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# Homology, Homotopy and Applications

## Volume 16 (2014)

### Number 2

### Homology operations in symmetric homology

Pages: 239 – 261

DOI: http://dx.doi.org/10.4310/HHA.2014.v16.n2.a13

#### Author

#### 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.

#### Keywords

symmetric homology, cyclic homology, homology operation, $E_{\infty}$ algebra

#### 2010 Mathematics Subject Classification

13D03, 55N35

Published 30 November 2014