The h–principle for broken Lefschetz fibrations

Williams, Jonathan¹

¹ Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712

Geometry & topology, Tome 14 (2010) no. 2, pp. 1015-1061.

Résumé

It is known that an arbitrary smooth, oriented $4$–manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken Lefschetz fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken Lefschetz fibration in a given homotopy class of smooth maps. One notable application is that adding an additional “projection" move generates all broken Lefschetz fibrations, regardless of homotopy class. The paper ends with further applications and open problems.

Détail

DOI : 10.2140/gt.2010.14.1015

Keywords: broken, Lefschetz fibration, $4$–manifold, stable map

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