4-Manifolds as covers of the 4-sphere branched over non-singular surfaces

Abstract

We prove the long-standing Montesinos conjecture that any closed oriented PL 4-manifold M is a simple covering of S4 branched over a locally flat surface (cf [12]). In fact, we show how to eliminate all the node singularities of the branching set of any simple 4-fold branched covering M → S4 arising from the representation theorem given in [13]. Namely, we construct a suitable cobordism between the 5-fold stabilization of such a covering (obtained by adding a fifth trivial sheet) and a new 5-fold covering M → S4 whose branching set is locally flat. It is still an open question whether the fifth sheet is really needed or not.