Noncommutative divergence and the Turaev cobracket

Taniguchi, Toyo

doi:10.2140/agt.2025.25.3103

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ISSN (electronic): 1472-2739

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Noncommutative divergence and the Turaev cobracket

Toyo Taniguchi

Algebraic & Geometric Topology 25 (2025) 3103–3132

DOI: 10.2140/agt.2025.25.3103

Abstract

The divergence map, an important ingredient in the algebraic description of the Turaev cobracket on a connected oriented compact surface with boundary, is reformulated in the context of noncommutative geometry using a flat connection on the space of $1$ -forms on a formally smooth associative algebra. We then extend this construction to homologically smooth associative algebras, which allows us to give a similar algebraic description of the Turaev cobracket on a closed surface. We also look into a relation between the Satoh trace and the divergence map on a free Lie algebra via geometry over the Lie operad.

Keywords

noncommutative geometry, divergence maps, formal smoothness, flat connections, loop operations

Mathematical Subject Classification

Primary: 57K20

Secondary: 16D20, 58B34

References

Publication

Received: 6 May 2024

Revised: 12 June 2024

Accepted: 25 July 2024

Published: 17 September 2025

Authors

Toyo Taniguchi
Graduate School of Mathematical Sciences The University of Tokyo Tokyo Japan

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Volume 25, issue 5 (2025)