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In this paper we use May's algebraic approach to Steenrod operation in conjunction with Munkholm's notion of shc-algebra in order to introduce what we call a -shc-algebra. ( is a cyclic group of order a ÿxed prime p.) We prove that the homology and the Hochschild homology of a -shc-algebra have natural Steenrod operations. Our main topological example is the free loop space. We prove that the Jones' isomorphism preserves Steenrod operations.

Let K be a field of characteristic p ≥ 0 and S 1 the unit circle. We prove that the shc-structure on a cochain algebra (A, d A ) induces an associative product on the negative cyclic homology HC − * A. When the cochain algebra (A, d A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in K, then HC − * A is isomorphic as a graded algebra to H − * S 1 (LX; K) the S 1 -equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85-106 (2002) to compute the S 1 -equivariant cohomology algebras of the free loop space of the complex projective space CP(n) when n + 1 = 0 [p] and of the even spheres S 2n when p = 2.Keywords Hochschild homology · Cyclic homology · Free loop space · Borel fibration · shc-algebra · Algebra of divided powers · S 1 -equivariant cohomology Mathematics Subject Classification (2000) 57T30 · 54C35 · 55S20 0 Introduction Throughout this paper, K is a field of characteristic p ≥ 0 and X a simply connected topological space.

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