Elimination of cusps in dimension 4 and its applications

Stefan Behrens, Kenta Hayano

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Abstract

We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S1where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.

Original languageEnglish
Pages (from-to)674-724
Number of pages51
JournalProceedings of the London Mathematical Society
Volume113
Issue number5
DOIs
Publication statusPublished - 2016 Jan 1

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ASJC Scopus subject areas

  • Mathematics(all)

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