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Meyer’s signature cocycle and hyperelliptic fibrations. (English) Zbl 0948.57013

In [W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbündeln, Bonn. Math. Schr. 53 (1972; Zbl 0243.58004), Math. Ann. 201, 239-264 (1973; Zbl 0241.55019)], important properties for the signature of surface bundles over surfaces are obtained, through the introduction of Meyer’s signature cocycle \(\tau_g:{\mathcal M}_g\times{\mathcal M}_g \to\mathbb{Z}\) (where, as usual, \({\mathcal M}_g\) denotes the mapping class group of the closed oriented surface of genus \(g)\) and the computation of the order of its cohomology class in the cohomology group \(H^2({\mathcal M}_g, \mathbb{Z})\) (which is finite only for \(g=1\) and \(g=2)\).
The present paper proves that the cohomology class of the signature cocycle \(\tau^{\mathcal H}_g\) restricted to the hyperelliptic mapping class group \({\mathcal H}_g\) [J. S. Birman and H. M. Hilden, Adv. Theory Riemann Surfaces, Proc. 1969 Stony Brook Conf., 81-115 (1971; Zbl 0217.48602)] in the cohomology group \(H^2({\mathcal H}_g,\mathbb{Z})\) is always equal to \(2g+1\).
Moreover, the paper introduces the notion of local signature of a singular fibre of a hyperelliptic locally analytic fibration of arbitrary genus and determines its values on Lefschetz singular fibers (compare with [Y. Matsumoto, Proc. Japan Acad., Ser. A 59, 100-103 (1983; Zbl 0532.55020), Lefschetz fibrations of genus two – a topological approach, in “Proceedings 37th Taniguchi Symposium on Topology and Teichmüller Spaces, 123-148 (1996; Zbl 0921.57006)], where a local signature related to \(\tau_g\) was defined and analyzed). Finally, the author compares his local signature with another local signature, which arises from a study about degeneration of hyperelliptic curves in algebraic geometry [see Arakawa and Ashikaga, Local splittings families of hyperelliptic pencils I (preprint)] and presents Terasoma’s theorem proving that the values of these two kinds of local signature always coincide.

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
14J29 Surfaces of general type
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
55S37 Classification of mappings in algebraic topology
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