A Cellular Diagonal Approximation On The Sequence Of Associahedra

The chain complexes on the sequence of associahedra de?ne a differential graded A1 operad. A diagonal approximation on the associahedra is therefore equivalent to exhibiting an A1 structure on a product of A1 algebras. In the following we construct a direct formula for the cell decomposition on the associahedra, corresponding to the diagonal, and give formulae for the number of cells in the resulting decomposition. Examples of the results in low dimension are also provided. We then apply the formula for the diagonal in two ways; firstly to investigate the explicit topological operad structure on the Loday realisation of the associahedra, and secondly to construct twisting cochains from an A1 coalgebra to an A1 algebra. We also consider possible approaches to measuring the obstruction to coassociativity inherent in the diagonal approximation

however we do not give any new results. In the appendix we also provide an algorithm for a alternative labelling of triangulations of the associahedra to that appearing in [Lod07b].