2023/11/28 by Matthias Franz, Franz, Matthias
Mathematics · #16E45 #55R20 (Primary) #55T20 (Secondary) #Advanced Topics in Algebra #Algebra over a field #Algebraic Topology (math.AT) #Algebraic structure #Algebraic structures and combinatorial models #Associative property #Cohomology #Commutative property #FOS: Mathematics #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Mathematical analysis #Mathematics #Multiplicative function #Product (mathematics) #Pure mathematics #Torsion (gastropod)
paper · pdf · doi:10.48550/arxiv.2311.16947
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2023/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct an A_∞-structure on the two-sided bar construction involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative product defined by Carlson and the author and generalizes the dga structure on the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is promoted to a quasi-isomorphism of A_∞-algebras. We also show that the resulting product on the differential torsion product involving cochain algebras agrees with the one defined by Eilenberg-Moore and Smith, for all triples of spaces. This is a consequence of the following result, which is of independent interest: The strongly homotopy commutative (shc) structure on cochains inductively constructed by Gugenheim-Munkholm agrees with the one previously defined by the author for all hgas.