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Hurwitz complete sets of factorizations in the modular group and the classification of Lefschetz elliptic fibrations over the disk. (English) Zbl 1311.57005

If \(\Sigma\) is a compact, connected, and oriented smooth two dimensional manifold, and \(M\) is a compact, connected, and oriented four dimensional smooth manifold, then a surjection \(f:M\to\Sigma\) is called a topological elliptic fibration over \(\Sigma\) if \(f(\text{int}(M))=\text{int}(\Sigma)\), \(f(\partial M)=\partial\Sigma\), \(f\) has a finite number of critical values \(q_1,...,q_n\in\text{int}(\Sigma)\), and \(f\) is locally holomorphic. A topological elliptic fibration \(f:M\to\Sigma\) is called: (i) relatively minimal if none of its fibers contains an embedded sphere with self-intersection \(-1\), (ii) Lefschetz if for each critical point \(p\) of \(f\) there exist charts relative to which \(f\) takes the form \((z_1,z_2)\mapsto z_1^2+z_2^2\), and the restriction of \(f\) to the set of critical points is injective.
In this paper, the authors consider the basic notions concerning topological elliptic fibrations over the unit disk and the classification of relatively minimal Lefschetz fibrations. They relate the problem of classifying all special elliptic fibrations over the disk to the problem of classifying their monodromy representations, up to conjugation and Hurwitz equivalence, in the modular group. Also, they study the relationship between \(u\)-special factorizations in \(PSL(2,\mathbb Z)\) and their liftings to \(SL(2,\mathbb Z)\), and present a combinatorial study of Hurwitz equivalence of \(s_1\)-special factorization in the modular group. Finally, the authors describe an algorithm for generating a relatively simple set that is \(H\)-complete in the set of \(s_1\)-special factorizations of any given element in the modular group.

MSC:

57M07 Topological methods in group theory
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
55R55 Fiberings with singularities in algebraic topology
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