Paper detail

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

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 [J M Montesinos, 4-manifolds, 3-fold covering spaces and ribbons, 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 [R Piergallini, Four-manifolds as 4-fold branched covers of S^4, 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 --> S^4 whose branching set is locally flat. It is still an open question whether the fifth sheet is really needed or not.