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

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 $Fp$ satisfying some extra conditions. When $A$ denotes the singular cochains with coefficients in $Fp$ 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 $Fp$. We conclude by performing some calculations on the negative cyclic homology.