Geometry & Topology, Vol. 6 (2002) Paper no. 13, pages 393--401.

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

Massimiliano Iori , Riccardo Piergallini

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 [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.

Keywords. 4--manifolds, branched coverings, locally flat branching surfaces

AMS subject classification. Primary: 57M12. Secondary: 57N13.

DOI: 10.2140/gt.2002.6.393

E-print: arXiv:math.GT/0203087

Submitted to GT on 30 April 2001. Paper accepted 9 July 2002. Paper published 21 July 2002.

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Massimiliano Iori , Riccardo Piergallini
Dipartimento di Matematica e Informatica
Universita di Camerino -- Italia

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