English

Constructing Lefschetz fibrations via Daisy Substitutions

Geometric Topology 2021-02-17 v3 Algebraic Geometry Symplectic Geometry

Abstract

We construct new families of non-hyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words (c1c2c2g1c2gc2g+12c2gc2g1c2c1)2=1(c_1c_2 \cdots c_{2g-1}c_{2g}{c_{2g+1}}^2c_{2g}c_{2g-1} \cdots c_2c_1)^2 = 1, (c1c2c2gc2g+1)2g+2=1(c_1c_2 \cdots c_{2g}c_{2g+1})^{2g+2} = 1, and (c1c2c2g1c2g)2(2g+1)=1(c_1c_2 \cdots c_{2g-1}c_{2g})^{2(2g+1)} = 1 in the mapping class group Γg\Gamma_{g} of the closed orientable surface of genus gg, and study the sections of these Lefschetz fibrations. Furthemore, we show that the total spaces of some of these Lefschetz fibraions are irreducible exotic 44-manifolds, and compute their Seiberg-Witten invariants. By applying the knot surgery to the family of Lefschetz fibrations obtained from the word (c1c2c2gc2g+1)2g+2=1(c_1c_2 \cdots c_{2g}c_{2g+1})^{2g+2} = 1 via daisy substitutions, we also construct an infinite family of pairwise non-diffeomorphic irreducible symplectic and non-symplectic 44-manifolds homeomorphic to (g2g+1)CP2#(3g2g(k3)+2k+3)CP2(g^2 - g + 1){\mathbb{CP}}{}^{2} \# (3g^{2} - g(k-3) + 2k + 3)\overline{\mathbb{CP}}{}^{2} for any g3g \geq 3, and k=2,,g+1k = 2, \cdots, g+1.

Keywords

Cite

@article{arxiv.1405.6669,
  title  = {Constructing Lefschetz fibrations via Daisy Substitutions},
  author = {Anar Akhmedov and Naoyuki Monden},
  journal= {arXiv preprint arXiv:1405.6669},
  year   = {2021}
}

Comments

27 pages, 6 figures. minor revisions for publication

R2 v1 2026-06-22T04:23:33.401Z