Homology Operations in Symmetric Homology

TL;DR

This paper demonstrates that symmetric homology of unital associative algebras admits rich algebraic structures, including homology operations and a Pontryagin product, by constructing an explicit $E_{}$ structure on the chain complexes.

Contribution

It introduces an explicit $E_{}$ structure on chain complexes computing symmetric homology, establishing homology operations and a Pontryagin product.

Findings

$HS_*(A)$ has homology operations.

$HS_*(A)$ possesses a Pontryagin product.

$HS_*(A)$ forms an associative commutative graded algebra.

Abstract

The symmetric homology of a unital associative algebra $A$ over a commutative ground ring $k$, denoted $HS_*(A)$, is defined using derived functors and the symmetric bar construction of Fiedorowicz. In this paper we show that $HS_*(A)$ admits homology operations and a Pontryagin product structure making $HS_*(A)$ an associative commutative graded algebra. This is done by finding an explicit $E_{\infty}$ structure on the standard chain groups that compute symmetric homology.