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.
Notes on file formats
Massimiliano Iori , Riccardo Piergallini
Dipartimento di Matematica e Informatica
Universita di Camerino -- Italia
Email: riccardo.piergallini@unicam.it
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