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 a quasi-algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper on the Goldman bracket to construct torsors of solutions to 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.
Highlights
Denote the set of free homotopy classes of maps S1 → X in a topological space X by λ(X) and the free R-module it generates by Rλ(X)
The cobracket was first defined by Turaev [33] on Rλ(M )/R and lifted to Rλ(M ) for framed surfaces in [34, §18] and [3]
If ξ is a quasi-algebraic framing of X, Qλ(X)∧, {, }, δξ is a “twisted” completed Lie bialgebra in the category of pro-mixed Hodge structures
Summary
Denote the set of free homotopy classes of maps S1 → X in a topological space X by λ(X) and the free R-module it generates by Rλ(X). If ξ is a quasi-algebraic framing of X, Qλ(X)∧, { , }, δξ is a “twisted” completed Lie bialgebra in the category of pro-mixed Hodge structures. Our solutions of the Kashiwara–Vergne problem have the property that the corresponding splitting of the filtrations are compatible with those of the Lie algebra of the relative completion of the mapping class group constructed in [12]. Letting the stabilizer of ξo in the mapping class group of (S, P ) act on the complex structure φ by precomposition, we obtain a larger a larger torsor of solutions to the KV-problem. The proof of Theorem 1 is completed in Section 7 where it is shown that each map in the factorization of the cobracket is a morphism of MHS for each choice of a complex structure. We assume familiarity with the sections of that paper on rational K(π, 1) spaces, iterated integrals, and Hodge theory
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