Steenrod operations in SHC-algebras

Abstract

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.

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

“…which induces a graded algebra isomorphism HC − * X ∼ = H − * S 1 (LX, F p ), [18]. Following the lines of [21], consider a π-shc differential graded algebra (N * X, µ X ,κ X ) endowed with the structural map θ X = φ N * X • C −κ X defining the algebraic Steenrod operations on HC − * (N * X) and W ⊗ (N * ( X )) ⊗p Γ X −→ N * ( X ) the structural map defining Steenrod operations on N * ( X )), [15, 7.5]. Define the map γ X as the composite…”

“…B. Ndombol and Jean-Claude Thomas [21] have introduced the notion of π-shcalgebra and have proved that • The normalized singular cochain complex with coefficients in F p of a connected space X, C * (X; F p ), is a π-shc-algebra [21].…”

“…that defines a nonassociative differential graded algebra structure on Hom(W, A). Proposition 2.1 ( [21]). If A = {A i } i∈Z is differential graded algebra such that A i = 0 if i < 0 and θ the structural map as in 2.1, then:…”

“…where the functors X −→ C − * ((N * X) ⊗p ) and X −→ Hom(W ; K[u] ⊗ N * ( X )) defined on the category Top with models M = {O n , O n = p 0 ( p+1 ( n × p )); n ∈ N} preserve the units and are respectively acyclic and corepresentable. We obtain from the equivariant cyclic model theorem (see [21,Appendix B] and [18]) that there exists a π-linear natural transformation C − * ((N * X) ⊗p )…”

On the Steenrod operations in cyclic cohomology

“…Some special operations (dot product, bracket ) on Hochschild complex that induce a structure of graded algebra on the cohomology have been considered in [16]. Steenrod operations on the Hochschild homology have been studied in [13]. There are also operations in K-theory, for instance [8], and λ-operations in orthogonal Ktheory [3].…”