The h–principle for broken Lefschetz fibrations

Williams, Jonathan

doi:10.2140/gt.2010.14.1015

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The $h$–principle for broken Lefschetz fibrations

Jonathan Williams

Geometry & Topology 14 (2010) 1015–1061

DOI: 10.2140/gt.2010.14.1015

Abstract

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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57M50, 57N13

Secondary: 57R70, 57R17

References

Publication

Received: 1 July 2009

Accepted: 23 February 2010

Published: 31 March 2010

Proposed: Ron Fintushel

Seconded: Yasha Eliashberg, Simon Donaldson

Authors

Jonathan Williams
Department of Mathematics The University of Texas at Austin Austin, Texas 78712
http://ma.utexas.edu/jwilliam

Volume 14, issue 2 (2010)