$4$-manifolds, $3$-fold covering spaces and ribbons

Abstract:

It is proved that a PL, orientable 4-manifold with a handle presentation composed by 0-, 1-, and 2-handles is an irregular 3-fold covering space of the 4-ball, branched over a 2-manifold of ribbon type. A representation of closed, orientable 4-manifolds, in terms of these 2-manifolds, is given. The structure of 2-fold cyclic, and 3-fold irregular covering spaces branched over ribbon discs is studied and new exotic involutions on ${S^4}$ are obtained. Closed, orientable 4-manifolds with the 2-handles attached along a strongly invertible link are shown to be 2-fold cyclic branched covering spaces of ${S^4}$. The conjecture that each closed, orientable 4-manifold is a 4-fold irregular covering space of ${S^4}$ branched over a 2-manifold is reduced to studying $\gamma \# {S^1} \times {S^2}$ as a nonstandard 4-fold irregular branched covering of ${S^3}$.