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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 S4 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 S4 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 S4 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
Mathematical Subject Classification 2000
Primary: 57M12
Secondary: 57N13
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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