Framed Matrices and $A_{\infty}$-Bialgebras

TL;DR

This paper develops the theory of $A_{ abla}$-bialgebras using biassociahedra and related structures, extending their applications to homology and specific topological spaces, providing new algebraic and geometric insights.

Contribution

It constructs and realizes biassociahedra and free matrad structures, defines $A_{ abla}$-bialgebras, and extends the Bott-Samelson isomorphism to this context, with applications to homology of loop spaces.

Findings

Homology of $A_{ abla}$-bialgebras admits induced structures.

Extended Bott-Samelson isomorphism to $A_{ abla}$-bialgebras.

Identified nontrivial $A_{ abla}$-bialgebra operations in specific spaces.

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