On Meyer's function of hyperelliptic mapping class groups
In [Math. Ann. 201, 239-264 (1973; Zbl 0245.55017)], \textit{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. \textit{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 [\textit{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 \textit{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.
scientific article; zbMATH DE number 1943302