The Torelli geometry and its applications

Abstract

Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod±(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skew-symmetric, bilinear form on H = H1(S,Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g,Z) and the action of ModS on H leads to a natural surjective homomorphism