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arXiv · 1807.09209

Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem

Richard Hain
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Abstract

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.

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BibTeXRIS

Richard Hain. 2018-07-24. Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem. https://arxiv.org/abs/1807.09209

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