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arXiv · 2002.06596

Hodge decomposition of string topology

Yuri Berest·Ajay C. Ramadoss·Yining Zhang
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Abstract

Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ of the free loop space of $X$ preserves the Hodge decomposition of $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in our earlier work.

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Yuri Berest, Ajay C. Ramadoss, Yining Zhang. 2020-02-18. Hodge decomposition of string topology. https://doi.org/10.1017/fms.2021.26

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