On Meyer's function of hyperelliptic mapping class groups

Abstract

In this paper, we consider Meyer's function of hyperelliptic mapping class groups of orientable closed surfaces and give certain explicit formulae for it. Moreover we study geometric aspects of Meyer's function, and relate it to the h- invariant of the signature operator and Morita's homomorphism, which is the core of the Casson invariant of integral homology 3-spheres. where G n denotes the mapping class group of S 2 leaving n points invariant. As is known, Dg ¼ Gg if g ¼ 1;2 and Dg 0Gg for gb3. The group Dg is called the hyperelliptic mapping class group. Our main object in the present paper is Meyer's signature cocycle (Me), which is a group 2-cocycle of the Siegel modular group Spð2g;ZÞ. Topologi- cally, this presents the signature of total spaces of surface bundles over a surface. Since the group Dg is acyclic over Q (cf. (C), (K)), the restriction of the pull-back of the signature cocycle via the classical representation