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 Xn and identify an induced nontrivial A∞-bialgebra operationω2n: H*(ΩXn; Z2)⊗2 → H*(ΩXn; Z2)⊗n. Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 15(4) (2022), pp. 41-140 |