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

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

Mathematical Subject Classification 2000

Primary: 57M12

Secondary: 57N13

References

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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

Volume 6, issue 1 (2002)

Massimiliano Iori and Riccardo Piergallini