Framed Matrices and A?-Bialgebras

We complete the construction of the biassociahedra KK, construct the free matrad W infinity, realize W infinity as the cellular chains of KK, and define an A infinity-bialgebra as an algebra over W infinity. We construct the bimultiplihedra JJ, construct the relative free matrad rW infinity as a W infinity-bimodule, realize rW infinity as the cellular chains of JJ, and define a morphism of A infinity-bialgebras as a bimodule over W infinity. We prove that the homology of every A infinity-bialgebra over a commutative ring with unity admits an induced A infinity-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A infinity-bialgebras and determine the A infinity-bialgebra structure of H* (1 sigma X; Q). For each n > 2, we construct a space Xn and identify an induced nontrivial A infinity-bialgebra operation omega 2n : H* (1Xn; Z2)circle times 2 H* (1 Xn; Z2)circle times n.