# Formal descriptions of Turaev's loop operations

Research paper by **Gwenael Massuyeau**

Indexed on: **07 Dec '15**Published on: **07 Dec '15**Published in: **Mathematics - Geometric Topology**

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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.