JournalsqtVol. 9, No. 1pp. 39–117

Formal descriptions of Turaev's loop operations

  • Gwénaël Massuyeau

    Université de Strasbourg and Université Bourgogne Franche-Comté, Dijon, France
Formal descriptions of Turaev's loop operations cover

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Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra Q[π]\mathbb Q[\pi] of the fundamental group π\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 Q[π]\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)T(H) of H:=H1(π;Q)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 pp punctures. Here we consider the identification between the completions of Q[π]\mathbb Q[\pi] and T(H)T(H) that arises from a Drinfeld associator by embedding π\pi into the pure braid group on (p+1)(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 22-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