On the Connes operator in Hochschild homology

by

Bitjong Ndombol

Abstract. Let 𝕂 be a field of characteristic p ≥ 0 and X a topological space. If N^{∗}X is the algebra of normalized singular cochains on X with coefficients in 𝕂, then a product is defined on the negative Hochschild complex ℭ_{∗}X := ℭ_{∗}N^{∗}X ([2]) which induces a natural commutative graded algebra structure on Hℭ_{∗}X := HH_{∗}X. We prove that if X is simply connected, the Connes operator B : ℭ_{∗}X → ℭ_{∗+1}X is an algebra derivation in homology.

AMS classification: 57T30; 54C35; 55S20

Keywords: Hochschild homology, Connes operator, free loop space.

Introduction.

Throughout this note, 𝕂 is a field of characteristic p ≥ 0. The classical definition of the normalized negative Hochschild complex ℭ_{∗}A of an algebra A naturally extends to a cochain algebra.

Let (A, d_{A}) be a cochain algebra over 𝕂. The negative Hochschild complex of the cochain algebra (A, d_{A}) with coefficients in (A, d_{A}) is ℭ_{∗}(A, d_{A}) = (A ⊗ BA, D) where B denotes the reduced bar construction functor and D is defined in section 1. By definition HH_{∗}A = H_{∗}ℭ_{∗}(A, d_{A}) is the negative Hochschild homology of (A, d_{A}) with coefficients in (A, d_{A}). If (A, d_{A}) = N^{∗}X the algebra of normalized cochains on the topological space X, then we note ℭ_{∗}X (resp. HH_{∗}X) instead of ℭ_{∗}N^{∗}X (resp. HH_{∗}N^{∗}X) and call it the negative normalized Hochschild complex (resp. the negative Hochschild homology) of X.

In [2], theorem 1, a natural product is defined on ℭ_{∗}X such that HH_{∗}X becomes a commutative graded algebra.

For any (cochain) algebra, we have (see [3] or [8]) the Connes operator B : ℭ_{∗}A → ℭ_{∗+1}A satisfying DB + BD = 0 and B ∘ B = 0. Thus, in particular, it defines a linear map of lower degree 1 in homology, say B_{∗} : HH_{∗}A → HH_{∗+1}A. In this note, we prove the following result.

Theorem. If X be a simply connected topological space, then 1- B_{∗} : HH_{∗}X → HH_{∗+1}X is an algebra derivation, 2- B_{∗} ∘ B_{∗} = 0. Note that property 2 is obvious and is true even if X is not simply connected.

We adopt the convention that negative lower degree is positive upper degree. The degree of an homogeneous element x is denoted |x|.

1- Hochschild homology.

1.1 Let DA and DC denote respectively the category of connected cochain algebras and the category of connected cochain coalgebras. That is, in particular, the differential is of upper degree one. The reduced bar and cobar constructions are a pair of adjoint functors B : DA ↔ DC : Ω, [4]. This adjunction yields, for a cochain algebra A, a natural homomorphism α_{A} : ΩBA → A of cochain algebras which induces an isomorphism in homology [4]. The elements 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 linear map ι_{A} : A → 𝕂 ⊕ Ā, ι_{A}(1) = 1 and ι_{A}(a) = ⟨[a]⟩, a ∈ Ā commutes with the differentials, but is not a morphism of cochain algebras. In any case, it satisfies α_{A}∘ι_{A} = id_{A} and 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.

1.2 Let (A, d_{A}) be a cochain algebra. We denote by d_{BA} the differential of the reduced bar construction BA. The tensor product (A, d_{A}) ⊗ (BA, d_{BA}) is then a differential module whose differential is denoted d_{A⊗BA}. The Hochshild differential, denoted by D, is defined by :

Da_{0}[a_{1}|....|a_{n}] = (d_{0} − d_{n})a_{0}[a_{1}|....|a_{n}] + d_{A⊗BA}a_{0}[a_{1}|....|a_{n}]

where d_{0}a_{0}[a_{1}|....|a_{n}] = (−1)^{|a_{0}|}a_{0}a_{1}[a_{2}|...|..|a_{n}] and

d_{n}a_{0}[a_{1}|....|a_{n}] = (−1)^{(|a_{n}|+1)(|a_{0}|+...+|a_{n−1}|+n−1)}a_{n}a_{0}[a_{1}|...|..|a_{n−1}].

By definition,

ℭ_{∗}A = (A ⊗ BA, D)

is the normalized negative Hochschild complex of (A, d_{A}) with coefficients in (A, d_{A}) and HH_{∗}A = Hℭ_{∗}A is the negative Hochschild homology of the cochain algebra (A, d_{A}) with coefficients in (A, d_{A}) . It is clear that ℭ_{∗}A is concentrated in non-negative total upper degrees. Hence, so is HH_{∗}A.

If (A, d_{A}) = N^{∗}X is the algebra of normalized singular cochains on the topological space X, then ℭ_{∗}N^{∗}X := ℭ_{∗}X is the normalized negative Hochschild complex of X and HH_{∗}N^{∗}X := HH_{∗}X is the negative Hochschild homology of X.

1.3 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}] = ∑_{i=0}^{n}(−1)^{ϵ_{i}}1[a_{i}|..|a_{n}|a_{0}|....|a_{i−1}] where ϵ_{i} = |a_{0}| + (|a_{0}| + |a_{1}| + .. + |a_{i−1}| + i)(|a_{i}| + .. + |a_{n}| + n − i + 1).

The Connes operator satisfies:

B^{2} = 0 and BD + DB = 0. So it induces a linear map of lower degree 1, B_{∗} : HH_{∗}A → HH_{∗+1}A.

2- Proof of the theorem.

2.1 Let X and Y be topological spaces. One has the Eilenberg-Zilber homomorphism of complexes EZ : C_{∗}X ⊗ C_{∗}Y → C_{∗}(X × Y) and the Alexander-Whitney homomorphism of complexes AW : C_{∗}(X × Y) → C_{∗}X ⊗ C_{∗}Y. The diagonal map △_{top} : X → X × X and the Alexander-Whitney homomorphism define a diagonal △ = AW ∘ C_{∗}△_{top} : C_{∗}X → C_{∗}S^{1} ⊗ C_{∗}X, and C_{∗}X is a graded coalgebra. Thus C_{∗}S^{1} is a graded coalgebra.

Recall that the set of continous maps from the unit circle S^{1} to X endowed with the compact-open topology is the free loop space of X and is denoted LX. There is an action of S^{1} on LX given by the rotation of the loops.

The action ν̄ : S^{1} × LX → LX yields an action ν = C_{∗}ν̄ ∘ EZ : C_{∗}S^{1} ⊗ C_{∗}LX → C_{∗}LX.

2.2 Let z ∈ C_{∗}S^{1} be a representative of the fundamental class of S^{1} and consider the map I : C^{n}LX → C^{n−1}LX, n ≥ 1 defined in [7], section 4, by

⟨I(x), σ⟩ = (−1)^{|x|}⟨x, ν(z ⊗ σ)⟩, σ ∈ C_{n−1}LX.

The map I satisfies I^{2} = 0 and dI = −Id where d is the differential of C^{∗}LX.

If △_{C_{∗}S^{1}} denotes the diagonal on C_{∗}S^{1}, for degree reasons, we have △z = z ⊗ 1 + 1 ⊗ z. That is, z is primitive. The following result is the cornerstone in the proof of the theorem stated.

2.3 Lemma The map I is a derivation when C^{∗}LX is endowed with the usual cup product.

Proof. The diagonal map on C_{∗}S^{1} ⊗ C_{∗}LX, say △_{C_{∗}S^{1}⊗C_{∗}LX} is given by

△_{C_{∗}S^{1}⊗C_{∗}LX} : C_{∗}S^{1} ⊗ C_{∗}LX ⟶^{△_{C∗S^{1}}⊗△_{C∗LX}} (C_{∗}S^{1})^{⊗2} ⊗ (C_{∗}LX)^{⊗2} ⟶^{id⊗T ⊗id} (C_{∗}S^{1} ⊗ C_{∗}LX)^{⊗2}

where T : A⊗B → B⊗A is the interchange isomorphism defined by T (a⊗b) = (−1)^{|a||b|}b⊗a, id is the identity map and M^{⊗2} = M ⊗ M.

Let j : C_{∗}LX → C_{∗}S^{1} ⊗ C_{∗}LX denote the map defined by j(σ) = z ⊗ σ. It is clear that I = (ν ∘ j)^{∨} = j^{∨} ∘ ν^{∨} where ^{∨} denotes the dual of a map between the dual vector spaces.

Recall that ν^{∨} = (EZ)^{∨} ∘ C^{∗}ν̄ is a homomorphism of graded algebras.

To finish the proof, it is now sufficient to prove that j^{∨} is a derivation of algebras or equivalently that j is a coderivation of coalgebras.

From the definition of △_{C_{∗}S^{1}⊗C_{∗}LX}, it is obvious that △_{C_{∗}S^{1}⊗C_{∗}LX} ∘ j = (j ⊗ Id) ∘ △_{C_{∗}LX} + (Id ⊗ j) ∘ △_{C_{∗}LX}.

2.4 In [7] (lemma 5.5 and proofs of theorems A and B), J. D.S. Jones has constructed a natural chain map Ψ : ℭ_{∗}X → C^{−∗}LX such that IΨ − ΨB is homotopic to 0 ( [7], theorem 4.1). Since dI = −Id, I induces I^{∗} : H^{∗}(LX; 𝕂) → H^{∗−1}(LX; 𝕂) and H_{∗}(Ψ) ∘ B_{∗} = I^{−∗} ∘ H^{−∗}(Ψ).

2.5 In [2] theorem 1 and Part II 1.1, it is proved that there exists a natural product on ℭ_{∗}X such that Ψ induces an isomorphism of graded algebras in homology when C^{∗}LX is endowed with the usual cup product.

Apply lemma 2.3 and 2.4 to end the proof of the theorem.

References

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[2] Bitjong Ndombol and J-C. Thomas , On the cohomology algebra of free loop spaces, Topology 41 (2002) 85-106.

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Bitjong Ndombol Université de Dschang Faculté des Sciences Dschang-Cameroun

e-mail: bitjongndombol@yahoo.fr

[AMA - Algebra Montpellier Announcements - 02-2005] [August 2005]
Received May 2005.