STEENROD OPERATIONS ON THE NEGATIVE CYCLIC HOMOLOGY OF THE SHC-COCHAIN ALGEBRAS

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1 Homology, Homotopy and Applications, vol. 11(1), 2009, pp STEENROD OPERATIONS ON THE NEGATIVE CYCLIC HOMOLOGY OF THE SHC-COCHAIN ALGEBRAS CALVIN TCHEKA (communicated by Charles Weibel) 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 F p satisfying some extra conditions. When A denotes the singular cochains with coefficients in 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 F p. We conclude by performing some calculations on the negative cyclic homology. 1. Introduction Since their construction by N. Steenrod [22], Steenrod operations have played a central role in homotopy theory and in representation theory. In the topological setting, Steenrod operations P i } i N are stable natural transformations P i : H H +i ( ; F p ) if p = 2 ( ; F p ) H +(p 1)i ( ; F p ) if p is an odd prime where H ( ; F p ) denotes the singular cohomology functor with coefficients in the prime field F p. When p = 2, P i is called an i-steenrod square and usually denoted by Sq i, while when p is an odd prime, P i is called an i-steenrod power. These transformations satisfy the following properties: 1. P 0 = id. 2. P i H k (,F p) = 0, (resp. ξ) if i > k (resp. i = k) p = 2 i > 2k (resp. i = 2k) p is an odd prime. 3. P k ( ) = i+j=k P i P j, (the Cartan formula). 4. The Adem relations (see [16], pages 129 and 367). Here id (respectively 0) denotes the identity transformation (respectively the constant transformation whose value is 0) while ξ denotes the Frobenius transformation x Received February 22, 2008, revised December 16, 2008; published on June 12, Mathematics Subject Classification: 55S20, 57T30, 54C35, 13D03. Key words and phrases: Hochschild homology, negative cyclic homology, bar and cobar construction, shc-algebra and π-shc-algebra. This article is available at Copyright c 2009, International Press. Permission to copy for private use granted.

2 316 CALVIN TCHEKA x p. A. Dold [4] defined them in a more general context replacing the singular chains on a topological space by an arbitrary simplicial coalgebra. Later P. May [15] gave a purely algebraic construction of Steenrod operations which leads to the notion of E -algebra, as recently developed by Mandell [14] in homotopy theory, and to the construction of Steenrod operations in other settings. For example, there exist Steenrod operations on the cohomology of a commutative F p -Hopf algebra due to A. Liulevicius [12], after the paper of Dold quoted above; the cohomology of a restricted p-lie algebra due to P. May [15]; the cohomology of non commutative p-differential forms due to M. Karoubi [11]; the cyclic cohomology of a commutative F p -Hopf algebra due to M. Elhamdadi and Y. Gouda [5]. Let us recall here that if V i } i N is a graded F p -vector space and if T c (sv ) denotes the free coalgebra generated by the suspension of V, denoted sv, then: V is an A -algebra if there exists a degree 1 coderivation D on T c (sv ) such that D D = 0 and D T 0 (sv ) = 0. V is a B -algebra if it is an A -algebra and if there exists a product on T c (sv ) such that T c (sv ) is a differential graded Hopf algebra. V is a C -algebra if it is an A -algebra such that T c (sv ) is a differential graded Hopf algebra for the shuffle product. While it is possible to define the Hochschild homology (and the negative cyclic homology) of an A -algebra [9, 8], the lack of associativity of these operadic algebras complicates explicit computations. Hopefully, strongly homotopy commutative algebras (shc for short), as introduced by H.J. Munkholm [17], will considerably simplify the above mentioned calculations. They are associative B -algebras. Moreover, it is known that: The normalized singular cochain complex with coefficients in F p of a connected space X, C (X; F p ), is a shc-algebra [17]. The Hochschild homology of a shc-algebra A with coefficients in A, HH (A; A), is a graded algebra [20]. The negative cyclic homology of a shc-algebra A, HC (A), is a graded algebra [18]. Let X be a 1-connected space and LX be the free loop space. That is, LX = Map(S 1, X) is the space of continuous maps from S 1 to X endowed with the compact open topology. The Jones isomorphism HH (C (X, F p ); C (X, F p )) H (LX; F p ) is a homomorphism of graded algebras [20]. Let X be a 1-connected space and LX the free loop space. The Jones isomorphism HC (C (X, F p )) H S 1 (LX; F p ) is a homomorphism of graded algebras [18]. Here H S 1 (LX; F p ) denotes the S 1 -equivariant cohomology of LX with coefficients in F p. B. Ndombol and Jean-Claude Thomas [21] have introduced the notion of π-shcalgebra and have proved that

3 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 317 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]. There exist Steenrod operations on the Hochschild homology of a π-shc-algebra A with coefficients in A. Let X be a 1-connected space and LX be the free loop space. The Jones isomorphism HH (C (X, F p ); C (X, F p )) H (LX; F p ) respects the Steenrod operations. In this paper we complete the above result in proving: Theorem 1.1. Let ((A, d A ), µ A, κ A ) be a π-shc cochain algebra as in [21]. 1. The negative cyclic homology of a differential graded π-shc algebra A with coefficients in A, HC (A), has algebraic Steenrod operations. 2. Let X be a 1-connected space and LX be the free loop space. The Jones isomorphism HC (C (X, F p )) H S 1 (LX; F p ) respects the Steenrod operations. (See [18, 10].) The Steenrod operations, considered in our theorem, are defined at the chain level and satisfy the properties: 1. P 0 (1) = 1 if 1 denotes the unit of the graded algebra HC (A). 2. P i i > k (resp. i = k) p = 2 = 0, (resp. ξ) if HC k (,F p) i > 2k (resp. i = 2k) p is an odd prime. 3. P k ( ) = i+j=k P i P j, the Cartan formula. Except if A = C (X; F p ), the Steenrod operations constructed by Ndombol-Thomas or those considered in part 1 of our theorem do not in general satisfy the Adem relations. An operadic construction as in [2] or [1] allows us to define an action of the large Steenrod algebra on the Hochschild homology of a E -algebra. Such an action on the negative cyclic homology remains an open question. In these notes, quasiisomorphism means a homomorphism which is an isomorphism in (co)homology. The paper is organized as follows. Section 2 is a recollection of definitions. Part 1 (respectively Part 2) of Theorem 1.1 is proved in Section 3 (respectively Section 4). Recalling the π-shc-minimal model and explicit computations are the subjects of Section 5 and Section 6 respectively. This paper is a part of my thesis supervised by professors B. Ndombol and J. C. Thomas of Yaounde I University, Cameroon, and Angers University, France respectively. Convention Throughout this paper, we use the Kronecker convention: an object with lower negative graduation has upper non-negative graduation. 2. Preliminaries Let π be any finite group and p a fixed prime. Throughout this paper, we work over the field F p equipped with the trivial action of π. The ring group F p [π] is an augmented algebra.

4 318 CALVIN TCHEKA 2.1. Algebraic Steenrod operations The material involved here is contained in [17]. Let π = 1, τ,..., τ p 1 } be the cyclic group of order p. Let W ε W F p be a projective resolution of F p over F p [π]; that δ i is W = (W i ) i0 ; W i Wi 1 ; W 0 F p [π], where each W i is a right projective F p [π]- η W module and δ i is π-linear. We choose F p W such that εw η W = id Fp. Necessarily η W ε W id W. Let A = A} i Z be a differential graded algebra (not necessarily associative). We denote by m (p) A (resp. (Hm A) (p) ) the iterated product a 1 a 2 a p a 1 (a 2 ( a p )) (resp. the iterated product induced on HA by m (p) A ). Identify τ π with the p-cycle (p, 1,..., p 1) and assume that π acts trivially on A, thus π acts diagonally on A p and on W A p. If the natural map H(W A p ) = H(A) p (Hm A) (p) HA lifts to a π-linear chain map θ : W A p A, then for any i Z and x H n A, there exists a well defined class P i H n+i (A) if p = 2 (x) H n+2i(p 1) (A) if p > 2, such that: If p = 2, If p > 2, P i (1 HA ) = 0 if i 0. P i (x) = 0 if i > n P i (x) = x 2 if i = n. P i (x) = 0 if 2i > n P i (x) = x p if n = 2i. Moreover these classes do not depend on the choice of the resolution W nor η and are compatible with algebra homomorphisms commuting with structural map θ. These operations do not in general satisfy P i (x) = 0 if i < 0, P 0 (x) = x, the Cartan formulas and the Adem relations Cartan formula Let us consider the differential graded algebra A = A i } i Z such that A i = 0 for i < 0. A homogeneous map W f A has a degree k if f Hom k (W, A) = i0 Hom(W i, A k i ) = k i=0 Hom(W i, A k i ). The differential of f is D(f) = d f ( 1) f f δ; π acts on each Hom k (W, A) by (σf)(w) = f(wσ); the evaluation map Hom(W, A) ev0 A is a homomorphism of chain complexes. f ev 0 (f) = f(e 0 )

5 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 319 Let W ψ W W W be any diagonal approximation and denote by m A the product on the algebra A. We have the cup-product k Hom(W, A) Hom(W, l A) Hom(W, k+l A) f g f g = m A (f g) ψ W 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: 1. The structural map W A p θ A induces a π-chain map A p θ Hom(W, A) u W θ(u) A w θ(u)(w) = ( 1) u w θ(w u) such that ev 0 θ = m (p) A and H(ev 0) H( θ) = H(m A ) (p). 2. If we assume that H( θ) respects the products, the algebraic Steenrod operations defined by θ satisfy the Cartan formula P i (xy) = j+i=i P j (x)p k (y), x, y H A Review of the construction of Steenrod operations We consider the standard small free resolution of π = 1, τ,..., τ p 1 }: W = (W i ) i0 ; W i = e i F p [π]; W 0 F p [π], δ W i i Wi 1, δ(e 2i+1 ) = (1 + τ)e 2i, δ(e 2i ) = (1 + τ + + τ)e 2i 1, [21] W ε W K e i ε W (e i ) = 0 if i 0 1 K if i = 0. Note that this standard free resolution equipped with its diagonal approximation ψ W has a coalgebra structure. Let θ π : W π A p A be the map induced by the structural map θ and denote by θ the homomorphism H(θ π ). Observe that any section ρ of a natural projection A kerd H(A) lifts to a π-linear chain map ρ: W (HA) p W A p and thus to a chain map ρ π : W π (HA) p W π A p. Since W is a semifree F p - module in the sense of [7], ρ = H(ρ π ) is an isomorphism. The algebraic Steenrod operations are defined as follows [15]; for x H n (A), each e k x p is a cocycle in W π (HA) p.

6 320 CALVIN TCHEKA If p = 2, and if p is odd, Sq i (x) = θ ρ (cl(e n i π x p )) = cl(θ π ρ π (e n i π x p )) = cl(θ π (e n i π ρ π (x p ))) = cl(θ π (e n i π ρ π (x) p )) = cl( θ(ρ(x) p ))(e n i ) P i (x) = ( 1) i ν(n)θ ρ (cl(e (n 2i)(p 1) 1 x p )) = ( 1) i ν(n)cl( θ(ρ(x) p ))(e (n 2i)(p 1) ) βp i (x) = ( 1) i ν(n)cl( θ(ρ(x) p ))(e (n 2i)(p 1) 1 ) where ν(n) = ( 1) j (( p 1 2 )!)ɛ if n = 2j + ɛ, ɛ = 0, 1 and θ the π-chain map defined in Proposition Hochschild homology and negative cyclic homology Here, Hochschild homology and negative cyclic homology are recalled. Let DA and DC denote respectively the category of connected cochain algebras and the category of connected cochain coalgebras. The reduced bar and cobar construction are a pair of adjoint functors B : DA DC : Ω (see [6]). The generators of BA (resp. ΩC) are denoted [a 1 a 2 a k ] B k A (resp. c 1 c 2 c l Ω l C) and [ ] = 1 B 0 A K (resp. = 1 Ω 0 C K). The adjunction mentioned above yields for a cochain algebra (A, d A ), a natural quasi-isomorphism of cochain algebras α A : ΩBA A [17]. The linear map ι A : A ΩBA such that ι A (1) = 1, ι A (a) = [a], a Ā is a chain complex quasiisomorphism. In any case, it satisfies α A ι A = id A, id ΩBA ι A α A = d ΩBA h + h d ΩBA for some chain homotopy h: ΩBA ΩBA such that α A h = 0, h ι A = 0, h 2 = 0. Let (A, d A ) be a cochain algebra. Recall also that the normalized Hochschild chain complex of (A, d A ) is a graded vector space C k A} k0, C k A = A B k A where the generators of C k A are of the form a 0 [a 1 a 2... a k ] if k > 0 and a[ ] if k = 0. We set ɛ i = a 0 + sa 1 + sa sa i 1, i 1 and define the Hochschild differential d = d 1 + d 2 by d 1 (a 0 [a 1 a k ]) = d A (a 0 )[a 1 a 2 a k ] i = 1 k ( 1) ɛ ia 0 [a 1 a 2 d A (a i ) a k ] d 2 (a 0 [a 1 a k ]) = ( 1) a0 a 0 a 1 [a 2 a k ] k + a 0 [a 1 a 2 a i 1 a i a k ] i=2 ( 1) sa k ɛ k a k a 0 [a 2 a k ]. The Hochschild differential decreases the degree by one (see [13] or [20] for more details).

7 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 321 By definition, HH A := H CA is the Hochschild homology of the cochain algebra (A, d A ). It is clear that CA is concentrated in non-negative total degrees. Hence so is HH A. If (A, d A ) = (N (X; K), d N (X;K)) is the algebra of normalized singular cochains on the topological space X, then C N (X; K) is the normalized Hochschild chain complex of X and HH X := HH N (X; K) is the Hochschild homology of X. For the cochain algebra (A, d A ), the Connes operator is the linear map B : C A C +1 A defined by Ba 0 [a 1 a n ] = n i=0 ( 1)ɛ i 1[a i a n a 0 a i 1 ], where ɛ i = a 0 + ( a 0 + a a i 1 + i)( a i + + a n + n i + 1). Consider the polynomial algebra K[u] on the single generator u of upper degree +2 and form the complex C A = K[u] C A with differential D defined by D(u l a 0 [a 1 a n ]) = u l d(a 0 [a 1 a n ]) + u l+1 B(a 0 [a 1 a n ]). The chain complex C A is the negative cyclic chain complex of the cochain algebra (A, d A ) (see [10]). Let L and M be two graded F p -modules; L ˆ M will denote the tensor product defined by (L ˆ M) n = Li M n i. Generally for a differential graded algebra A, C A = F p [u] ˆ Fp A. So, for example, an element of degree d is given by an infinite sum of the form u i e i where e i A d+2i [3]. If A is positively graded, C A = F p [u] ˆ Fp A = F p [u] Fp A, and its homology HC A is the negative cyclic homology of (A, d A ). Again, it is clear that C A is concentrated in non-negative total degrees and so is HC A. If (A, d A ) = N (X; K) is the algebra of normalized singular cochains on the topological space X, then C N (X; K) is the negative cyclic chain complex of X, and HC X := HC N (X; K) the associated negative cyclic homology. If (A; d A ) is commutative (in the graded sense), then the multiplication m A : A A A is a homomorphism of DG-algebras. Thus the composite Cm A sh: CA CA CA defines a multiplication on CA which makes it into a commutative algebra [13, 4.2.2], where sh: CA CA C(A A) denotes the shuffle map Homotopy 1. Recall that f, g DA(A, A ) are homotopic in DA if there exists a linear map h: A A such that f g = d A h + h d A and h(xy) = h(x)g(y) + ( 1) x f(x)h(y) with x, y A. If f, g DA(A, A ) are homotopic, we write f DA g. 2. Let (A, d A ) (resp. (C, d C )) be a differential graded algebra (resp. coalgebra). Let T (C, A) = t Hom 1 (C, A): Dt = t t, t η C = 0 = ε A t} be the twisting cochain space as in [17, 1.8], where D denotes the differential in Hom(C, A) and the usual cup-product on Hom(C, A). The universal twisting cochains t, t T (C, A) are homotopic in T C(C, A) if there exists a linear map h Hom 1 (C, A) such that Dh = t h h t, h η C = η A and εa h = ε C and we write t T t [17, 1.11]. 3. Denote by π-dm the category whose objects are differential graded modules over F p equipped with an action of the cyclic group π and whose morphisms are

8 322 CALVIN TCHEKA F p [π]-linear. If the F p [π]-linear maps f and g are homotopic with a F p [π]-linear homotopy, we write f π DM g. Denote by π-da the subcategory whose objects are differential graded algebras over F p equipped with an action of the cyclic group π and whose morphisms are linear morphisms of F p [π]-differential graded algebras. If the maps f, g π-da are homotopic with a F p [π]-linear homotopy, we write f π DA g. Lemma 2.2. Let (T V, d V ) be a differential graded algebra and assume that a finite group π acts freely on V. Let (A, d A ) be a F p [π]-differential graded algebra and f, g π-da(tv,a). if f π DA g then C f π DM C g. Proof. We consider as in [21, Lemma A.6], the cylinder object I(T V, d) := (T (V 0 V 1 sv )) on (T V, d): (T V, d) δ 0 I(T V, d) δ 1 (T V, d) p (T V, d) with δ 0 (V ) = V 0, δ 1 (V ) = V 1, p(v 0 ) = p(v 0 ) = v, p(sv 0 ) = 0, D = d on V 0 and V 1, Dsv = dsv where S is the unique (δ 0, δ 1 )-derivation S : T V T (V 0 V 1 sv ) extending the graded isomorphism s: V sv. The free π-action on T V naturally extends to a free π-action on I(T V, d) so that I(T V, d) is a π-free algebra and the maps p, δ 0, δ 1 are π-equivariant quasi-isomorphisms of differential graded algebras. Moreover from the free π-action on the cylinder object I(T V, d), we have a free π-action on the negative cyclic complex C I(T V ) of the cylinder object I(T V ) by the following rule: For any u l x 0 [x 1 x 2 x n 1 x n ] C I(T V ) and σ π, σ u l x 0 [x 1 x 2 x n 1 x n ] = u l σ x 0 [σ x 1 σ x 2 σ x n 1 σ x n ]. Thus C I(T V ) is an object of the category π-dm. By definition, f DA g (resp. f π DA g) if and only if there exists H DA(I(T V ), A) (resp. H π DA(I(T V ), A)) such that H δ 0 = f and H δ 1 = g. If H π-da( I(TV)) is a homotopy between f and g, then C H is a π-linear homotopy between C f and C g. Hence C f π DM C g.

9 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS A strongly homotopy commutative algebra (see [17, 4]) A strongly homotopy commutative algebra (shc-algebra for short) is a triple (A, d A, µ A ) with (A, d A ) ObjDA and µ A DA(ΩB(A A), ΩBA) satisfying 1. α A µ A ι A A = m A, where m A is the product on A; 2. α A µ A ΩB(id A η A ) ι A = α A µ A ΩB(η A id A ) ι A = id A, where K η A A is the unit in A; i.e. ηa is the unit up to homotopy for µ A ; 3. µ A ΩB(α A id A ) ΩB(µ A id A ) χ (A A) A DA µ A ΩB(id A α A ) ΩB(id A µ A ) χ A (A A) ; i.e. µ A is associative up to homotopy; 4. µ A ΩBT DA µ A ; where T denotes the interchange map T (x y) = ( 1) x y y x; i.e. µ A is commutative up to homotopy. The following natural homomorphisms of DG-algebras are defined in [17, 2.2]; ΩB(ΩB(A A) A) χ (A A) A and satisfy ΩB(A A A) χ A (A A) ΩB(A ΩB(A A)), α (A A) A χ (A A) A = α A A A = α A (A A) χ A (A A). Consider A and A in ObjDA. The map f DA(A, A ) is a shc-map from (A, d A, µ A ) to (A, d A, µ A ) if 1. α A ΩBf ι A = f; 2. α A ΩBf η ΩBA = η A ; 3. ΩBf µ A DA µ A ΩB(f f). As proved by [17], an example of shc-cochain algebra is the algebra N (X; K) of normalized singular cochains of a topological space X. On the other hand it is proved in [20] that if (A, d A, µ A ) is a shc-cochain algebra, then 1. BA is a differential graded Hopf algebra and H BA is a commutative graded Hopf algebra; 2. CA is a (non associative) graded algebra such that HH A is a commutative graded algebra; 3. if f : (A, d A, µ A ) (A, d A, µ A ) is a morphism of shc-cochain algebras, we have A i CA ρ BA the commutative diagram f Cf Bf, where i, i, ρ, A i CA ρ BA ρ and Cf are homomorphisms of cochain algebras and Bf is a homomorphism of differential graded Hopf algebras. Remark 2.3. We recall the following facts given in [21, A.2]: 1- If ((A, d A ), µ A ) is a shc differential graded algebra, so is ΩB(A), with the shc structural map µ ΩB(A) given by the composite µ ΩB(A) = θ ΩB(A) µ A ΩB(α A α A ), where θ ΩB(A) DA(ΩB(A), ΩBΩB(A)) is the unique section of α ΩB(A) DA(ΩBΩB(A), ΩB(A)) such that α A α ΩB(A) θ ΩB(A) = α A.

10 324 CALVIN TCHEKA 2- If ((A, d A ), µ A ) is a shc differential graded algebra and ((W, δ W ), ψ W ) a standard small free resolution of π equipped with its differential graded coalgebra structure, then Hom(W ; A) is a shc differential graded algebra with the shc structural map µ Hom(W ;A) DA(ΩB([Hom(W ; A)] 2 ), ΩB([Hom(W ; A)])) defined by µ Hom(W ;A) = ΩB(Hom(W, α A µ A )) θ Hom(W ;A 2 ) ΩB(ψ A ), where [Hom(W ; A)] 2 ψ A [Hom(W ; A 2 )] the map defined by ψ A (f g) = (f g) ψ W is the homomorphism of differential graded algebras satisfying Hom(W, m A ) ψ A = and θ Hom(W,A 2 ) DA(ΩB Hom(W, A A), ΩB(Hom (W, ΩB(A A)))) the unique homomorphism of differential graded algebras such that Hom(W, α A 2) α Hom(W,ΩB(A 2 )) θ Hom(W,A 2 ) = α Hom(W,A A) Shc-equivalence and shc-formality [20, 5] The shc cochain algebras (A, d A, µ A ) and (A, d A, µ A ) are said to be shc-equivalent (A shc A ) if there exists a sequence of shc morphisms A A 1 A which are quasi-isomorphisms. One particular case of shc-equivalence is the shc-formality. Recall that the cohomology algebra of a shc cochain algebra is commutative. Every commutative cochain algebra is a shc algebra with shc structural map µ A = ΩB(m A ): ΩB(A A) ΩBA, where m A is the product on A. A shc cochain algebra A is shc-formal if it is shc-equivalent to its cohomology algebra H A A π-strongly homotopy commutative algebra; see [21, 1.6] Let (A, d A, µ A ) and (A, d A, µ A ) be two shc differential graded algebras. There exists a natural homomorphism µ A A DA(ΩB((A A ) 2 ); ΩB(A A )) such that (A A, d A A ; µ A A ) is a shc differential graded algebra. In particular, if (A, d A, µ A ) is a shc differential graded algebra, then for any n 2, there exists a homomorphism of differential graded algebras called the shc iterated structural map ΩB(A n ) µ(n) A ΩB(A) such that µ (2) = µ and α A µ (n) i A n m (n) A (see [21, Lemma A.3]. A shc-algebra (A, d A, µ A ) is a π-strongly homotopy commutative algebra (a π-shcalgebra for short) if there exists a map ΩB(A p ) κ A Hom(W, A) that is a π-linear homomorphism of differential graded algebras such that ev 0 κ A DA α A µ (p) A, where p is the prime number characteristic of F p. The action of S p on B(A p ) (resp. ΩB(A p )) is defined by the rule: For any σ S p σ[a 1 a 2 a p 1 a p ] = [a σ(1) a σ(2) a σ(p 1) a σ(p) ], a i A p (resp. σ < x 1 x 2 x p 1 x p >=< σx 1 σx 2 σx p 1 σx p >, x i B(A p )). f Recall that a strict π-shc homomorphism ((A, d A ), µ A, κ A ) ((A, d A ), µ A, κ A ) is a strict shc homomorphism such that the following homomorphisms of differential graded algebras κ A ΩB(f p ) and Hom(W, f) κ A are π-linear homotopic.

11 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS Proof of the first part of Theorem 1.1 As explained in [13, 4.3], a (p, q)-cyclic shuffle is a permutation σ(1),..., σ(p), σ(p + 1),..., σ(p + q)} in S p+q obtained as follows: Perform a cyclic permutation of any order on the set 1,..., p} and perform a cyclic permutation of any order on the set p + 1,..., p + q}. We shuffle the two results to obtain σ(1),... σ(p), σ(p + 1),..., σ(p + q)} in S p+q. The permutation obtained in that way is a cyclic shuffle if 1 appears before p + 1; we denote by C p,q the set of (p, q)-cyclic shuffles. A map : C p (A, A) C q (A, A) C p+q (A 2, A 2 ) is defined by a 0 [a 1 a 2 a p 1 a p ] b 0 [b 1 b 2 b q 1 b q ] = ( 1) ε(σ) a 0 b 0 [c σ(1) c σ(2) c σ(p+q 1) c σ(p+q) ], σ P C (p,q) where ε(σ) is the signature of σ and a σ(i) 1 if 1 i p c σ(i) = 1 b σ(i) p if p + 1 i p + q. The cyclic shuffle (see [13, 4.3.2]) is a linear map defined by C (A, A) C (A, A) sh C + +2(A 2, A 2 ) sh (a 0 [a 1 a 2 a p 1 a p ] b 0 [b 1 b 2 b q 1 b q ]) = 1[a 0 a 1 a 2 a p 1 a p ] 1[b 0 b 1 b 2 b q 1 b q ]. It is clear from the definition that if a 0 = 1 or b 0 = 1 then sh (a 0 [a 1 a 2 a p 1 a p ] b 0 [b 1 b 2 b q 1 b q ]) = 0. Proposition 3.1. (see [13, 4.3.7]) The following identities are satisfied: d sh = sh d; sh B = B sh ; B sh sh B + d sh sh d = 0, where d and B are the Hochschild differential and the Connes operator respectively. When K[u] C(A) C(A) and K[u] C(A A) are equipped with the obvious differentials denoted respectively by D and ˆD, the linear map Sh: K[u] C(A) C(A) K[u] C(A A) defined by Sh = id K[u] sh + u(id K [u] sh ) satisfies D Sh = Sh ˆD. Let (A, d A, µ A ) be a shc-algebra. The chain map m C A given by the composite K[u] C(A) K[u] C(A) id T id K[u] K[u] C(A) C(A) m K[u] id K[u] C(A) C(A) Sh K[u] C(A A) S A A K[u] C(ΩB(A A)) C (µ A) C (ΩB(A)) C (α A) C (A)

12 326 CALVIN TCHEKA defined a product on C (A) associative up to homotopy; where S A A is a linear section induced by the surjective quasi-isomorphism C (ΩB(A A)) C (α A A) C (A A), T the interchange isomorphism and m K[u] the product on K[u]. Precomposing H (m C (A)) by the Künneth isomorphism yields an associative product on HC (A). Together with this product, HC (A) is an associative graded algebra (see [18]). Proposition 3.2. Let ((A; d A )) be a cochain algebra and ((W ; δ W ); Ψ W ) a standard free resolution of π equipped with its coassociative coalgebra structure. There exists a natural homomorphism of chain complexes φ A such that the following diagram commutes. φ A C (Hom(W, A)) Hom(W ; C (A)) C (ev0) ev 0 C (A) Proof. Let us prove the existence of the map φ A. We define φ A by φ A : K[u] C(Hom(W ; A)) Hom(W ; K[u] CA) such that u l f 0 [f1 f 2 f k 1 f k ] φ A (u l f 0 [f1 f 2 f k 1 f k 1 ]) φ A (u l f 0 [f1 f 2 f k 1 f k 1 ]) where W Ψ(1) W :=Ψ W map defined by such that = (Id Id s k ) (g(u l ) f 0 f1 f 2 f k 1 f k ) Ψ (k+1) W, g(u l ): W K[u] W W ; W Ψ(k+1) W W k+2 denotes the iterated diagonal, and g the K[u] g Hom(W ; K[u]) u l g(u l ) e i g(u l )(e i ) = g(u l )(τ j e i ) u l k if i = 2k, 0 k l; = 0 if not.

13 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 327 Here we check in detail that φ A commutes with the differentials. Consider for this purpose D, the differential in C (Hom(W ; A)) defined by D(u l f 0 [f 1 f 2 f k 1 f k ]) = u l d C (Hom(W ;A))(f 0 [f 1 f 2 f k 1 f k 1 ]) + u l+1 B(f 0 [f 1 f 2 f k 1 f k 1 ]) and D the differential in Hom(W ; C A) defined by D(f) = D f ( 1) f f δ where D denotes the differential in C A; f Hom(W ; C A). We have to prove that D φ A = φ A D. φ A D(u l f 0 [f1 f 2 f k 1 f k ]) = φ A (u l d C (Hom(W ;A))(f 0 [f1 f 2 f k 1 f k ]) + u l+1 B(f 0 [f1 f 2 f k 1 f k ])). From the definition of the Hochschild differential, one has φ A (u l d 1 (f 0 [f 1 f k ])) = φ A [u l df 0 [f 1 f k ] k ( 1) εi u l f 0 [f 1 d(f i ) f k ]] i=1 = φ A (u l d(f 0 )[f 1 f k ]) k ( 1) εi φ A (u l f 0 [f 1 d(f i ) f k ]) i=1 = (Id Id s k ) (g(u l ) df 0 f 1 f k ) Ψ (k+2) W k ( 1) εi (Id Id s k )(g(u l ) f 0 f 1 df i i=1 f k ) Ψ (k+2) W = (Id Id s k )(g(u l ) (d A f 0 ( 1) f0 f 0 δ W ) f 1 f k ) Ψ (k+2) W k ( 1) εi (Id Id s k )(g(u l ) f 0 f 1 (d A f i ( 1) fi f i δ W ) i=1 f k ) Ψ (k+2) W = (Id Id s k )(Id d A Id Id Id d A k) (g(u l ) f 0 f 1 f k ) Ψ (k+2) W (Id δ W k+1) Ψ k+1 W ( 1) (P k 0 f i )+k (Id Id s k ) (g(u l ) f 0 f 1 f k ) = (Id d 1 ) (Id Id s k )(g(u l ) f 0 f 1 f 2 f k 1 f k ) Ψ (k+2) W ( 1) ε k (Id Id s k )(g(u l ) f 0 f 1 f k ) (Id δ W ) ϕ k+1 W = (Id d 1 ) φ A (u l f 0 [f 1 f k ]) ( 1) ε k φ A (u l f 0 [f 1 f k ]) δ.

14 328 CALVIN TCHEKA This result follows from the fact that ((W, δ W ), ψ W ) is a differential graded coalgebra: φ A (u l d 2 (f 0 [f 1 f 2 f k 1 f k ])) = φ A (( 1) f0 u l (f 0 f 1 )[f 2 f 3 f k 1 f k ]+ k ( 1) ε i u l f 0 [f 1 f 2 f i 1 f i f k 1 f k ] i=2 ( 1) ( f k +1)ε k u l (f k f 0 )[f 1 f 2 f k 2 f k 1 ]) = φ A (( 1) f0 u l m A (f 0 f 1 ) Ψ W [f 2 f 3 f k 1 f k ]+ k ( 1) εi u l f 0 [f 1 f 2 m A (f i 1 f i ) Ψ W f k 1 f k ] i=2 ( 1) ( f k +1)ε k u l m A (f k f 0 ) ψ W [f 1 f 2 f k 2 f k 1 ]) = ( 1) f 0 (Id Id s (k 1) ) [g(u l ) m A (f 0 f 1 ) ψ W f 2 f 3 f k 1 f k ] Ψ (k) W + k ( 1) εi (Id Id s (k 1) ) i=2 [g(u l ) f 0 f 1 m A (f i 1 f i ) Ψ W f k ] ϕ k W ( 1) ( f k +1)ε k (Id Id s (k 1) ) [g(u l ) m A (f k f 0 ) Ψ W f 1 f 2 f k 2 f k 1 ] = (Id Id s (k 1) ) [(Id m A Id)+ k ( 1) i (Id Id m A Id) + (Id (m A Id) σ k )] 2 (Id Id s (k 1) ) 1 φ A (u l f 0 [f 1 f 2 f k 1 f k ]) = (Id d 2 C(A) ) φ A(u l f 0 [f 1 f 2 f k 1 f k ]). We have the result from the definition of the cup-product on Hom(W, A) and the fact that ((W, δ W ), ψ W ) is a differential graded coalgebra. Let us verify that φ A (u l+1 B(f 0 [f1 f 2 f k 1 f k ])) = (Id B) φ A (u l+1 f 0 [f1 f 2 f k 1 f k ]). Since k B(f 0 [f1 f 2 f k 1 f k ]) = ( 1) ε i 1[f i f i+1 f k f 1 f i 2 f i 1 ] with j=0 i=0 i 1 k i+1 ε i = ( f j + 1)( f j + k i + 1), j=i

15 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 329 then φ A (u l+1 B(f 0 [f1 f 2 f k 1 f k ])) k = φ A (u l+1 ( 1) ε i 1[f i f i+1 f k f 1 f i 2 f i 1 ]) = φ A ( = = = i=0 k ( 1) εi u l+1 1[f i f i+1 f k f 0 f i 2 f i 1 ]) i=0 k ( 1) εi φ A (u l+1 1[f i f k f 0 f i 1 ]) i=0 k ( 1) εi (Id Id s (k+1) ) i=0 [g(u l+1 ) 1 f i f i+1 f k f 0 f i 1 ] Ψ (k+2) W k ( 1) εi g(u l+1 ) (Id s (k+1) ) i=0 [1 f i f i+1 f k f 0 f i 1 ] Ψ (k+1) W k = (g(u l+1 ) ( 1) εi (Id s (k+1) ) i=0 [1 f i f i+1 f i+2 f k f 0 f i 2 f i 1 ]) Ψ (k+2) W = g(u l+1 ) B((Id s k ) [f 0 f 1 f 2 f k ]) ψ (k+2) W = (Id B) (g(u l+1 ) (Id s k ) [f 0 f 1 f k ]) Ψ (k+1) W = (Id B) φ A (u l+1 f 0 [f 1 f k 1 f k ]). From the above calculations, we conclude that φ A is a chain complex homomorphism such that the diagram above commutes. Lemma 3.3. If (A, d A, µ A, κ A ) is a π-shc-algebra, then (i) κ A is a strict homomorphism of shc-algebras; and (ii) H (φ A ): HC Hom(W, A) H (Hom(W, C A)) preserves the natural products. Proof. (i) was proved in [21, A.5].

16 330 CALVIN TCHEKA In order to establish (ii), consider the following commutative diagrams A and B: [C Hom(W, A)] 2 φ 2 A [Hom(W ; C A)] 2 ψ C A Hom(W, (C A) 2 ) Sh (A) Hom(W,Sh) C ((Hom(W, A)) 2 ) C (ψ A) C Hom(W, A 2 ) φ A 2 Hom(W ; C (A 2 )) C (α Hom(W,A) 2 ) C (α Hom(W,A 2 ) ) C (ΩB(Hom(W, A)) 2 ) C (ΩBψA) C (ΩB(Hom(W, A 2 ))) and C (Hom(W, A 2 )) φ A 2 Hom(W, C (A 2 )) C (Hom(W,α A 2 )) (B) Hom(W,C α A 2 ) C (Hom(W, ΩB(A 2 ))) φ ΩB(A 2 ) Hom(W, C (ΩB(A 2 ))). From F 3, F 8 and [21, Lemma A.4(b)], we deduce that there exist homomorphisms of differential graded algebras 1. ΩB(Hom(W, A)) θ A Hom(W, ΩBA) 2. Hom(W, A 2 )) θ A 2 Hom(W, ΩBA 2 ) 3. ΩB(Hom(W, A 2 )) θ A 2 ΩB Hom(W, ΩBA 2 ) such that Hom(W, α A ) θ A = α Hom(W,A), Hom(W, α A 2) α Hom(W,ΩB(A 2 )) θ A 2 = α Hom(W,A 2 ) and Hom(W, α A 2) θ A 2 = α Hom(W,A 2 ), since µ Hom(W,A) = ΩB Hom(W, α µ A ) θ Hom(W,A 2 ) ΩBψ A, where θ Hom(W,A 2 ) denotes the unique homomorphism of differential graded algebras such that Hom(W, α A 2) α Hom(W,ΩB(A 2 )) θ Hom(W,A 2 ) = α Hom(W,A A).

17 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 331 Then we obtain the following commutative diagrams C and D: C (ΩB((Hom(W, A)) 2 )) C ΩB(ψA) C C ΩB(Hom(W, A 2 θ A )) 2 C ΩB(Hom(W, ΩB(A 2 ))) C (µ Hom(W,A) ) (C) C α Hom(W,ΩB(A)) C (ΩB Hom(W,µ A)) C (ΩB Hom(W, A)) C θ A C (Hom(W, ΩB(A))) and C C (ΩB(Hom(W, A 2 (θ A ))) 2 ) φ C ΩB(A (Hom(W, ΩB(A 2 2 ) ))) Hom(W, C (ΩB(A 2 ))) C (θ A 2 ) C (α Hom(W,ΩB(A 2 )) ) C ΩB(Hom(W, ΩB(A 2 ))) (D) Hom(W,C µ A) C α Hom(W,ΩB(A)) C ΩB Hom(W,µ A) C (Hom(W, ΩBA)) φ ΩBA Hom(W, C ΩB(A)) By choosing linear sections of C α (Hom(W,A 2 )) (resp. Hom(W, C α A 2 )), one can define the product m C (Hom(W,A)) on C (Hom(W, A)) as in 2.3 (resp. m Hom(W,C A) on Hom(W, C A)). Thus by gluing together diagrams A, B, C and D, we have the following commutative diagram: [C (Hom(W, A))] 2 φ 2 A [Hom(W, C A)] 2 m C (Hom(W,A)) m Hom(W,C A) C Hom(W, A) φ A Hom(W, C A) which commutes up to linear homotopy. This proves that H φ A preserves the natural products End of the proof of the first part of Theorem 1.1 Let ((A, d A ), µ, κ A ) be a π-shc algebra. Since for any p 2, ΩB(A p ) is a shc algebra with α A µ (p) A i A p = m(p) A (see 2.1.7), hence we deduce from the definition of the π-shc algebra that the following diagram commutes up to homotopy in the

18 332 CALVIN TCHEKA category π-dm: ΩB(A p ) κ A Hom(W ; A) α A p (1) ev 0 A p m (p) A A. Let S p be the symmetric group on 1, 2,..., p} and consider the action of S p on C (A p ) (resp. on C (ΩB(A p ))) defined by the following rule: σ(u l b 0 [b 1 b 2 b s 1 b s ]) = u l σb 0 [σb 1 σb 2 σb s 1 σb s ], b i A p (resp. b i ΩB(A p )) so that C A p, C ΩB(A p ) are π-chain complexes and ΩB(A p ) α A p A p, C (ΩB(A p )) C (α A p ) C (A p ) are quasi-isomorphisms of π-chain complexes. Sh (p) Consider on the other hand the π-chain complex homomorphism [C A] p C (A p ), called a p-iterated cyclic shuffle map and defined by induction as follows: Sh = Sh (mk[u] Id) (id T id), Sh (2) = Sh ( Sh id) and for all p 2, Sh (p) = Sh (p) ( Sh (p 1) id) (where Sh = (id sh) + u(id sh ) denotes the cyclic shuffle map). Indeed we have for any x = x n1 x n2 x np 1 x np [C A] p, (x ni = u l n i a n i 0 [ani 1 ani 2 ani s i 1 an i s i ] C A) p i=1 Sh (p) (x) = u ln 1 + +ln p (a n 1 0 an p 0 ) ( 1) ɛ σ [a n an 1 s a n σ 1 a n 2 s a n p an p s p ] + u ln 1 + +ln p +p σ ( 1) ɛ σ [a n an p an a n1 s a n a n2 s a np s p ] with σ a (n 1, n 2,..., n p )-shuffle and σ a (n 1, n 2,..., n p )-cyclic shuffle. Since for every σ S p, x (C (A)) p, σx = x σ(n1 ) x σ(n2 ) x σ(n p), the sets of (n 1, n 2,..., n p )-shuffles and (n 1, n 2, n p )-cyclic shuffles coincide respectively with the set of (σ(n 1 ), σ(n 2 ),..., σ( n p))-shuffles and the set of (σ(n 1 ), σ(n 2 ),..., σ( n p))-cyclic shuffles, then the p-iterated cyclic shuffle map Sh (p) is π linear and we obtain the following sequence of π-quasi-isomorphisms: C (ΩB(A p )) C (α A p ) C (A p ) Sh (p) [C A] p.

19 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 333 Let φ A be the map defined in the proof of Proposition 3.2. By applying the functor C ( ) to the diagram (1) above and using Lemma 2.2 and the definition of the product m (p) on C A C A, we obtain the following the diagram commutative up to homotopy; C (ΩB(A p )) φ A C ( κ A) Hom(W ; C (A)) C (α A p ) C (A p ) C α A C µ(p) A ev 0 Sh (p) [C (A)] p m (p) C A C (A) which induces the following commutative diagram in homology: HC (ΩB(A p )) H (Hom(W ; C H (φ A C κ (A))) A) HC (S A p ) HC ( Sh p ) H (ev 0 ) [HC (A)] p (H m) (p) HC (A). Thus C (A) together with the structural map θ = φ A C ( κ A ) is a Dold quasialgebra. From the lines of the proof of May[15, Proposition 2.3], this diagram defined algebraic Steenrod operations on HC (A). Following the same lines, Proposition 2.1 can still be used to prove the Cartan formulas [21]. 4. Proof of the second part of Theorem 1.1 Let us begin this section by giving the proof of the following result due to Bitjong Ndombol and Jean-Claude Thomas [21, Theorem B]. Proposition 4.1. Let X be a topological space. The algebra N (X) is a natural π- shc-algebra. Proof. Here we replace the ground field F p by F p [π]. Munkholm established in [17, 4 7] that there exists a natural transformation µ X : ΩB(N (X) N (X)) ΩB(N (X) such that α N (X) µ X i N (X) N (X) = m N (X), where m N (X) denotes the usual product on N (X). Furthermore (N (X), µ X ) is a shc-algebra.

20 334 CALVIN TCHEKA Let denote the topological diagonal. From [21, A.2], F 3, and the fact that the Eilenberg-Zilber map EZ is a trivialized extension, we obtain the following commutative diagram: ΩB((N X) 2 ) θ N (X) ΩB(N (X 2 )) ΩB(N( )) ΩB(N X) α (N X) 2 α N X 2 α N X N ( ) (N X) 2 EZ N (X 2 ) N X, from which we define the following homomorphism of differential graded algebras µ X = ΩB(N( )) θ N (X). Moreover there exists a homomorphism of differential graded algebras ΩB(N X) θ N X Hom(W, N X) deduced from [21, Lemma A.4(a)] and F 3, which rise to the commutative diagram ΩB((N X) 2 κ X ) Hom(W, N X) µ X ev 0 θ N X ΩB(N α N ( ) X) N X. Thus we obtain a F p [π]-homomorphism of differential graded algebras κ X defined by κ X = θ N X µ X such that ev 0 κ X = α N ( ) µ X. To end this proof it is enough to establish that µ X DA µ X. We remark that µ X DA µ X if and only if t X T t X where t X, t X T (B(N (X)) 2, ΩB(N (X))) denote the universal twisting cochains associated respectively to µ X and µ X [17]. Finally, it is necessary to find H Hom 1 (ΩB((N X) 2 ), ΩBN X) such that H : t X T t X. i.e., DH = t X H H t X and H η BN X p = η BN X; ε ΩBN X H = ε BN X. We define H by the following induction formula: H = h( t X H H t X ) + η ΩBN X ε BN X p H i 0 = H η BN X p = η ΩBN X, where h: 1 ΩBN X i N X α N X and i 0 = η B(N X) p. Since (I) t X H = m ΩB(N X) ( t X H) B(N X) p and H t X = m ΩB(N X) (H t X ) B(N X) p, then H = h m ΩB(N X) ( t X H H t X ) B(N X) p + η ΩB(N X) ε ΩB(N X) p.

21 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 335 and so Notice that ε BN X p η B(N X) p = Id K; B(N X) p η B(N X) p = η B(N X) p η B(N X) p t X η B(N X) p = t X η B(N X) p = 0, H i 0 = H η B(N X) p = h m ΩB(N X) ( t X η B(N X) p Hη B(N X) p Hη B(N X) p t X η B(N X) p) + η ΩB(N X) = η ΩB(N X). More generally, suppose that H i j can be written as (I) for j k and let us prove that H i k+1 can be written by formula (I) and H i j (j k < k ). where H i k = h m ΩB(N X)( t X H H t X ) B(N X) p i k + η ΩB(N X) ε ΩB(N X) p i k = h m ΩB(N X) [ t X H H t X ][η B(N X) p i k + i ν i k ν + i k η B(N X) p] + η ΩB(N X) ε ΩB(N X) p i k k ν=1 (s(ia)) ν i ν BA (s(a 1 ) s(a 2 ) s(a ν )) i ν (s(a 1 ) s(a 2 ) s(a ν )) = [a 1 a ν ] (A = (N X) p ). In particular i k (s(a)) = [a] and 0 if k > 0 ε N X([a 1 a k ]) = 1 if k = 0 and ε N X p i k(sa 1 sa 2 sa k ) = B(N X) i k = η B(N X) p i k + k Thus we deduce that 0 if k > 0 1 if k = 0 ν=1 i ν i k ν + i k η B(N X) p. H i k = h m ΩB(N X)( t X H H t X )(i 0 i k + k 1 ν=1 i ν i k ν + i k i 0 ) k k = h m ΩB(N X)[ ( t X i ν ) (H i k ν) H i ν t X i k ν] ν=0 k 1 = h m ΩB(N X)[ ( t X i ν ) (H i k ν) (H i ν ) (t X i k ν)]. k ν=1 Let us verify that DH = t X H H t X. ν=0 ν=0

22 336 CALVIN TCHEKA Since DH = D[h( t X H H t X )] + D(η ΩB(N X) ε ΩB(N X) p with D(η ΩB(N X) ε ΩB(N X) p) = d ΩB(N X)η ΩB(N X) ε ΩB(N }} X) p 0 η ΩB(N X) ε ΩB(N X) pd ΩB(N X) } p, } 0 = 0 we also have Dh = 1 ΩB(N X) i N X α N X and Df = df ( 1) f fd. Thus DH = (Dh)[ t X H H t X ] + h(d[ t X H H t X ]) = (1 ΩB(N X) i N X α N X)[ t X H H t X ] + h(d[ t X H H t X ]). Since α N X h = 0 and D[ t X H H t X ] = 0, we deduce that DH = t X H H t X, hence t X T t X End of the proof of the second part of the theorem Consider the simplicial model K of the unit cycle S 1 and the cosimplicial model space X defined by X = Map(K(n), X) whose geometric resolution X is homeomorphic to LX, [20, Part II-3]. In [10, Lemma 5.5, Proof of Theorem A and Theorem B], J.D.S. Jones has constructed the F p [u]-modules quasi-isomorphism C N X Ψ F p [u] N ( X ) which induces a graded algebra isomorphism HC X = H S (LX, F 1 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 ˆθ 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 W (F p [u] N p id (id T id) p ( X )) W (F p [u]) p (N ( X )) id m(p) Fp[u] p id W (F p [u]) (N ( X )) p T id (F p [u]) W (N ( X )) p id Γ X F p [u] N ( X ); γ X is the structural map defining Steenrod operations on (F p [u] N ( X )) and inducing the chain map γ X (see Proposition 2.1). Consider the following diagram: C (ΩB(N X) p ) C (α (N X) p ) C ((N X) p ) Sh p (C ((N X))) p φ N X (C κ X ) (1) T X (2) Ψ p X Hom(W ; C (N X)) Hom(W ;Ψ X ) Hom(W ; K[u] N ( X )) γ X (K[u] N ( X )) p

23 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 337 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 = p0 ( 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 ) T X Hom(W ; K[u] N ( X )) such that T C (α (N X) p) π Hom(W ; Ψ X ) φ N X (C κ X ). Consequently the Jones isomorphism respects Steenrod operations. 5. π-shc models, [21, 3] 5.1. Minimal algebra Let V = V i } i1 be a graded vector space and let (T V, d V ) denotes the free differential graded algebra generated by V : T r V = V V V (r times) and v 1 v 2 v k (T V ) n if k i=1 v i = n. The differential d V on T V is the unique degree 1 derivation on T V defined by a given linear map V T V and such that d V d V = 0. The differential d V : T V T V decomposes as d V = d 0 + d 1 + with d k V T k+1 V. If we assume that V 1 = 0 and d 0 = 0 then (T V, d V ) is called a 1- connected minimal algebra. For any differential graded algebra (T U, d U ) such that H 0 (T U, d U ) = F p and H 1 (T U, d U ) = 0, there exists a sequence of homomorphisms of differential graded algebras, (T U, d U ) P V (T V, d V ) ϕ V (T U, d U ) where (T V, d V ) denotes a 1-connected minimal algebra, ϕ V P V DA id and P V ϕ V = id such that V = H(U, d U ) (see [20, 3.1] or [21, 6.4]). Moreover (T V, d V ) is unique up to isomorphism Minimal model of a product Assume that (A, d A ) is a differential graded algebra such that H 0 (A) = F p and H 1 (A) = 0 and let (T U[n], d U[n] ) = Ω((BA) n ), n 1. Following the discussion above, we obtain a sequence with (T U[n], d U[n] ) = Ω((BA) n ) P V [n] (T V [n], d V [n] ) ϕ V [n] (T U[n], d U[n] ) V [n] = s 1 (H((BA) n )) = s 1 ((H(BA)) n ) = s 1 ((F p sv ) n ) n = ( ((F p ) k 1 V (F p ) n k )) s 1 (sv sv sv ). k=1 For n = 1, V [1] = V = s 1 H(BA) and the composite ψ V = α A ϕ V : (T V, d V ) A is a quasi-isomorphism. The algebra (T V, d V ) is called a 1-connected minimal model of A. For n 2, consider the homomorphism q bv : (T V [n], d V [n] ) (T V, d V ) n defined by q bv (y) = 1 k 1 y 1 n k, if y V k := F k 1 p V F n k p, k 1; 2;... n}

24 338 CALVIN TCHEKA and q bv (y) = 0 if y V [n] n i=1 V i. The composite (T V [n], d V [n] ) q V b (T V, dv ) n (ψ V ) n A n is a quasi-isomorphism ([20]). Therefore (T V [n], d V [n] ) is a minimal model of A n π-shc minimal models For any n > 1, the cyclic group S n acts on V = V [n] s 1 (H(BA)) n. This action extends diagonally on T V [n] so that d V [n] and the homomorphism (ψ V ) n q V [n] are S n -linear. Since α A n is a S n -equivariant quasi-isomorphism, we deduce from [21, Lemma 3.3] that the composite (ψ V ) n q V [n] lifts to a homomorphism of differential graded algebra L: (T V [n]d V [n] ) ΩB(A n ) which is S n -equivariant and α A n L = (ψ V ) n q V [n]. Let ((A, d A ), µ A ) be an augmented shc-algebra and assume that H 0 (A) = F p and H 0 (A) = 0. Define the composite µ (n) V = P V µ (n) A L: (T V [n], d V [n]) (T V, d V ) such that µ (2) V = µ V : (T V [2], d V [2] ) (T V, d V ). The triple ((T V, d V ), µ V ) is called a shc-minimal model for ((A, d A ), µ A ) [20, Section 6]. Let ((A, d A ), µ A, κ A ) be an augmented π-shc-algebra and assume that H 0 (A) = F p and H 0 (A) = 0. Following [21, Lemma 3.3], the composite κ A L lifts to S p -equivariant homomorphism of algebras ˆκ A : (T V [p], d V [p] ) Hom(W, ΩBA). Hence the composite κ V = Hom(W, P V ) ˆκ A : (T V [p], d V [p] ) Hom(W, T V ) is a S p -equivariant homomorphism of algebras and the triple ((T V, d V ), µ V, κ V ) is called the π-shcminimal model for the ((A, d A ), µ A, κ A ) [21, 3.4]. 6. Examples In this section, the characteristic of the field F p is p = Projective space CP Let X = CP, H (X; K) = Λ(x) = F 2 [x] with x = 2. Thus the graded algebras HC (N CP ; F 2 ) and HC (Λ(x)) are isomorphic. It also follows that the algebraic Steenrod operations on HC (N CP ; F 2 ) are computed by those on HC (Λ(x)). Since ((Λ(x), 0), µ Λ(x), κ Λ(x) ) is a 1-connected π-shc differential graded algebra together with finite generated cohomology groups such that the shc structural map µ Λ(x) = ΩB((m Λ(x) )) and the π-shc structural map ΩB(Λ(x) 2 ) κ Λ(x) Hom(W ; Λ(x)) defined on the generic elements as follows: we have b i j = 2(m j + p j ). For any y = < c 1 c 2 c r 1 c r >, c i = [b i 1 b i 2 b i l i 1 b i l i ], b i j = (x x x) (x x x x) }}}} m j p j,

25 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 339 Thus y = 2α + q, 2α + q, and finally: α = r li i=1 j (m j + p j ); κ Λ(x) (y)(e k τ) = x α β if 2β = k q 0 if not. q = r r i=1 l i and κ Λ(x) (y) = In particular, for any y <[x 1]>, <[1 x]>}, we obtain: κ Λ(x) (y)(e k τ) = x if k = 0 0 if k > 0, and for y = <[x x]> x 2 if k = 0 κ Λ(x) (y)(e k τ) = x if k = 2 0 if k / 0, 2}. Then ((Λ(x), 0), µ Λ(x), κ Λ(x) ) has a 1-connected π-shc minimal model ((T V, d V ), µ V, κ V ) where V = xf 2, κ V : T V Hom(W, T V ) is the map such that V = V [2] := x F 2 x F 2 x F 2 #x F 2 and κ ˆV (x )(e i ) = κ ˆV (x )(e i τ) = κ ˆV (x )(e i ) = κ ˆV (x x if i = 0 )(e i τ) = 0 if i > 0 κ ˆV (x x )(e i ) = κ ˆV (x x x if i = 1 )(e i τ) = 0 if not. Let T ˆV q ˆV (T (x)) 2 be the surjective quasi-isomorphism of differential graded algebras defined on the generic elements as follows: q ˆV (x ) = x 1; q ˆV (x ) = 1 x; and q ˆV (x x ) = 0, and inducing the chain complex quasi-isomorphism C T ˆV C q ˆV C (T (x)) 2. We have the following diagram Hom(W, A) W π A 2 Hom(W, A) Hom(W,ϕ 2) Hom(W, C T (x)) 1 W Ψ 2 1 W π [C T (x)] 2 Hom(W,ϕ 2) Hom(W, C T (x) θ W C T V S 1 W Sh W π C ((T (x)) 2 ) 1 W C q bv θ W C T V

26 340 CALVIN TCHEKA where A = F p[u] [F p p 1 i=1 F p<z r >] Λ(y) < u z r, r p 1 > = H S 1(LCP( ), F p) = HC (N (CP( ), F p )) = HC (Λ(x)) = HC T (x), with u = 2, y = 2p, z r = 2r + 1 and F p <z r > the graded vector space generated by z r (see [19, Theorem 2]). Sh is the homomorphism of chain complexes defined in Section 1, S a linear section of 1 W C q ˆV ; the structural map θ is defined by θ = φ Λ(x) C κ ˆV and ϕ 2 is defined as follows: The linear differential map factors through K[u] [K ( p 1 i=0 K<z ϕ r>)] Λ(x) K[u] Λ(x) Λ(sx) A K[u] Λ(x) Λ(sx) and K[u] Λ(x) Λ(sx) ψ A where ϕ is a differential graded algebras quasi-isomorphism defined in [19, 4.1] or [18, 4.6]. Consider the differential graded algebras quasi-isomorphisms K[u] Λ(x) Γ(sx) τ θ K[u] C(Λ(x)) = C Λ(x) (see [18, 4] or [19, 3.2]). From this we define the chain complex homomorphisms A ϕ 1 such that ϕ 1 = ϕ τ and ϕ 2 = ψ θ. More precisely we have: ϕ 2 C (Λ(x)) (i) ϕ 1 (u) = τ ϕ(u) = τ(u 1) = u 1[ ] ϕ 1 (y) = τ ϕ(1 y) = τ(1 x p ) = 1 x p [ ] r 0,..., p 1}, ϕ 1 (z r ) = τ ϕ(z r ) = τ(x r sx) = 1 x r [x]. (ii) ϕ 2 (u l x n [x k ]) = ψ θ(u l x n [x k ]) = kψ(u l x k+n 1 sx)

27 STEENROD OPERATIONS ON CYCLIC HOMOLOGY OF SHC-COCHAIN ALGEBRAS 341 for q > 0 ϕ 2 (u l x n [x k1 x k2 x kq 1 x kq ]) = 0. ϕ 2 (u l x n [ ]) = ψ θ(u l x n u l y k if n = kp [ ]) = 0 if n kp Finally, observe that S, the linear section of 1 W C q ˆV, is only defined in low degrees as follows: 1. S(e i u l 1[ ]) = e i u l 1[ ] 2. S(e i u l (x k 1)[ ]) = e i u l x k [ ] 3. S(e i u l (1 x k )[ ]) = e i u l x k [ ] 4. S(e i u l (x k x k )[ ]) = e i u l (x k x k )[ ] 5. S(e i u l (x k 1)[x k 1]) = e i u l x k [x k ] 6. S(e i u l (1 x k )[1 x k ]) = e i u l x k [x k ] 7. S(e i u l 1[x k 1]) = e i u l 1[x k ] 8. S(e i u l 1[1 x k ]) = e i u l 1[x k ] 9. S(e i u l (x k 1)[1 x k ]) = e i u l x k [x k ] 10. S(e i u l (1 x k )[x k 1]) = e i u l x k [x k ] 11. S(e i u l 1[1 x k 1 x k ]) = e i u l 1[x k x k ] 12. S(e i u l 1[x k 1 x k 1]) = e i u l 1[x k x k ] 13. S(e i u l 1[1 x k x k 1]) = e i u l 1[x k x k ] 14. S(e i u l 1[x k 1 1 x k ]) = e i u l 1[x k x k ] 15. S(e i u l 1[x k x k ]) = e i u l 1[x k x k ] (k > 1) 16. S(e i u l 1[x 1 1 x]) = e i u l 1[x x ] + e i u l 1[x x ] (k = 1) 17. S(e i u l 1[x k 1 1 x k x k 3 1 x k 4 ]) = e i u l 1[x k 1 x k 2 x k 3 x k 4 ] 18. S(e i u l 1[1 x x x x 1]) = e i u l 1[x x x x ] 19. S(e i u l 1[x x x 1 1 x]) = e i u l 1[x x x x ] + e i u l 1[x x x x ] 20. S(e i u l 1[x x x 1 1 x]) = e i u l 1[x x x x ] + e i u l 1[x x x x ] 21. S(e i u l 1[x x x 1 1 x]) = e i u l (x 2 x )[x ] + e i u l (x x )(x x )[ ] 22. S(e i u l 1[x x x 1 1 x]) = e i u l (x x 2 )[x ] + e i u l (x x )(x x )[ ] Now we define algebraic Steenrod operations on H q (A) by the formula Sq i (x) = cl( θ(s (id W Sh)(e n i x x))), x H (A) 1. For x = u, (a) Sq 0 (u) = cl[ϕ 2 (u 1[ ])] = u = x. (b) Sq 1 (u) = cl[ϕ 2 (g(u 2 )(e 1 ) κ ˆV (1)(e 0 )[ ] + g(u 2 )(e 0 ) κ ˆV (1)(e 1 )[ ])] = 0. (c) Sq 2 (u) = cl[ϕ 2 (g(u 2 )(e 0 ) κ ˆV (1)(e 0 )[ ])] + cl[ϕ 2 (u 2 1[ ])] = x 2.

28 342 CALVIN TCHEKA (d) for i > 2, Sq i (u) = For x = y, (a) Sq 0 (y) = cl[ϕ 2 (g(1)(e 0 ) κ ˆV (x 2 x 2 )(e 4 )[ ])] = cl[ϕ 2 (1 x 2 [ ])] = 1 1 y. (b) Sq 1 (y) = cl[ϕ 2 (Id Id)(g(1) κ ˆV (x 2 x 2 )) ψ W (e 3 )] = 0. (c) Sq 2 (y) = cl[ϕ 2 (Id Id)(g(1) κ ˆV (x 2 x 2 )) ψ W (e 2 )] = cl[ϕ 2 (1 x 3 [ ])] = 0. (d) Sq 3 (y) = cl[ϕ 2 (Id Id)(g(1) κ ˆV (x 2 x 2 )) ψ W (e 1 )] = 0. (e) Sq 4 (y) = cl[ϕ 2 (Id Id)(g(1) κ ˆV (x 2 x 2 )) ψ W (e 0 )] = cl[ϕ 2 (1 x 4 [ ])] = y 2. (f) for i > 4 = 2p, Sq i (y) = For x = z r H 2r+1 (A) with 0 r p 1, since p = 2 then r 0, 1}. (a) x = z 0 i. Sq 0 (z 0 ) = cl[ϕ 2 ((Id Id s 2 )(g(1) κ ˆV (1) κ ˆV (x ) κ ˆV (1) κ ˆV (x ) κ ˆV (x )))ψ (3) W (e 1) + Id Id s 2 )(g(1) κ ˆV (1) κ ˆV (x x ))ψ (2) W (e 1)] = Sq 0 (z 0 )cl[ϕ 2 (1 1[x])] = z 0. ii. Sq 1 (z 0 ) = cl[ϕ 2 ((Id Id s 2 )(g(1) κ ˆV (1) κ ˆV (x ) κ ˆV (x ) + iii. for i > 1, Sq i (z 0 ) = 0. (b) x = z 1. i. Sq 0 (z 1 ) = z 1. g(1) κ ˆV (1) κ ˆV (x ) κ ˆV (x )))ψ (3) W (e 0) + (Id Id s 2 )(g(1) κ ˆV (1) κ ˆV (x x ))ψ (2) W (e 0)] = cl[ϕ 2 (2(1 1[x x]))] = z 2 0 = 0. ii. Sq 1 (z 1 ) = cl[ϕ 2 ((Id Id s 2 )(g(1) κ ˆV (x x ) κ ˆV (x ) κ ˆV (x ) + g(1) κ ˆV (x x ) κ ˆV (x ) κ ˆV (x )))ψ (3) W (e 2) + (Id Id s 2 )(g(1) κ ˆV (x x ) κ ˆV (x x ))ψ (2) W (e 2)] = cl[ϕ 2 (2(1 x[x x]))] = 0. iii. Sq 2 (z 1 ) = cl[ϕ 2 ((Id Id s 2 )(g(1) κ ˆV (x x ) κ ˆV (x ) κ ˆV (x ) + g(1) κ ˆV (x x ) κ ˆV (x ) κ ˆV (x )))ψ (3) W (e 1) + (Id Id s 2 )(g(1) κ ˆV (x x ) κ ˆV (x x ))ψ (2) W (e 1)] = cl[ϕ 2 (1 x 2 [x x])] = y z 0. iv. Sq 3 (z 1 ) = cl[ϕ 2 (2(1 x 2 [x x]))] = 0. v. for i > 3, Sq i (z 1 ) = 0.