Algebraic characterization of simple closed curves via Turaev's cobracket

The vector space V generated by the conjugacy classes in the fundamental group of an orientable surface has a natural Lie cobracket δ : V → V ⊗ V. For negatively curved surfaces, δ can be computed from a geodesic representative as a sum over transversal self-intersection points. In particular, δ is zero for any power of an embedded simple closed curve. Denote by Turaev(k) the statement that δ(x^k) = 0 if and only if the non-power conjugacy class x is represented by an embedded curve. Computer implementation of the cobracket δ unearthed counterexamples to Turaev(1) on every surface with negative Euler characteristic except genus zero surfaces. Computer search has verified Turaev(2) for hundreds of millions of the shortest classes. In this paper, we prove Turaev(k) for k = 3, 4, 5, ... for surfaces with boundary. Turaev himself introduced the cobracket in the 1980s and wondered about the relation with embedded curves, in particular asking if a statement equivalent to Turaev(1) might be true. We give an application of our result to the curve complex. We show that a permutation of the set of free homotopy classes that commutes with the cobracket and the power operations is induced by an element of the mapping class group.