Torelli theorem [SEP] named after
Torelli theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) "C" is determined by its Jacobian variety "J"("C"), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus "J"("C"), with certain 'markings', is enough to recover "C". The same statement
functorial and Homeo acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the "Torelli group", named after the Torelli theorem.
In the case of orientable surfaces, this is the action on first cohomology "H"(Σ) ≅ Z. Orientation-preserving maps are precisely those that act trivially on top cohomology "H"(Σ) ≅ Z. "H"(Σ) has a symplectic structure, coming from the