On a function on the mapping class group of a surface of genus 2
The mapping class group \({\mathcal M}_g\) of a surface \(\Sigma_g\) of genus \(g \geq 2\) admits a natural action on \(H^1(\Sigma_g, Z)\). Such an action preserves the intersection pairing on the surface which is a standard symplectic form, and induces the classical representation \(\rho: {\mathcal M}_g \to \Aut H^1(\Sigma, Z)\). By fixing a symplectic basis of \(H^1(\Sigma_g, Z)\), we can identify \(\Aut H^1(\Sigma, Z)\) with the Siegel modular group \(\operatorname {Sp}(2g, Z)\). Hence \(\rho: {\mathcal M}_g \to \text{Sp}(2g, Z)\) is a non-trivially surjective homomorphism with the Torelli group \({\mathcal T}_g\) defined to be the kernel of \(\rho\). The author extends the above representation to a twist-version as the following. Any nonzero class \(w \in H^1(\Sigma_g, Z_2) = \Hom(\pi_1(\Sigma, \star), Z_2)\) defines a nontrivial flat \(O(2)\)-bundle \(P\) over \(\Sigma_g\) (for \(w = 0\), it defines a trivial \(O(2)\)-bundle over \(\Sigma_g\) and everything is back to the classical situation). Let \(G\) be the automorphisms of \(P\) which preserve an orientation of the fiber at the base point \(\star\), and \(M_P\) be the moduli space of flat \(O(2)\)-connections modulo the gauge group \(G\). Let \({\mathcal M}_{g^{\star}}^w\) be the diffeomorphisms of \(\Sigma_g\) preserving the base point, the orientation of \(\Sigma_g\) and the cohomology class \(w\). For \(f \in {\mathcal M}_{g^{\star}}^w\), there is a lift \(f_{\star}\in G\) which provides an action on \(M_P\). This action is independent of the choice of a lift and a representation of the isotopy class of \(f\). The moduli space \(M_P\) can be identified with \(T^{2(g-1)}\) and the action of \({\mathcal M}_{g^{\star}}^w\) gives \[ \rho_{g^{\star}}^w: {\mathcal M}_{g^{\star}}^w \to \Aut H^1(\pi_1(\Sigma_g, \star), Z_w) = \operatorname {Sp}(2(g-1), Z) \] with twisted Torelli group \({\mathcal T}_{g^{\star}}^w\) as the kernel of \(\rho_{g^{\star}}^w\). The main result of the paper is to prove that \(\rho_{g^{\star}}^w\) is indeed a non-trivially surjective representation. This representation is different from Morita's one constructed from \({\mathcal T}_{g, 1}\) to \(\bigwedge^3 H^1(\Sigma_g, Z)\), and factors through Looijenga's representation constructed from finite Abelian coverings. As an application, the author constructs a function on \({\mathcal M}_2\) from \(\rho_{2^{\star}}^w\) which is related to Atiyah, Patodi and Singer's twisted \(\rho\)-invariant, and the function on \({\mathcal T}_2\) gives another interpretation of the Meyer function. It would be interesting to extend \(Z_2\) to \(Z_p\) for \(p\) odd and prime for the similar construction. Let \(K_g\) be the subgroup of \({\mathcal M}_g\) generated by all the Dehn twists along separating simple closed curves. Then \(K_g \subset {\mathcal T}_g \subset {\mathcal M}_g\) and \(K_{g^{\star}} \subset {\mathcal T}_{g^{\star}} \subset {\mathcal M}_{g^{\star}}^w\). Morita showed that the Casson invariant of \(H_g \cup_{l_g \phi} H_g\) (\(l_g \in {\mathcal M}_g\) with \(S^3 = H_g \cup_{l_g} H_g\) and \(\phi \in K_g\)) is a mapping \(K_g \to Z\) related to Hirzebruch's signature defect by secondary characteristic classes of surface bundles. It is interesting to find out whether there is a twisted version of Morita's construction to obtain twisted-Casson invariants for the corresponding 3-manifolds \(H_g \cup_{l_g \phi} H_g\) with \(\phi \in K_{g^{\star}}\).
scientific article; zbMATH DE number 1421596
scientific article