N ov 2 00 7 A combinatorial formula for Earle ’ s twisted 1 - cocycle on the mapping class group M g , ∗

We present a formula expressing Earle’s twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita’s twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution. Introduction and statement of the result. Let (Σg, ∗) be a closed oriented C -surface of genus g ≥ 2 with a fixed base point ∗ and let Mg,∗ be the mapping class group of (Σg, ∗), namely the group of all orientation preserving diffeomorphisms of (Σg, ∗) modulo isotopies fixing the base point ∗. The group Mg,∗ naturally acts on the first homology group H = H1(Σg;Z). In [1], C. Earle discovered a twisted 1-cocycle ψ : Mg,∗ → 1 2g−2 H . This cocycle is complex analytic by nature. In fact, he discovered this cocycle in the study of the action of Mg,∗ = mod(Γ) on J(V ), using his notation, the family of Jacobi varieties over the Teichmüller space of compact Riemann surfaces of genus g. We call ψ Earle’s twisted 1-cocycle. The construction of ψ will be recalled in section 2. In view of [3], ψ gives rise to a generator of the first cohomology group H(Mg,∗;H) ∼= H(Mg,1;H) ∼= Z. Here Mg,1 is the mapping class group of Σg relative to an embedded disc, see section 1. Other than ψ, there have been known various ways of constructing cocycles representing a generator of this cohomology group; see S. Morita [3, 4, 5, 6] and T. Trapp [7]. Among others, there is a combinatorial one: Morita’s twisted 1-cocycle f : Mg,∗ → H defined in [3]. Although ψ naturally arises it seems more abstract than other known cocycles. The motivation of the present paper is to understand ψ more concretely. We compare ψ with f . As a product we obtain a formula expressing Earle’s cocycle ψ, which appeared in the context of complex analysis, using Morita’s cocycle f , more combinatorial one. To state the result, let us fix the notation. Let A1, . . . , Ag, B1, . . . , Bg be a fundamental system of generators of the fundamental group π1(Σg, ∗) and fix it throughout this paper. Then the group π1(Σg, ∗) is isomorphic to the group Γ =< A1, . . . , Ag, B1, . . . , Bg|ζ = 1 > . Here, ζ = ∏g k=1[Ak, Bk] = [A1, B1] · · · [Ag, Bg] and [Ak, Bk] = AkBkA −1 k B −1 k . The natural projection θ : Γ → H gives the abelianization of Γ and H is identified with Z by the direct decomposition H = Z · θ(A1)⊕ · · · ⊕ Z · θ(Bg).