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In this note, we use the pairing induced by the interchange map in conjunction with the strongly homotopy commutative algebra structure to define products on the Eilenberg–Moore differential Tor and give a simplified proof of an improved outcome of Jones's result due to Ndombol and Thomas. As a result, we establish an isomorphism of graded algebras between the Hochschild homology and the free loop space cohomology of a simply connected topological space. [ABSTRACT FROM AUTHOR]

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 S¹-equivariant cohomology of the free loop space on X with coefficients in Fp. We conclude by performing some calculations on the negative cyclic homology. [ABSTRACT FROM AUTHOR]