Let \(m,n\in\mathbb{N},mn\geq2\). The biassociahedron \(KK_{n,m}\) is a contractible \((m+n-3)\)-dimensional polytope with a single top dimensional cell \(e^{m+n-1}\), particularly \[ KK_{1,n}\cong KK_{n,1} \] being Stasheff’s associahedron \(K_{n}\) [J. D. Stasheff, Trans. Am. Math. Soc. 108, 275–292, 293–312 (1963; Zbl 0114.39402)]. The authors’ construction of \(KK_{n,m}\) in [S. Saneblidze and R. Umble, Homology Homotopy Appl. 13, No. 1, 1–57 (2011; Zbl 1222.55006)], which is valid for all \(m\) when \(n\leq3\) and for all \(n\) when \(m\leq3\), extends M. Markl’s construction for \(m+n\leq6\) in [M. Markl, J. Pure Appl. Algebra 205, No. 2, 341–374 (2006; Zbl 1116.18011); J. Homotopy Relat. Struct. 10, No. 2, 205–238 (2015; Zbl 1351.18011)].
This paper introduces and applies the theory of framed matrices to construct \(KK_{m,n}\) for all \(m\) and \(n\). A framed matrix is an equivalence class of paths of generalized bipartition matrices whose entries involve augmented bipartitions, which are pairs of partitions \((A\mid\cdots\left\vert A_{r},B_{1}\right\vert \cdots\mid B_{r})\) of finite sets of positive integers in which \(A_{i}\) and \(B_{j}\) are possibly null. The paper constructs the bimultiplihedra \(JJ\), constructs the relative free matrad \(r\mathcal{H}_{\infty}\) as a \(\mathcal{H}_{\infty}\)-bimodule, realizes \(r\mathcal{H}_{\infty}\) as the cellular chains of \(JJ\), and defines a morphism of \(\mathcal{A}_{\infty}\)-bialgebras as a bimodule over \(\mathcal{H}_{\infty}\).
The main result is the following theorem.
Theorem 11.5. Let \(A\) be a free \(R\)-module, let \(B\) an \(\mathcal{A}_{\infty}\)-bialgebra, and let \(g:A\rightarrow B\) be a homology isomorphism. Then

(i)

(Existence) \(g\) induces an \(\mathcal{A}_{\infty}\)-bialgebra structure \(\omega_{A}\) on \(A\) and extends to a map \(G:\) \(A\Rightarrow B\) of \(\mathcal{A}_{\infty}\)-bialgebras, and

(ii)

(Uniqueness) \((\omega_{A},G)\) is unique up to isomorphism.

The proof follows from a new transfer algorithm based on the interpretation of \(JJ_{n,m}\) as a subdivision of \(KK_{n,m}\times I\).
As an application, the authors extend the Bott-Samelson isomorphism to an isomorphism of \(\mathcal{A}_{\infty}\)-bialgebras, determining the \(\mathcal{A}_{\infty}\)-bialgebrastructure of \(H_{\ast}(\Omega\Sigma X;\mathbb{Q})\) (Theorem 12.2). The \(\mathcal{A}_{\infty}\)-bialgebra structure of \(H_{\ast}(\Omega\Sigma X;\mathbb{Q})\) provides the first nontrivial rational homology invariant for \(\Omega\Sigma X\) (Corollary 12.3). For each \(n\geq2\), they construct a space \(X_{n}\), identifying an induced nontrivial \(\mathcal{A}_{\infty}\)-bialgebra operation \[ \omega_{2}^{n}:H^{\ast}(\Omega X_{n};\mathbb{Z}_{2})^{\otimes 2}\rightarrow H^{\ast}(\Omega X_{n};\mathbb{Z}_{2})^{\otimes n} \]