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Hodge Filtration and Operations in Higher Hochschild (Co)homology and Applications to Higher String Topology

  • Grégory Ginot
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2194)

Abstract

This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics and aims to present the theory of higher Hochschild (co)homology and its application to higher string topology. There is an emphasis on explicit combinatorial models provided by simplicial sets to describe derived structures carried or described by Higher Hochschild (co)homology functors. It contains detailed proofs of results stated in a previous note as well as some new results. One of the main result is a proof that string topology for higher spheres inherits a Hodge filtration compatible with an (homotopy) E n+1-algebra structure on the chains for d-connected Poincaré duality spaces. We also prove that the E n -centralizer of maps of commutative (dg-)algebras are equipped with a Hodge decomposition and a compatible structure of framed E n -algebras. We also study Hodge decompositions for suspensions and products by spheres, both as derived functors and combinatorially.

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Authors and Affiliations

  1. 1.Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13VilletaneuseFrance

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