  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_{\ast}\left( \Omega\Sigma X;\mathbb{Q}\right) $.
For each $n\geq2$, we construct a space $X_{n}$ and identify an induced
nontrivial $A_{\infty}$-bialgebra operation $\omega_{2}^{n}:
H^{\ast}\left(\Omega X_{n};\mathbb{Z}_{2}\right) ^{\otimes2}\rightarrow
H^{\ast}\left(\Omega X_{n};\mathbb{Z}_{2}\right) ^{\otimes n}$.
