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# Homology, Homotopy and Applications

## Volume 11 (2009)

### Number 1

### Steenrod operations on the negative cyclic homology of the shc-cochain algebras

Pages: 315 – 348

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n1.a13

#### Author

#### Abstract

In this paper we prove that the Steenrod operations act naturally on the negative cyclic homology of a differential graded algebra $A$ over the prime field $\mathbb{F}_p$ satisfying some extra conditions. When $A$ denotes the singular cochains with coefficients in $\mathbb{F}_p$ of a 1-connected space $X$, these extra conditions are satisfied. The Jones isomorphism identifies these Steenrod operations with the usual ones on the $S^1$-equivariant cohomology of the free loop space on $X$ with coefficients in $\mathbb{F}_p$. We conclude by performing some calculations on the negative cyclic homology.

#### Keywords

Hochschild homology, negative cyclic homology, bar and cobar construction, shc-algebra

#### 2010 Mathematics Subject Classification

13D03, 54C35, 55S20, 57T30

Published 1 January 2009