Homology, Homotopy and Applications

Diagonals on the permutahedra, multiplihedra and associahedra

Samson Saneblidze and Ronald Umble

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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

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

First available in Project Euclid: 13 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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

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