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Regular version of the site

Article

Symplectic surgeries along certain singularities and new Lefschetz fibrations

Advances in Mathematics. 2020. Vol. 360. P. 1-23.
Katzarkov L., Akhmedov A.

We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (ta0ta1ta2⋯ta2g+1)2g+1=tb1tb2gtb3, 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 (tc1tc2⋯tc2g−1tc2gtc2g+12tc2gtc2g−1⋯tc2tc1)2n=1, (tc1tc2⋯tc2gtc2g+1)(2g+2)n=1and (tc1tc2⋯tc2g−1tc2g)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.