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

Morifuji, Takayuki

J. Math. Soc. Japan 55, No. 1, 117-129 (2003).

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.

Reviewer: Cynthia L.Curtis (Ewing)

MSC:

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

Keywords:

mapping class group; signature cocycle; eta invariant; Casson invariant

Citations:

Zbl 0245.55017; Zbl 0648.58035; Zbl 0684.57008; Zbl 0747.57010