Periodic Cyclic Cohomology of Group Rings

Alejandro Adem^{(1)} and Max Karoubi

Department of Mathematics University of Wisconsin Madison WI 53706

Let G be a discrete group and R any commutative ring. According to [B] and [KV] (see also [L]), the cyclic cohomology HC^{∗}(R[G]) can be expressed as a direct product ∏_{⟨u⟩} H^{∗}(BG(u))^{(2)}. Here ⟨u⟩ runs through the set of all conjugacy classes of

elements u in G; if we denote by Z(u) the centralizer of u, ℤ is acting on Z(u) by the translation by u and G(u) is the topological group

G(u) = ℝ ×_{ℤ} Z(u).

On the other hand, it is well known that the Hochschild cohomology HH^{∗}(R[G]) can be expressed as the direct product ∏_{⟨u⟩} H^{∗}(BZ(u)).

Let us consider the double complex

C^{2}(G) ⟶^{1−t} C^{2}(G) ⟶^{N}
↑  ↑
C^{1}(G) ⟶^{1−t} C^{1}(G) ⟶^{N}
↑  ↑
C^{0}(G) ⟶^{1−t} C^{0}(G) ⟶^{N}

(1) Partially supported by an NSF grant.

(2) The cohomology groups are taken with coefficients in R (unless stated otherwise) and BH denotes as usual the classifying space of the topological group H. defining cyclic cohomology HC^{∗}(G) (see [L]). In this double complex, C^{n}(G) means the R–module of functions f : G^{n+1} → R. We define HC_{f}^{∗}(R[G]), the locally finite cyclic cohomology, as the cohomology obtained by restricting ourselves to functions f such that f(g_{0}, g_{1}, …, g_{n}) = 0 if g_{0}g_{1} … g_{n} = 1 outside a finite number of conjugacy classes. We can define HH_{f}^{∗}(R[G]) in an analogous way. It follows immediately from [B] and [KV] that HH_{f}^{∗}(R(G)) and HC_{f}^{∗}(R(G)) are respectively isomorphic to

⨁_{⟨u⟩} H^{∗}(BZ(u)) and ⨁_{⟨u⟩} H^{∗}(BG(u)).

The obvious exact sequence of groups

Z(u) → G(u) → 𝕊^{1}

gives rise to the sphere fibration

𝕊^{1} → BZ(u) → BG(u).  (F)

In the associated Gysin exact sequence

H^{n}(BG(u))⟶^{χ}H^{n+2}(BG(u)) → H^{n+2}(BZ(u)) → H^{n+1}(BG(u))

χ is the cup-product with the Euler class of the complex line bundle induced by (F). This sequence is a reinterpretation of the fundamental exact sequences of A. Connes:

\begin{matrix}HC^{n}(R[G])⟶^{s}HC^{n+2}(R[G]) → HH^{n+2}(R[G]) → HC^{n+1}(R[G]) \\ HC_{f}^{n}(R[G]) → HC_{f}^{n+2}(R[G]) → HH_{f}^{n+2}(R[G]) → HC_{f}^{n+1}(R[G]) \end{matrix}

The purpose of this paper is to compute the periodic locally finite cyclic cohomology HP_{f}^{n}(R[G]) = lim_{→} HC_{f}^{n+2r}(R[G]) in terms of algebraic invariants related to classical group cohomology. In order to achieve this goal, one has to analyze more closely the cohomology of BG(u). More precisely, we define HP^{n}(BG(u)) = lim_{→} H^{n+2r}(BG(u)). Due to the fact that direct sums commute with direct limits, we have that

HP_{f}^{n}(R[G]) ≅ lim_{→} ⨁_{<u>} H^{n+2r}(BG(u)) ≅ ⨁_{<u>} HP^{n}(BG(u)).

We compute the summands in this decomposition, obtaining complete results in some instances. This requires precise information about extension data from the cohomology of certain subgroups of G.

§1 THE HYPERBOLIC CASE

Let us assume first that u is of infinite order (u hyperbolic with the terminology of [B]). We reformulate this case in a manner similar to what was done in [B] and [KV]. Then the fibration BZ(u) → BG(u) is homotopically equivalent to the fibration BZ(u) → B\overline{Z}(u) with fiber BC(u) ≃ Bℤ ≃ 𝕊^{1}. Here C(u) denotes the cyclic group generated by u and \overline{Z}(u) = Z(u)/C(u). More precisely, one has the commutative diagram

ℤ  ⟶^{i}  C(u)
j ↓  ↓
Z(u) ⟶^{γ} ℝ × Z(u) ⟶^{pr_{2}}  Z(u)
↓  ↓  ↓
G(u) = ℝ ×_{ℤ} Z(u) ⟶ Z(u)/C(u) = \overline{Z}(u)
↓
𝕊^{1}

where i(1) = u, j(1) = (0, u). This shows that the homotopy fiber of BZ(u) → BG(u) may be identified with Bℤ ≃ BC(u) ≃ 𝕊^{1}. The cup product with θ ∈ H^{2}(B\overline{Z}(u)), image of the generator of H^{2}(B𝕊^{1}) by the induced map H^{2}(B𝕊^{1}) → H^{2}(B\overline{Z}(u)) defines an inductive system of cohomology groups

→ H^{n}(B\overline{Z}(u)) → H^{n+2}(B\overline{Z}(u)) → · · · .

Note that if R = ℤ, then θ ∈ H^{2}(B\overline{Z}(u), ℤ) is precisely the class associated to the extension for Z(u) in the diagram above.

If (n) ∈ ℤ/2, let H^{(n)}(B\overline{Z}(u))(θ^{−1}) be the inductive limit of this system. This limit corresponds to the part of locally finite periodic cyclic cohomology associated to a conjugacy class ⟨u⟩, with u hyperbolic.

We have therefore proved

Theorem 1.1:

Let u be a hyperbolic element, θ ∈ H^{2}(B\overline{Z}(u)) the class defined above. Then HP^{n}(BG(u)) ≅ H^{(n)}(B\overline{Z}(u))(θ^{−1}).

The following example illustrates how this works.

Example 1.2:

Let K denote the fundamental group of the Klein bottle. It can be described via the presentation < a, b | bab^{−1} = a^{−1} >. Note that b^{2} is a central element in K, and that hence we can express K as the extension

1 →< b^{2} >→ K → D_{∞} → 1

where D_{∞} ≅ ℤ/2 ∗ ℤ/2 is the infinite dihedral group. If x, y are generators for H^{2}(D_{∞}, ℤ) ≅ ℤ/2 ⊕ ℤ/2, such that xy = 0, then we may assume that x is the cohomology class associated with the Klein group. If we take u = b^{2} and K = Z(u), then we obtain

H^{∗}(BG(u); ℤ) ≅ H^{∗}(B\overline{Z}(u); ℤ) ≅ H^{∗}(BD_{∞}; ℤ) ≅ ℤ[x, y]/(2x, 2y, xy)

and

HP^{∗}(BG(u), ℤ) ≅ ℤ[x, y]/(2x, 2y, xy) (x^{−1}) ≅ ℤ/2[x, x^{−1}].

§2 THE ELLIPTIC CASE

Let us consider now the case where u is of finite order n (u elliptic with the terminology of [B]). In that case, we have the isomorphism

G(u) = ℝ ×_{ℤ} Z(u) ≈ 𝕊^{1} ×_{ℤ/n} Z(u)

where ℤ/n is embedded in 𝕊^{1} as roots of unity and embedded in Z(u) as C(u), the cyclic group generated by u. Consider the cohomology spectral sequence of the fibration

B𝕊^{1} → BG(u) → B\overline{Z}(u)

where, as in the hyperbolic case, we denote by \overline{Z}(u) the quotient group Z(u)/C(u). This spectral sequence degenerates and we have H^{∗}(BG(u)) ≃ H^{∗}(B𝕊^{1})⊗H^{∗}(B\overline{Z}(u)) if the restriction map H^{2}(BG(u)) → H^{2}(B𝕊^{1}) is onto. This map is included in the exact sequence

H^{2}(B\overline{Z}(u)) → H^{2}(BG(u)) → H^{2}(B𝕊^{1})⟶^{d_{3}}H^{3}(B\overline{Z}(u)) → H^{3}(BG(u))

where d_{3} is the first non-trivial differential of the spectral sequence.

Lemma 2.1:

Let R = ℤ and let θ be the generator of H^{2}(B𝕊^{1}; ℤ). Then d_{3}(θ) = β(e), where where e ∈ H^{2}(B\overline{Z}(u); C(u)) ≃ H^{2}(B\overline{Z}(u); ℤ/n) is the cohomology class associated to the central extension

1 → C(u) → Z(u) → \overline{Z}(u) → 1

and β : H^{2}(B\overline{Z}(u); ℤ/n) → H^{3}(B\overline{Z}(u); ℤ) is the Bockstein.

Proof. We have the commutative diagram of groups extensions

𝕊^{1} → G(u) → \overline{Z}(u)
↑  ↑  ∥
C(u) → Z(u) → \overline{Z}(u)

which induces a commutative diagram of topological spaces

B𝕊^{1} → BG(u) → B\overline{Z}(u)
↑  ↑  ∥
BC(u) → BZ(u) → B\overline{Z}(u)
∥  ↓  ↓
BC(u) → EBC(u) → B^{2}C(u)

(because BC(u) is a simplicial abelian group for instance). The class of the extension in H^{2}(B\overline{Z}(u); C(u)) can be thought of as the image of the tautological element in H^{2}(B^{2}C(u); C(u)) ≃ H^{1}(BC(u), C(u)) by the induced map

H^{2}(B^{2}C(u); C(u)) → H^{2}(B\overline{Z}(u); C(u)).

On the other hand, the map d_{3} in the integral cohomology spectral sequence associated to the fibration

BC(u) → EBC(u) → B^{2}C(u)

is an isomorphism H^{2}(BC(u)) ≃ H^{3}(B^{2}C(u)) and we have the commutative diagram

H^{2}(BC(u)) → H^{3}(B\overline{Z}(u))
∥  ↑
H^{2}(BC(u)) →^{≃} H^{3}(B^{2}C(u))

Since the generator of H^{2}(BC(u); ℤ) ≃ ℤ/n is the Bockstein of the canonical generator of H^{1}(BC(u); C(u)) ≃ Z/n, the lemma follows.

We will be able to finish our computation of H^{∗}(BG(u)) under the hypothesis that β(e) = 0. This hypothesis is fulfilled for instance if Z(u) is abelian (cf [KV]). This is due to the fact that G(u) ≃ 𝕊^{1} × \overline{Z}(u), the isomorphism being given by a group map

𝕊^{1} ×_{C(u)} Z(u) → 𝕊^{1} ×_{C(u)} \overline{Z}(u)

defined by (z, s) ↦ (zϵ(s)^{−1}, \overline{s}) where s ∈ Z(u), \overline{s} its class in \overline{Z}(u) and ϵ : Z(u) → 𝕊^{1} any extension of the inclusion C(u) → 𝕊^{1} sending u to e^{\frac{2iπ}{n}}. However, our hypothesis is more general than the one formulated in [KV]: one can choose G to be a central extension

1 → ℤ/2 → G → Q_{8} → 1

corresponding to a non-trivial element in H^{2}(Q_{8}, ℤ/2) ≅ ℤ/2⊕ℤ/2 (see [AM], page 132). Clearly the canonical inclusion C(u) → 𝕊^{1} cannot be extended to Z(u) but we have however that H^{3}(B\overline{Z}(u); ℤ) = 0 in this case (see [AM], page 170).

Theorem 2.2:

Let R be any ring of coefficients and assume that the image of the element β(e) ∈ H^{3}(B\overline{Z}(u); ℤ) in H^{3}(B\overline{Z}(u); R) is zero. Then

H^{∗}(BG(u)) ≃ H^{∗}(B\overline{Z}(u)) ⊗_{R} H^{∗}(BS^{1}).

Proof. We have to choose an element ṽ ∈ H^{2}(BG; ℤ) which maps onto the generator θ of H^{2}(B𝕊^{1}; ℤ) ≅ ℤ. Let v ∈ H^{2}(BG; R) be its image in the cohomology with coefficients in R. Because the fiber is totally homologous to zero, the obvious map H^{∗}(B\overline{Z}(u)) ⊗_{R} H^{∗}(B𝕊^{1}) → H^{∗}(BG(u)) is an isomorphism.

Corollary 2.3: If n = 0 in R, then we have in general that HP^{∗}(BG(u)) ≅ H^{∗}(B\overline{Z}(u))[θ, θ^{−1}].

Theorem 2.4:

With the notation and hypothesis above, let us write

f^{∗}(θ) = nθ + w = σ

where f : G(u) → 𝕊^{1} is the canonical map,

f^{∗} : H^{∗}(BS^{1}) → H^{∗}(BG(u)) ≃ H^{∗}(B𝕊^{1}) ⊗ H^{∗}(B\overline{Z}(u))

and w ∈ H^{2}(B\overline{Z}(u)). Then, the part of the periodic cyclic cohomology corresponding to the elliptic element u is isomorphic to the localization of H^{∗}(BG(u)) with σ inverted.

It is not hard to show that if R = ℤ, then the mod n reduction of w in H^{2}(B\overline{Z}(u); ℤ/n) is the class of the extension 1 → C(u) → Z(u) → \overline{Z}(u) → 1 (which of course by construction is in the image of the reduction map).

Next we provide an interesting example where β(e) ≠ 0.

Example 2.5:

Let Ã_{5} denote the double cover of the alternating group A_{5}. Let u be the central element of order 2. As A_{5} is simple, we have H^{2}(A_{5}; ℤ) = 0. On the other hand H^{2}(A_{5}; ℤ/2) ≅ ℤ/2 (see [AM], page 96), hence the Bockstein is a monomorphism on this group, and so β(e) ≠ 0. Using the fact that H^{∗}(Ã_{5}; ℤ) ≅ ℤ[u_{4}]/120u_{4} (see [AM], page 280), we see that the spectral sequence for the fibration

BÃ_{5} → BG(u) → B𝕊^{1}

collapses at the E_{2} stage. There is an extension problem however. Using coefficents in ℤ/2 instead, we still have this collapse but in this case we can write down the ring structure:

H^{∗}(BG(u); ℤ/2) ≅ ℤ/2[v_{4}] ⊗ E(z_{3})[θ]

where as before θ is the image of the generator in H^{∗}(B𝕊^{1}). This is very different from the cohomology of B𝕊^{1} × BA_{5}. Finally, we obtain the periodic version,

HP^{∗}(BG(u); ℤ/2) ≅ ℤ/2[v_{4}] ⊗ E(z_{3})[θ, θ^{−1}].

Remark 2.6:

As a matter of fact, we can prove more precise results if R = ℤ, since we have the following sequence of homotopy fibrations:

B𝕊^{1} → BG(u) → B\overline{Z}(u) →^{θ} B^{2}𝕊^{1} = K(ℤ, 3).

Here the map θ is defined by the Bockstein of the cohomology class of the extension 1 → C(u) → Z(u) → \overline{Z}(u) → 1. Therefore, if the homotopy class of θ is zero, BG(u) has the homotopy type of

B\overline{Z}(u) × Ω(B^{2}𝕊^{1}) ≃ B\overline{Z}(u) × B𝕊^{1}.

Now let {A_{α}} be a family of abelian groups; we define A = ∏′(A_{α}[t, t^{−1}]) as the direct limit of the numerable system (B_{n}), where B_{n} = ∏_{α} A_{α}[t] and the canonical map B_{n} → B_{n+1} is multiplication by t; an element of A may be identified formally with a family of Laurent polynomials a_{α}(t) ∈ A_{α}[t, t^{−1}], where the negative exponent of t is bounded from below. The following is a direct consequence of the previous observations:

Theorem 2.7:

For each conjugacy class ⟨u⟩ ∈ G, let us assume that the map B\overline{Z}(u) →^{θ} K(ℤ, 3) defined in 2.6 is homotopically trivial. Then

HC^{∗}(ℤ[G]) ≅ ∏_{⟨u⟩}′H^{∗}(B\overline{Z}(u); ℤ)[t, t^{−1}].

(Note that HC_{f}^{∗}(ℤ[G]) ≅ ⨁_{⟨u⟩} H^{∗}(B\overline{Z}(u); ℤ)[t, t^{−1}].)

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