a pseudo-spine which is obtained by attaching k B2ʹs to an S1 by maps of the form z → zn. This result recovers the fact that the Mazur 4-manifold has a disk pseudo-spine (which may then be shrunk to an arc). To prove this result, the mapping swirl (a “swirled” mapping cylinder) of a map to a circle is introduced, and a fundamental property of mapping swirls is established: homotopic maps to a circle have homeomorphic mapping swirls. l conjectures concerning the existence of pseudo-spines in compact 4-manifolds are stated and discussed, including the following two related conjectures: every compact contractible 4-manifold has an arc pseudo-spine, and every compact contractible 4-manifold has a handlebody decomposition with no 3- or 4-handles. It is proved that an important class of compact contractible 4-manifolds described by Poénaru satisfies the latter conjecture.