Formal descriptions of Turaev's loop operations (Q1708581)

In [Mat. Sb., Nov. Ser. 106(148), 566--588 (1978; Zbl 0384.57004); translation in Math. USSR, Sb. 35, 229--250 (1979)], \textit{V. G. Turaev} associated to a surface with non-empty boundary two loop operations defined on the algebra of the fundamental group \(\pi\) of the surface. One is a bilinear map called intersection and the other is a linear map called self-intersection. Turaev used them to give conditions for the realizability of an element of \(\pi\) by a simple loop and for the realizability of a finite number of elements of \(\pi\) by non-intersecting loops. In [\textit{G. Massuyeau} and \textit{V. Turaev}, Ann. Inst. Fourier 63, No. 6, 2403--2456 (2013; Zbl 1297.57005)], Turaev and the author gave a formal description of the intersection map, which generalizes the formal description of the Goldman bracket via symplectic expansions of the algebra of the group \(\pi\) given in [\textit{N. Kawazumi} and \textit{Y. Kuno}, Quantum Topol. 5, No. 3, 347--423 (2014; Zbl 1361.57027)]. In the paper under review, a similar algebraic description is given for Turaev's self-intersection map when the surface is a disk with \(p\) punctures. In the first part of the paper, Fox pairings and Turaev's loop operations are reviewed and three-dimensional formulas for these operations are shown, in the setting of general surfaces with non-empty boundary. Then, in the case of a disk with \(p\) punctures, special expansions of the algebra of the group \(\pi\) are defined and such expansions are constructed from the Kontsevich integral via an embedding of \(\pi\) in the group of pure braids with \((p+1)\) strands. The main result of the paper (Theorem 7.1) describes the self-intersection map, via these latter expansions, in terms of tensor operations involving the Drinfeld associator underlying the construction of the Kontsevich integral. If the Drinfeld associator is assumed to be even, these tensor operations are proved to be independent of it. A formal description of the so-called Turaev co-bracket is deduced.