Homology, Homotopy and Applications

Volume 26 (2024)

Number 1

Comparing diagonals on the associahedra

Pages: 141 – 149

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n1.a9

Authors

Samson Saneblidze (A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University 2, Tbilisi, Georgia)

Ronald Umble (Department of Mathematics, Millersville University of Pennsylvania, Millersville, Penn., U.S.A.)

Abstract

We prove that the formula for the diagonal approximation $\Delta_K$ on J. Stasheff’s $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in $\href{ https://dx.doi.org/10.4310/HHA.2004.v6.n1.a20}{[7]}$ agrees with the “magical formula” for the diagonal approximation $\Delta^\prime_K$ derived by Markl and Shnider in $\href{ https://www.ams.org/journals/tran/2006-358-06/S0002-9947-05-04006-7/ }{[5]}$, by J.-L. Loday in $\href{ https://doi.org/10.1007/978-0-8176-4735-3_13 }{[4]}$, and more recently by Masuda, Thomas, Tonks, and Vallette in $\href{ https://doi.org/10.5802/jep.142}{[6]}$.

Keywords

associahedron, permutahedron, diagonal approximation, magical formula

2010 Mathematics Subject Classification

Primary 55P48, 55P99. Secondary 52B05, 52B11.

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Dedicated to the memory of Jean-Louis Loday

Received 22 August 2022

Received revised 22 March 2023

Accepted 26 March 2023

Published 20 March 2024