|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
Bialgebraic approach to
rack cohomology
Simon Covez, Marco Andrés Farinati, Victoria Lebed and Dominique Manchon |
|
Algebraic & Geometric Topology 23 (2023) 1551–1582
|
 |
Abstract |
|
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. |
Keywords
rack cohomology, quandle cohomology, differential graded
bialgebra, cup product, dendriform algebra, Zinbiel
algebra, homotopy
|
Mathematical Subject Classification 2010
Primary: 16T10, 20N02, 55N35, 57M27
|
References |
Publication
Received: 7 May 2019
Revised: 18 November 2021
Accepted: 6 December 2021
Published: 14 June 2023
|
Authors |
| Simon Covez | |
| Lycée Sainte-Pulchérie Istanbul Turkey |
|
| Marco Andrés Farinati | |
| Departamento de Matemática
FCEyN Universidad de Buenos Aires Buenos Aires Argentina |
|
| Victoria Lebed | |
| Normandie Univ UNICAEN, CNRS, LMNO Caen France |
|
| Dominique Manchon | |
| Laboratoire de Mathématiques Blaise
Pascal Université Clermont-Auvergne Clermont-Ferrand France |
|
| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
Open Access made possible by participating institutions via Subscribe to Open.
