Higher structures on homology groups

TL;DR

This paper extends the classical operad-based Gerstenhaber algebra structure on cohomology to a dual setting involving cooperads, resulting in Gerstenhaber coalgebra structures on homology theories.

Contribution

It introduces a dual framework where cooperads with comultiplication induce Gerstenhaber coalgebra structures on homology, broadening the algebraic structures applicable to homology theories.

Findings

Gerstenhaber coalgebra structures on Tor groups over bialgebras and Hopf algebras

Gerstenhaber coalgebra structures on Hochschild homology of Frobenius algebras

Dualization of operad structures to cooperad contexts

Abstract

We dualise the classical fact that an operad with multiplication leads to cohomology groups which form a Gerstenhaber algebra to the context of cooperads: as a result, a cooperad with comultiplication induces a homology theory that is endowed with the structure of a Gerstenhaber coalgebra, that is, it comes with a graded cocommutative coproduct which is compatible with a coantisymmetric cobracket in a dual Leibniz sense. As an application, one obtains Gerstenhaber coalgebra structures on Tor groups over bialgebras or Hopf algebras, as well as on Hochschild homology for Frobenius algebras.