Executive Summary : | Turaev cobracket is a loop operation on an oriented surface, introduced by Turaev to give a Lie bialgebra structure to Goldman Lie algebra on the free module generated by the free homotopy classes of non-contractible loops. This cobracket is a modification of a loop operation introduced by Turaev in his earlier work to obtain a relation between Milnor invariants of a link and the intersection forms of curves on a Seifert surface of the link. The goal of the proposed project is to use hyperbolic geometry of the oriented surface to study the properties of the Turaev cobracket and its relation with the Poisson algebra of functions on the moduli space of representations. The two main objectives of the proposal are: A) Obtaining a relation between self-intersection number of a curve and the number of terms in its Turaev cobracket. B) Obtaining a cobracket structure on the regular functions of GL(n,C) character variety which is equivalent to the topological Turaev cobracket. A typical approach to study the properties of Turaev cobracket involves use of combinatorial group theory together with surface topology. The use of hyperbolic geometry to study Turaev cobracket is a very recent topic of research and the existing literature is quite limited. Therefore the goal of the proposed project is twofold: 1) Firstly, use hyperbolic geometry to interpret properties of Turaev cobracket of a loop in terms of the properties of its geodesic representative. 2) Then use tools from hyperbolic geometry to study the properties of the geodesic representatives of the terms of the Turaev cobracket to derive results about Turaev cobracket. The possible outcome of the project includes obtaining new results about Turaev cobracket as well as obtaining alternative proofs of the existing theorems using tools from hyperbolic geometry of the surface. |