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 --> K and its factorization through J. We introduce the notion of a permutahedral set Z and lift \Delta_P to a diagonal on Z. We show that the double cobar construction \Omega^2(C_*(X)) is a permutahedral set; consequently \Delta_P lifts to a diagonal on \Omega^2(C_*(X)). Finally, we apply the diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal generality.

Abstract:
We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph associahedra.

Abstract:
We give a topological solution to the $\Ainf$ Deligne conjecture using associahedra and cyclohedra. For this we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally acts on the Hochschild cochains of an $\Ainf$ algebra yielding an explicit, topological and minimal solution to the $\Ainf$ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as a new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow--up construction.

Abstract:
Our aim is to construct a functorial tensor product of $A_\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of $A_\infty$-algebras.

Abstract:
We study a cofibrant E-infty operad generated by the Fox-Neuwirth cells of the configuration space of points in the Euclidean space. We show that, below the `critical dimensions' in which `bad cells' exist, this operad is modeled by the geometry of the Fulton-MacPherson compactification of this configuration space. We analyze the Tamarkin bad cell and calculate the differential of the corresponding generator. We also describe a simpler, four-dimensional bad cell. We finish the paper by proving an auxiliary result giving a characterization, over integers, of free Lie algebras.

Abstract:
We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday-Ronco Hopf algebra.

Abstract:
Let C_*(K) denote the cellular chains on the Stasheff associahedra. We construct an explicit combinatorial diagonal \Delta : C_*(K) --> C_*(K) \otimes C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We apply the diagonal \Delta to define the tensor product of A_\infty-(co)algebras in maximal generality.

Abstract:
Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.

Abstract:
A graph associahedron is a simple polytope whose face lattice encodes the nested structure of the connected subgraphs of a given graph. In this paper, we study certain graph properties of the 1-skeleta of graph associahedra, such as their diameter and their Hamiltonicity. Our results extend known results for the classical associahedra (path associahedra) and permutahedra (complete graph associahedra). We also discuss partial extensions to the family of nestohedra.