Symplectic surgeries along certain singularities and new Lefschetz fibrations

Anar Akhmedov, Ludmil Katzarkov

Research output: Contribution to journalArticle

Abstract

We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (ta0 ta1 ta2 ⋯ta2g+1 )2g+1=tb1 tb2 gtb3 , also known as the trident relations, in the mapping class group Γg,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (tc1 tc2 ⋯tc2g−1 tc2g tc2g+1 2tc2g tc2g−1 ⋯tc2 tc1 )2n=1, (tc1 tc2 ⋯tc2g tc2g+1 )(2g+2)n=1 and (tc1 tc2 ⋯tc2g−1 tc2g )2(2g+1)n=1 in the mapping class group Γg of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S2 with non-separating vanishing cycles.

Original languageEnglish (US)
Article number106920
JournalAdvances in Mathematics
Volume360
DOIs
StatePublished - Jan 22 2020

Keywords

  • 4-manifold
  • Lefschetz fibration
  • Mapping class group
  • Symplectic surgery
  • Trident relation

ASJC Scopus subject areas

  • Mathematics(all)

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