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 $S^4$ branched over a locally flat surface (cf [Trans. Amer. Math. Soc. 245 (1978) 453–467]). In fact, we show how to eliminate all the node singularities of the branching set of any simple 4–fold branched covering $M\to S^4$ arising from the representation theorem given in [Topology 34 (1995) 497–508]. 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 $M\to S^4$ whose branching set is locally flat. It is still an open question whether the fifth sheet is really needed or not.