# Formal descriptions of Turaev's loop operations

### Gwénaël Massuyeau

Université de Strasbourg and Université Bourgogne Franche-Comté, Dijon, France

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## Abstract

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb Q[\pi]$ of the fundamental group $\pi$ of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra $\mathbb Q[\pi]$ is completed with respect to powers of its augmentation ideal and is appropriately identified to the degree-completion of the tensor algebra $T(H)$ of $H:=H_1(\pi;\mathbb Q)$.

In this paper, we obtain a similar algebraic description for Turaev's self-intersection map in the case of a disk with $p$ punctures. Here we consider the identification between the completions of $\mathbb Q[\pi]$ and $T(H)$ that arises from a Drinfeld associator by embedding $\pi$ into the pure braid group on $(p+1)$ strands; our algebraic description involves a formal power series which is explicitly determined by the associator. The proof is based on some three-dimensional formulas for Turaev's loop operations, which involve $2$-strand pure braids and are shown for any surface with boundary.

## Cite this article

Gwénaël Massuyeau, Formal descriptions of Turaev's loop operations. Quantum Topol. 9 (2018), no. 1, pp. 39–117

DOI 10.4171/QT/103