Framed Matrices and $A_{\infty}$-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_{\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 $\omega_{2}^{n}: H^{\ast}\left(\Omega X_{n};\mathbb{Z}_{2}\right) ^{\otimes2}\rightarrow H^{\ast}\left(\Omega X_{n};\mathbb{Z}_{2}\right) ^{\otimes n}$.