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A reduction of the string bracket to the loop product

Katsuhiko Kuribayashi, Takahito Naito, Shun Wakatsuki and Toshihiro Yamaguchi

Algebraic & Geometric Topology 24 (2024) 2619–2654
Abstract

The negative cyclic homology for a differential graded algebra over the rational field has a quotient of the Hochschild homology as a direct summand if the S–action is trivial. With this fact, we show that the string bracket in the sense of Chas and Sullivan is reduced to the loop product followed by the BV operator on the loop homology provided the given manifold is BV-exact. The reduction is indeed derived from the equivalence between the BV-exactness and the triviality of the S–action. Moreover, it is proved that a Lie bracket on the loop cohomology of the classifying space of a connected compact Lie group possesses the same reduction. By using these results, we consider the nontriviality of string brackets. We also show that a simply connected space with positive weights is BV-exact. Furthermore, the higher BV-exactness is discussed featuring the cobar-type Eilenberg–Moore spectral sequence.

Keywords
string topology, string bracket, Hochschild homology, cyclic homology, positive weight, Eilenberg–Moore spectral sequence, BV-exactness
Mathematical Subject Classification 2010
Primary: 55P35, 55P50, 55T20
References
Publication
Received: 12 May 2022
Revised: 11 May 2023
Accepted: 12 July 2023
Published: 19 August 2024
Authors
Katsuhiko Kuribayashi
Department of Mathematical Sciences
Faculty of Science
Shinshu University
Matsumoto
Nagano
Japan
Takahito Naito
Nippon Institute of Technology
Gakuendai
Miyashiro-machi
Minamisaitama-gun
Saitama
Japan
Shun Wakatsuki
Department of Mathematical Sciences
Faculty of Science
Shinshu University
Matsumoto
Nagano
Japan
Toshihiro Yamaguchi
Faculty of Education
Kochi University
Akebono-cho
Kochi
Japan

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