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Hodge decomposition of string topology

Published online by Cambridge University Press:  13 April 2021

Yuri Berest
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA; E-mail:
Ajay C. Ramadoss
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA; E-mail:
Yining Zhang
Department of Mathematics, University of Colorado Boulder, Boulder, CO 80309, USA; E-mail:


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 {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {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 of [7].

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