Geometry & Topology

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

Massimiliano Iori and Riccardo Piergallini

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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 MS4 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 MS4 whose branching set is locally flat. It is still an open question whether the fifth sheet is really needed or not.

Article information

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

Received: 30 April 2001
Accepted: 9 July 2002
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

4–manifolds branched coverings locally flat branching surfaces


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.

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