Algebraic characterization of simple closed curves via Turaev's cobracket

TL;DR

This paper proves Turaev's conjecture that the cobracket vanishes for powers of simple closed curves if and only if the curve is embedded, for surfaces with boundary, advancing understanding of curve characterization.

Contribution

The paper establishes Turaev(k) for all k ≥ 3 on surfaces with boundary, confirming a long-standing conjecture and extending previous computational verifications.

Findings

Proved Turaev(k) for k=3,4,5,... for surfaces with boundary.

Confirmed Turaev(2) for a large class of shortest curves.

Connected the cobracket symmetry to mapping class group actions.

Abstract

The vector space $\V$ generated by the conjugacy classes in the fundamental group of an orientable surface has a natural Lie cobracket $\map{\delta}{\V}{\V\times \V}$. For negatively curved surfaces, $\delta$ can be computed from a geodesic representative as a sum over transversal self-intersection points. In particular $\delta$ is zero for any power of an embedded simple closed curve. Denote by Turaev(k) the statement that $\delta(x^k) = 0$ if and only if the nonpower conjugacy class $x$ is represented by an embedded curve. Computer implementation of the cobracket delta unearthed counterexamples to Turaev(1) on every surface with negative Euler characteristic except the pair of pants. Computer search have verified Turaev(2) for hundreds of millions of the shortest classes. In this paper we prove Turaev(k) for $k=3,4,5,\dots$ for surfaces with boundary. Turaev himself introduced the cobracket in the 80's and wondered about the relation with embedded curves, in particular asking if 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 operation is induced by an element of the mapping class group.