Algebraic characterization of simple closed curves via Turaev's cobracket (Q2797312)

Let \(\mathbb{V}\) be the vector space generated by the conjugacy classes in the fundamental group of an orientable surface. It has a natural Lie cobracket \(\delta : \mathbb{V} \longrightarrow \mathbb{V} \otimes \mathbb{V}\). ``Denote by \(\mathrm{Turaev}(k)\) the statement: \(\delta(x^k)=0\) iff the non-power conjugacy class \(x\) is represented by an embedded curve.''NEWLINENEWLINEThe main theorem of the article states: ``Let \(\mathcal{V}\) be a free homotopy class of curves on an oriented surface with boundary. Then \(\mathcal{V}\) contains a power of a simple curve if and only if the Turaev cobracket of \(\mathcal{V}^3\) is zero. Moreover, if \(\mathcal{V}\) is a non-power and \(p\) is an integer larger than three, then the number of terms of the cobracket of \(\mathcal{V}^p\) (counted with multiplicity) equals \(2p^2\) times the self-intersection number of \(\mathcal{V}\).''NEWLINENEWLINEIn the appendix examples and an application of the result to the curve complex are given.