Homology, Homotopy and Applications

Volume 6 (2004)

Number 1

Diagonals on the permutahedra, multiplihedra and associahedra

Pages: 363 – 411

DOI: http://dx.doi.org/10.4310/HHA.2004.v6.n1.a20

Authors

Samson Saneblidze (A. Razmadze Mathematical Institute, Georgian Academy of Sciences, Tbilisi, Republic of Georgia)

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

Abstract

We construct an explicit diagonal $\Delta_{P}$ on the permutahedra $P.$ Related diagonals on the multiplihedra $J$ and the associahedra $K$ are induced by Tonks’ projection $P\rightarrow K$ \cite{tonks} and its factorization through $J.$ We introduce the notion of a permutahedral set $% \mathcal{Z}$ and lift $\Delta_{P}$ to a diagonal on $\mathcal{Z}$. We show that the double cobar construction $\Omega^{2}C_{\ast}(X)$ is a permutahedral set; consequently $\Delta_{P}$ lifts to a diagonal on $% \Omega^{2}C_{\ast}(X)$. Finally, we apply the diagonal on $K$ to define the tensor product of $A_{\infty}$-(co)algebras in maximal generality.

Keywords

diagonal, permutahedron, multiplihedron, associahedron

2010 Mathematics Subject Classification

Primary 05A18, 05A19, 52B05, 55U05. Secondary 55P35.

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