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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
Kuribayashi, Katsuhiko 1 ; Naito, Takahito 2 ; Wakatsuki, Shun 1 ; Yamaguchi, Toshihiro 3
@article{10_2140_agt_2024_24_2619,
author = {Kuribayashi, Katsuhiko and Naito, Takahito and Wakatsuki, Shun and Yamaguchi, Toshihiro},
title = {A reduction of the string bracket to the loop product},
journal = {Algebraic and Geometric Topology},
pages = {2619--2654},
publisher = {mathdoc},
volume = {24},
number = {5},
year = {2024},
doi = {10.2140/agt.2024.24.2619},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2619/}
}
TY - JOUR AU - Kuribayashi, Katsuhiko AU - Naito, Takahito AU - Wakatsuki, Shun AU - Yamaguchi, Toshihiro TI - A reduction of the string bracket to the loop product JO - Algebraic and Geometric Topology PY - 2024 SP - 2619 EP - 2654 VL - 24 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2619/ DO - 10.2140/agt.2024.24.2619 ID - 10_2140_agt_2024_24_2619 ER -
%0 Journal Article %A Kuribayashi, Katsuhiko %A Naito, Takahito %A Wakatsuki, Shun %A Yamaguchi, Toshihiro %T A reduction of the string bracket to the loop product %J Algebraic and Geometric Topology %D 2024 %P 2619-2654 %V 24 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2619/ %R 10.2140/agt.2024.24.2619 %F 10_2140_agt_2024_24_2619
Kuribayashi, Katsuhiko; Naito, Takahito; Wakatsuki, Shun; Yamaguchi, Toshihiro. A reduction of the string bracket to the loop product. Algebraic and Geometric Topology, Tome 24 (2024) no. 5, pp. 2619-2654. doi: 10.2140/agt.2024.24.2619
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