Contents Online

# 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

#### 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.