Kyoto Journal of Mathematics

Constructing Lefschetz fibrations via daisy substitutions

Anar Akhmedov and Naoyuki Monden

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Abstract

We construct new families of nonhyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words ( c 1 c 2 c 2 g 1 c 2 g c 2 g + 1 2 c 2 g c 2 g 1 c 2 c 1 ) 2 = 1 , ( c 1 c 2 c 2 g c 2 g + 1 ) 2 g + 2 = 1 , and ( c 1 c 2 c 2 g 1 c 2 g ) 2 ( 2 g + 1 ) = 1 in the mapping class group Γ g of the closed orientable surface of genus g , and we study the sections of these Lefschetz fibrations. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic 4 -manifolds, and we compute their Seiberg–Witten invariants. By applying the knot surgery to the family of Lefschetz fibrations obtained from the word ( c 1 c 2 c 2 g c 2 g + 1 ) 2 g + 2 = 1 via daisy substitutions, we also construct an infinite family of pairwise nondiffeomorphic irreducible symplectic and nonsymplectic 4 -manifolds homeomorphic to ( g 2 g + 1 ) CP 2 # ( 3 g 2 g ( k 3 ) + 2 k + 3 ) CP ¯ 2 for any g 3 and k = 2 , , g + 1 .

Article information

Source
Kyoto J. Math., Volume 56, Number 3 (2016), 501-529.

Dates
Received: 1 July 2014
Revised: 9 March 2015
Accepted: 13 May 2015
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1471872278

Digital Object Identifier
doi:10.1215/21562261-3600148

Mathematical Reviews number (MathSciNet)
MR3542772

Zentralblatt MATH identifier
1351.57032

Subjects
Primary: 57R55: Differentiable structures
Secondary: 57R17: Symplectic and contact topology

Keywords
$4$-manifold Lefschetz fibration rational blowdown mapping class group daisy relation

Citation

Akhmedov, Anar; Monden, Naoyuki. Constructing Lefschetz fibrations via daisy substitutions. Kyoto J. Math. 56 (2016), no. 3, 501--529. doi:10.1215/21562261-3600148. https://projecteuclid.org/euclid.kjm/1471872278


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