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Bialgebraic approach to rack cohomology

Simon Covez, Marco Andrés Farinati, Victoria Lebed and Dominique Manchon

Algebraic & Geometric Topology 23 (2023) 1551–1582

We interpret the complexes defining rack cohomology in terms of a certain dg bialgebra. This yields elementary algebraic proofs of old and new structural results for this cohomology theory. For instance, we exhibit two explicit homotopies controlling structure defects on the cochain level: one for the commutativity defect of the cup product, and the other for the “Zinbielity” defect of the dendriform structure. We also show that, for a quandle, the cup product on rack cohomology restricts to, and the Zinbiel product descends to quandle cohomology. Finally, for rack cohomology with suitable coefficients, we complete the cup product with a compatible coproduct.

rack cohomology, quandle cohomology, differential graded bialgebra, cup product, dendriform algebra, Zinbiel algebra, homotopy
Mathematical Subject Classification 2010
Primary: 16T10, 20N02, 55N35, 57M27
Received: 7 May 2019
Revised: 18 November 2021
Accepted: 6 December 2021
Published: 14 June 2023
Simon Covez
Lycée Sainte-Pulchérie
Marco Andrés Farinati
Departamento de Matemática FCEyN
Universidad de Buenos Aires
Buenos Aires
Victoria Lebed
Normandie Univ
Dominique Manchon
Laboratoire de Mathématiques Blaise Pascal
Université Clermont-Auvergne

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