On Meyer’s function of hyperelliptic mapping class groups. (English) Zbl 1031.57017

In [Math. Ann. 201, 239-264 (1973; Zbl 0245.55017)], W. Meyer defines a signature cocycle which may be viewed as a function from \(\Delta_g\) to \(\frac{1}{2g+1} \mathbb Z\), where \(\Delta_g\) is the hyperelliptic mapping class group. M. Atiyah has done much to understand this function geometrically, relating it to Hirzebruch’s signature defect, the logarithmic monodromy of Quillen’s determinant line bundle, and the Atiyah-Patodi-Singer \(\eta\)-invariant [M. Atiyah, Math. Ann. 278, 335-380 (1987; Zbl 0648.58035)]. Meyer gave a formula for his function in the above paper in the case \(g = 1\).
In the current paper, the author gives some formulae for computing Meyer’s function on certain subgroups of \(\Delta_g\), primarily but not exclusively for the case \(g = 2\). He then relates Meyer’s function to \(\eta\)-invariants, extending Atiyah’s results. Finally, the function is related to S. Morita’s work [Topology 28, 305-323 (1989; Zbl 0684.57008) and Topology 30, 603-621 (1991; Zbl 0747.57010)], thereby relating the function to Casson’s invariant for integral homology spheres.


57M99 General low-dimensional topology
14H45 Special algebraic curves and curves of low genus
57M10 Covering spaces and low-dimensional topology
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R20 Characteristic classes and numbers in differential topology
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