  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.
