The category of $E_\infty$-coalgebras, the $E_\infty$-coalgebra structure on the homology, and the dimension completion of the fundamental group

Abstract

We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus we obtain a category of $E_\infty$-coalgebras. It turns out that if the homology of an $E_\infty$-coalgebra have no torsion, then there is a natural way to define an $E_\infty$-coalgebra structure on the homology so that the resulting coalgebra be isomorphic to the initial $E_\infty$-coalgebra in our category. We also discuss some invariants of the $E_\infty$-coalgebra structure on homology and relate them to the invariant formerly used by the author to distinguish the fundamental groups of the complements of combinatorially equivalent complex hyperplane arrangements.