Framed matrices and A_{∞}-bialgebrasS. Saneblidze, R. UmbleWe complete the construction of the biassociahedra KK, construct the free matrad H_{∞}, realize H_{∞} as the cellular chains of KK, and define an A_{∞}-bialgebra as an algebra over H_{∞}. We construct the bimultiplihedra JJ, construct the relative free matrad rH_{∞} as a H_{∞}-bimodule, realize rH_{∞} as the cellular chains of JJ, and define a morphism of A_{∞}-bialgebras as a bimodule over H_{∞}. We prove that the homology of every A_{∞}-bialgebra over a commutative ring with unity admits an induced A_{∞}-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A_{∞}-bialgebras and determine the A_{∞}-bialgebra structure of H_{*}(ΩΣX; Q). For each n≥2, we construct a space X_{n} and identify an induced nontrivial A_{∞}-bialgebra operationω_{2}^{n}: H^{*}(ΩX_{n}; Z_{2})^{⊗2} → H^{*}(ΩX_{n}; Z_{2})^{⊗n}. Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 15(4) (2022), pp. 41-140 |