The mapping class group and the Meyer function for plane curves

For each d ≥ 2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d = 4, using our Meyer function, we can define the local signature for 4dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given. Introduction. Let Σg be a closed orientable C ∞-surface of genus g ≥ 0 and let Γg be the mapping class group of Σg, namely the group of all isotopy classes of orientation preserving diffeomorphisms of Σg. In [12] W.Meyer discovered and studied a cocycle τg : Γg ×Γg → Z. For the sake of the reader a brief definition of τg will be given in Appendix. This cocycle is called the Meyer’s signature cocycle. In his paper W.Meyer showed that the cohomology class [τg] ∈ H(Γg;Z) is torsion for g = 1, 2 and has infinite order for g ≥ 3, and gave an explicit formula for the unique Q-valued 1-cochain of Γ1 cobounding τ1 using the Rademacher function ([12] p.259 Satz 4). Since the hyperelliptic mapping class group Γg , a subgroup of Γg, was shown to be Q-acyclic by F.Cohen[5] and N.Kawazumi[9] independently, it was known to specialists that there exists the unique 1-cochain of Γg cobounding τg restricted to Γ H g . In [7] H.Endo directly showed that the existence and the uniqueness of such a 1-cochain φg : Γ H g → 1 2g+1 Z using a finite presentation of Γg by J.Birman-H.Hilden[3]. He also defined the local signature for hyperelliptic fibrations using φg , and studied the geometry of hyperelliptic fibrations; for example, he derived a signature formula for such fibrations over a closed surface. His formula originates from Y.Matsumoto[11, Theorem 3.3] where genus 2 fibrations are discussed. For the study of the function φg , see also T.Morifuji’s paper[13]. The purpose of the present paper is to give another interesting example of these phenomena; the Meyer function on the mapping class group for plane curves. For d ≥ 2 a group Π(d) and a homomorphism ρ : Π(d) → Γg,where g = 1 2 (d− 1)(d− 2), will be constructed. The group Π(d) can be considered as the fundamental group of the classifying space for isotopy classes of continuous families of non-singular plane curves of degree d; the precise meaning of this statement will be given in Theorem 6.1 later. The main results of this paper are Theorem 4.1 and Theorem 4.2. As a consequence of them it follows that the pull back ρ[τg] vanishes in the rational cohomology H (Π(d);Q) and there exists the unique 1-cochain φ : Π(d) → Q such that δφ = ρτg. φ will be called the Meyer function for plane curves of degree d.