## Geometry & Topology

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

#### 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.

#### Article information

Source
Geom. Topol., Volume 6, Number 1 (2002), 393-401.

Dates
Accepted: 9 July 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882799

Digital Object Identifier
doi:10.2140/gt.2002.6.393

Mathematical Reviews number (MathSciNet)
MR1914574

Zentralblatt MATH identifier
1021.57003

Subjects
Primary: 57M12: Special coverings, e.g. branched
Secondary: 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]

#### Citation

Iori, Massimiliano; Piergallini, Riccardo. 4–manifolds as covers of the 4–sphere branched over non-singular surfaces. Geom. Topol. 6 (2002), no. 1, 393--401. doi:10.2140/gt.2002.6.393. https://projecteuclid.org/euclid.gt/1513882799

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