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On hyperelliptic \(C^\infty\)-Lefschetz fibrations of four-manifolds. (English) Zbl 0948.57018

A symplectic Lefschetz fibration (SLF) is said to be hyperelliptic if its monodromy representation [A. Kas, Pac. J. Math. 89, 89-104 (1980; Zbl 0457.14011)] is equivalent – by an inner automorphism of the mapping class group \(MC_g\) – to one taking values in \(HMC_g\subset MC_g\) (the hyperelliptic mapping class group [J. S. Birman and H. M. Hilden, in ‘Adv. Theory Riemann Surfaces, Proc. 1969 Stony Brook Conf.’, 81-115 (1971; Zbl 0217.48602)]). The present paper studies hyperelliptic SLF’s by means of branch loci in \(\mathbb{S}^2\)-fibrations over \(\mathbb{S}^2\), and proves that they are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. Among other properties, the authors show the symplecticity of the branch locus; this fact contradicts the possibility of factorizing any SLF by means of simple 3-fold branched covers of \(\mathbb{S}^2\)-fibrations over \(\mathbb{S}^2\) [see Fuller, Lefschetz fibrations and 3-fold branched covering spaces (preprint) and Lefschetz fibrations and 3-fold branched covering spaces II (preprint)]. Moreover, as a consequence of the main results of the paper, the classification of genus 2 fibrations is reduced to the classification of certain symplectic submanifolds in rational ruled surfaces.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
53D05 Symplectic manifolds (general theory)
57M12 Low-dimensional topology of special (e.g., branched) coverings
55R55 Fiberings with singularities in algebraic topology
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References:

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[4] DOI: 10.2307/1971050 · Zbl 0339.14024 · doi:10.2307/1971050
[5] DOI: 10.2140/pjm.1980.89.89 · Zbl 0457.14011 · doi:10.2140/pjm.1980.89.89
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[9] DOI: 10.1007/BF02774014 · Zbl 0533.57002 · doi:10.1007/BF02774014
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