Some algebraic aspects of the Turaev cobracket

Abstract

The Turaev cobracket, a loop operation introduced by V. Turaev \[52], which measures the self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself \[51], as well as a counterpart of the Goldman bracket \[22]. In this survey based on the author’s joint works with A. Alekseev, Y. Kuno and F. Naef, we review some algebraic aspects of the cobracket and its framed variants including their formal description, an application to the mapping class group of the surface and a relation to the (higher genus) Kashiwara–Vergne problem. In addition, we review a homological description of the cobracket after R. Hain \[24].