## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 6, Number 1 (2004), 363-411.

### Diagonals on the permutahedra, multiplihedra and associahedra

Samson Saneblidze and Ronald Umble

#### 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 $P\rightarrow K$ [19] 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 6, Number 1 (2004), 363-411.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139839559

**Mathematical Reviews number (MathSciNet)**

MR2118493

**Zentralblatt MATH identifier**

1069.55015

**Subjects**

Primary: 55U05: Abstract complexes

Secondary: 05A18: Partitions of sets 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] 55P35: Loop spaces 81T30: String and superstring theories; other extended objects (e.g., branes) [See also 83E30]

#### Citation

Saneblidze, Samson; Umble, Ronald. Diagonals on the permutahedra, multiplihedra and associahedra. Homology Homotopy Appl. 6 (2004), no. 1, 363--411. https://projecteuclid.org/euclid.hha/1139839559