#### Volume 6, issue 1 (2002)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
4–manifolds as covers of the 4–sphere branched over non-singular surfaces

### Massimiliano Iori and Riccardo Piergallini

Geometry & Topology 6 (2002) 393–401
 arXiv: math.GT/0203087
##### 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.

##### Keywords
4–manifolds, branched coverings, locally flat branching surfaces
Primary: 57M12
Secondary: 57N13
##### Publication
Received: 30 April 2001
Accepted: 9 July 2002
Published: 21 July 2002
Proposed: Robion Kirby
Seconded: Wolfgang Metzler, Ronald Stern
##### Authors
 Massimiliano Iori Dipartimento di Matematica e Informatica Università di Camerino Italy Riccardo Piergallini Dipartimento di Matematica e Informatica Università di Camerino Italy