- journal article metadata
European Mathematical Society Publishing House
2018-02-04 23:30:02
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
9
2018
1
Formal descriptions of Turaev's loop operations
Gwénaël
Massuyeau
Université de Strasbourg and Université Bourgogne Franche-Comté, Dijon, France
Loop operations, surfaces, braid groups, Drinfeld associators, Kontsevich integral
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.
Manifolds and cell complexes
Nonassociative rings and algebras
Group theory and generalizations
39
117
10.4171/QT/103
http://www.ems-ph.org/doi/10.4171/QT/103
2
1
2018