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

Iori, Massimiliano¹ ; Piergallini, Riccardo¹

¹ Dipartimento di Matematica e Informatica, Università di Camerino, Italy

Geometry & topology, Tome 6 (2002) no. 1, pp. 393-401.

Résumé

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.

Détail

DOI : 10.2140/gt.2002.6.393

Keywords: 4–manifolds, branched coverings, locally flat branching surfaces

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