Homology Operations in Symmetric Homology

Abstract

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

arXiv comment22 pages, 2 figures. To Appear in Homology, Homotopy and Applications