Framed Matrices and A∞-Bialgebras *

Abstract

We complete the construction of the biassociahedra $KK$, construct the free matrad ${\mathcal{H}}_{\infty }$, realize ${\mathcal{H}}_{\infty }$ as the cellular chains of $KK$, and define an $A_{\infty }$-bialgebra as an algebra over ${\mathcal{H}}_{\infty }$. We construct the bimultiplihedra $JJ$, construct the relative free matrad $r{\mathcal{H}}_{\infty }$ as a ${\mathcal{H}}_{\infty }$-bimodule, realize $r{\mathcal{H}}_{\infty }$ as the cellular chains of $JJ$, and define a morphism of $A_{\infty }$-bialgebras as a bimodule over ${\mathcal{H}}_{\infty }$. We prove that the homology of every $A_{\infty }$-bialgebra over a commutative ring with unity admits an induced $A_{\infty }$-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of $A_{\infty }$-bialgebras and determine the $A_{\infty }$-bialgebra structure of $H_{*}\left(\mathrm{\Omega }\mathrm{\Sigma }X;\mathrm{\mathbb{Q}}\right)$. For each $n\ge 2$, we construct a space $X_n$ and identify an induced nontrivial $A_{\infty }$-bialgebra operation ${\omega }_2^n:H^{*}{\left(\mathrm{\Omega }{X}_n;{\mathrm{\mathbb{Z}}}_2\right)}^{\otimes 2}\to H^{*}{\left(\mathrm{\Omega }{X}_n;{\mathrm{\mathbb{Z}}}_2\right)}^{\otimes n}$.