# Some algebraic aspects of the Turaev cobracket

## TL;DR

This paper reviews algebraic properties of the Turaev cobracket, its variants, and their applications to surface mapping class groups and the Kashiwara-Vergne problem, highlighting recent joint work and homological insights.

## Contribution

It provides a comprehensive survey of algebraic aspects of the Turaev cobracket and its framed variants, connecting them to mapping class groups and the Kashiwara-Vergne problem.

## Key findings

- Formal description of the cobracket and variants
- Application to the mapping class group
- Relation to the Kashiwara-Vergne problem

## Abstract

The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman bracket. 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.