Special handlebody decompositions of simply connected algebraic surfaces
| Autor: | Richard Mandelbaum |
|---|---|
| Rok vydání: | 1980 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 80:359-362 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1980-0577774-x |
| Popis: | In this article we prove that any nonsigular complete-intersection surface admits a handlebody decomposition with no 1and 3-handles. This generalizes results of Rudolph, Harer and Akbuluf on hypersurfaces of CP3. Introduction. Among the problems posed at the Stanford Conference (24th Summer Research Institute, August, 1976) is the following [K, Preliminary List]. Problem 50 (Kirby). Does every simply-connected closed 4-manifold have a handlebody decomposition without 1-handles? Without 1and 3-handles? Rudolph [R] shows that any nonsingular hypersurface of CP has a handlebody decomposition without 1-handles (or dually, without 3-handles), however, Rudolph's method does not allow one to eliminate both 1and 3-handles simultaneously. In December, 1976, at a lecture at the Institute for Advanced Study, Kirby exhibited a handlebody decomposition of the Kummer surface without both 1and 3-handles. See [HKK]. The Kummer surface is diffeomorphic to a nonsingular quartic in CP3 and we sketch here the following generalization of the [HKK] result exhibited by Kirby. THEOREM. Suppose X is a nonsingular complete intersection of k distinct hypersurfaces VI . . ., Vk in CPk+2. Then X has a handlebody deconWosition without 1and 3-handles. 1. Lefschetz fibrations. DEFINITION. Let V be an algebraic surface and suppose L is a pencil of curves on V. Then we shall say L is a Lefschetz pencil if and only if (1) the generic element of L is nonsingular and irreducible. (2) L has only a finite number of singular elements, each of which has only one ordinary double point as its singularity. We recall [W], [Z] that every algebraic surface admits Lefschetz pencils. (For more details on Lefschetz pencils see especially [AF], [WI.) Furthermore the Lefschetz pencil L gives rise to a rational map f of V to CP'. If this map is a morphism f: V-CP' we shall call L a Lefschetz fibration. It is clear that every Lefschetz pencil on V gives rise to a Lefschetz fibration f: VCP of V = { V blown up at the base points of L} onto CP'. (See [AF].) Received by the editors December 4, 1978. AMS (MOS) subject classifications (1970). Primary 57D55, 57A15, 14J99. 'Supported by an NSF grant. i 1980 American Mathematical Society 0002-9939/80/0000-0532/$02.00 |
| Databáze: | OpenAIRE |
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