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Hodge decomposition of string topology

Part of: Higher algebraic $K$-theory Homotopy theory

Published online by Cambridge University Press:  13 April 2021

Yuri Berest ,
Ajay C. Ramadoss  and
Yining Zhang
Yuri Berest
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY14853-4201, USA; E-mail: berest@math.cornell.edu
Ajay C. Ramadoss
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN47405, USA; E-mail: ajcramad@indiana.edu
Yining Zhang
Affiliation:
Department of Mathematics, University of Colorado Boulder, Boulder, CO80309, USA; E-mail: yining.zhang@colorado.edu

Abstract

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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].

MSC classification

Primary: 55P50: String topology
Secondary: 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e33
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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