K-THEORY OF THE INDICIAL ALGEBRA OF A MANIFOLD WITH CORNERS

PIERRE-YVES LE GALL AND BERTRAND MONTHUBERT

Abstract. We compute the K-theory groups of the C^{∗}-algebra of the groupoid of a manifold with corners, in which the analytic index takes its values.

1. Introduction

In [6], the second author associated a Lie groupoid G(X) to any manifold with corners X, following general ideas of Alain Connes ([2]). This definition allows one to define a pseudodifferential calculus for any manifold with corners, which is the pseudodifferential calculus of G(X) (cf. [8, 7, 10]), and an analytic index ind_{a} : K^{∗}(T X) → K_{∗}(C^{∗}(G(X))). The latter K-theory group fulfills an interesting property : if X and X′ are two manifolds with corners of same maximal codimension, then K_{∗}(C^{∗}(G(X))) ≃ K_{∗}(C^{∗}(G(X′))). It implies that the K-theory group in which the analytic index takes its values is universal among manifolds of the same codimension, a fact which is well known for manifolds without boundary, i.e. codimension 0 manifolds with corners.

In this article we compute these indicial K-theory groups. In section 2 we recall the main definitions, like that of G(X). In section 3 we show that there is a recurrence relation between the K-theory groups of G(X) and those of C_{0}(ℝ^{n}) ⋊ 𝔖_{n}. Using Baum-Connes’ Chern character, we compute (following ideas of Kroum Tzanev) the latter groups in section 4. This gives us the K-theory groups of the C^{∗}-algebra of the groupoid of a manifold with corners in section 5.

It’s a pleasure for us to thank Max Karoubi, André Legrand, Georges Skandalis and Kroum Tzanev for many enlightening discussions.

2. Main definitions

In this section, we recall some definitions and properties of [6]. We call codimension p manifold with corners a C^{∞} manifold modelled on ℝ_{+}^{n}, such that each point has a neighborhood diffeomorphic to some ℝ_{+}^{k}×ℝ^{n−k}, with k ≤ p, the point corresponding to the origin. We do not assume that each codimension 1 submanifold is embedded, unlike in the definition of R. Melrose ([5]).

Let X be a manifold with corners, and x a point of X; we denote by F (x) the connected component of the set of points having the same codimension as x which contains x, and by N_{x}F (x) = T_{x}X/T_{x}F (x) the normal space to the boundary at x. We can now define G(X):

G(X)={(x, y, α), x, y ∈X, codim(x)=codim(y), α:N_{y}F (y) →^{∼} N_{x}F (x)}.

In this definition, α is an affine isomorphism of quadrants; locally one can trivialize N_{y}F (y) ≃ ℝ_{+}^{k} and N_{x}F (x) ≃ ℝ_{+}^{k}, and α is given by a monomial matrix (it has one, and only one non-zero element on each line and each column, and this element is positive).

The groupoid structure of G(X) is given by r, s : G(X) → X with r(x, y, α) = x, and s(x, y, α) = y; the composition law is induced by the composition of the isomorphisms, in the sense that (x, y, α)(y, z, β) = (x, z, αβ). We endow it with a C^{∞} manifold structure as follows.

Let Ω and Ω′ be two charts of X of same codimension. Thus Ω ≃^{π} U ×ℝ_{+}^{A}

and Ω′ ≃^{π′} U′ × ℝ_{+}^{A′}, where A and A′ are the sets of local hyperfaces contained in Ω and Ω′; they have same cardinal. Let σ : A′ → A be a bijection.

This allows one to define a map

\begin{matrix}ψ_{σ} : U × U′ × ℝ_{+}^{A′} × ℝ_{+}^{∗}^{A′} & → & G(X) \\ (u, u′, t, λ) & ↦ & (π^{−1}(u, σ(λt)), π′^{−1}(u′, t), α_{σ,t,λ}) \end{matrix}

where α_{σ,t,λ} is an isomorphism of the normal bundle defined as follows. Let B′_{t} = {i ∈ A′, t_{i} = 0}, and B_{t} = σ(B). Let m_{σ,t,λ} be the product of the matrix of σ_{|B′} by the diagonal matrix composed of the λ_{i} for i ∈ B′. Then α_{σ,t,λ} is given through the trivialisations induced by π and π′:

α_{σ,t,λ} : N_{π′^{−1}(u′,t)}F (π′^{−1}(u′, t)) ⟶^{≃} N_{π^{−1}(u,σ(λt))}F (π^{−1}(u, σ(λt)))
≃ ↓  ≃ ↓
ℝ_{+}^{B′}  ⟶^{m_{σ,t,λ}}  ℝ_{+}^{B}

This map is injective, and one thus obtains a C^{∞} structure on G(X), which turns it into a Lie groupoid, whose fibers are submanifolds without boundary; it is amenable. Though this groupoid is not necessarily Hausdorff, one can build a pseudodifferential calculus associated to it, and an analytic index map ([8, 6])

ind_{a} : K^{∗}(T X) → K_{∗}(C^{∗}(G(X))).

Actually, G(X) is universal among the manifolds with corners of same codimension in the following sense: if X and X′ have the same codimension, then G(X) is equivalent to G(X′), (because X and X′ are open and closed subsets of X ∪ X′ that meet each orbit of G(X ∪ X′) so G(X ∪ X′)_{|X} = G(X) and G(X ∪ X′)_{|X′} = G(X′) are both equivalent to G(X ∪ X′)). This implies ([9]) that C^{∗}(G(X)) is Morita equivalent to C^{∗}(G(X′)), and finally K_{∗}(C^{∗}(G(X))) is canonically isomorphic to K_{∗}(C^{∗}(G(X′))).

To compute the groups in which the analytic index takes its values for a codimension n manifold with corners, it is thus enough to compute K_{∗}(C^{∗}(G_{n})), where G_{n} = G(ℝ_{+}^{n}).

Let’s give an explicit description of the stucture of G_{n}: Let’s recall that an isomorphism of ℝ_{+}^{n} is given by a matrix which on each row and each column has exactly one nonzero coefficient, and this coefficient is strictly positive; equivalently it is the product of a diagonal matrix with striclty positive coefficients by the matrix of a permutation on the vectors of the basis. The group of such isomorphisms is thus isomorphic to the semi-direct product ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n} where 𝔖_{n} acts on ℝ_{+}^{∗}^{n} by permutation. For every λ = (λ_{1}, ⋯, λ_{n}) ∈ ℝ_{+}^{∗}^{n} and σ ∈ 𝔖_{n}, one will denote by α_{σ,λ} the isomorphism of ℝ_{+}^{n} given by the product of diag(λ_{1}, ⋯, λ_{n}) and the matrix of σ.

If t = (t_{1}, ⋯, t_{n}) ∈ ℝ_{+}^{n}, then N_{t} = {x = (x_{1}, ⋯, x_{n}) ∈ ℝ_{+}^{n}; x_{i} = 0 if t_{i} ≠ 0}. So the isomorphism α_{σ,λ} restricts to an isomorphism from N_{t} onto N_{σ(λt)}. The local charts are thus defined by the maps ψ_{σ} : (t, λ) ↦ (σ(λt), t, (α_{σ,λ})_{|N_{t}}) from ℝ_{+}^{n} × ℝ_{+}^{∗}^{n} onto G(ℝ_{+}^{n}).

3. A recurrence relation

Let G′_{n} be the restriction of G_{n} = G(ℝ_{+}^{n}) to the set Y_{n} of points of dimension 0 and 1 in ℝ_{+}^{n}. The partitions ℝ_{+}^{n} = {0} ∪ X′ and Y_{n} = {0} ∪ Y′_{n} give rise to the following commutative diagram whose lines are exact sequences:

0 → C^{∗}(Y′_{n}×Y′_{n}×ℝ_{+}^{∗}^{n−1}⋊𝔖_{n−1}) ⟶^{j_{n}} C^{∗}(G′_{n}) ⟶^{q_{n}} C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n}) → 0
↑^{p_{n−1}}  ↑^{p′_{n}}  ↑^{=}
0 →  C^{∗}(G_{n}_{|X′})  ⟶^{i_{n}} C^{∗}(G_{n}) ⟶^{p_{n}} C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n}) → 0

(since C^{∗}(G_{n}_{|{0}}) = C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n})).

As X′ is open in ℝ_{+}^{n}, one has G_{n}_{|X′} = G(X′) which is Morita-equivalent to G_{n−1} since X′ is a manifold with corners of codimension n − 1. From now on, we will write G_{n−1} instead of G_{n}_{|X′} in the diagrams.

These exact sequences induce 6-terms exact sequences in K-theory, whose boundary maps will be denoted by ∂_{n} and δ_{n} respectively.

Theorem 3.1. For any n ∈ ℕ, the K-theory map i_{n}_{∗} = 0, and the following sequence is exact for (n ≥ 2):

0 ⟶ K_{∗}(C^{∗}(G_{n})) ⟶^{p_{n}_{∗}} K_{∗}(C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n})) ⟶^{δ_{n∗}} K_{1−∗}(C^{∗}(G_{n−1})) ⟶ 0

Proof. We will show this by induction. First when n = 1 one has the following exact sequence (where ℝ_{+}^{∗} × ℝ_{+}^{∗} denotes the groupoid of pairs):

0 ⟶ C^{∗}(ℝ_{+}^{∗} × ℝ_{+}^{∗}) ⟶ C^{∗}(G_{1}) ⟶ C^{∗}(ℝ_{+}^{∗}) ⟶ 0

whose (non trivial) boundary map is the Bott isomorphism (cf. [6]). So K_{∗}(C^{∗}(G_{1})) = 0 and, as a consequence, i_{1}_{∗} = 0 and i_{2}_{∗} = 0. Let’s assume that i_{k}_{∗} = 0 for k ≤ n − 1. As the following diagram commutes, one gets i′_{n}_{∗} = 0 and p′_{n}_{∗} is injective:

0 ↑

0 ⟶ C^{∗}(Y′_{n}×Y′_{n}×ℝ_{+}^{∗}^{n−1}⋊𝔖_{n−1}) ⟶^{j_{n}} C^{∗}(G′_{n}) ⟶^{q_{n}} C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n}) ⟶ 0
↑^{p_{n−1}}  ↑^{p′_{n}}  ↑^{=}
0 ⟶  C^{∗}(G_{n−1})  ⟶^{i_{n}} C^{∗}(G_{n}) ⟶^{p_{n}} C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n}) ⟶ 0
↑^{i_{n−1}}  ↑^{i′_{n}}
C^{∗}(G_{n−2})  = C^{∗}(G_{n−2})
↑ ↑
0  0

As p′_{n}_{∗}∘i_{n}_{∗} = j_{n}_{∗}∘p_{n−1}_{∗} one obtains i_{n}_{∗} = 0 if and only if Im(p_{n−1}_{∗}) ⊂ Ker (j_{n}_{∗}) . But by exactness Ker (j_{n}_{∗}) = Im(∂_{n}) .

Since by assumption i_{n−2}_{∗} = 0, p_{n−2}_{∗} is also injective and from the diagram below

0 → C^{∗}(Y′_{n−1}×Y′_{n−1}×ℝ_{+}^{∗}^{n−2}⋊𝔖_{n−2}) →^{j_{n−1}} C^{∗}(G′_{n−1}) →^{q_{n−1}} C^{∗}(ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1}) → 0
↑^{p_{n−2}}  ↑^{p′_{n−1}}  ↑^{=}
0 →  C^{∗}(G_{n−2})  →^{i_{n−1}} C^{∗}(G_{n−1}) →^{p_{n−1}} C^{∗}(ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1}) → 0

one gets that Im(p_{n−1}_{∗}) = Ker (δ_{n−1}) = Ker (∂_{n−1}) . It remains to show that Ker ∂_{n−1} ⊂ Im ∂_{n}, which is an easy consequence of the following lemma.

Lemma 3.1. There exists a map k_{n} : C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n−1}) → C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n}) such that

∂_{n} ∘ k_{n}_{∗} = τ_{n} + k_{n−1}_{∗} ∘ τ_{n−1}^{−1} ∘ ∂_{n−1} ∘ τ_{n},

where τ_{n} : K_{∗}(C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n−1})) → K_{1−∗}(C^{∗}(ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1})) is the Bott isomorphism.

To define the map k_{n}, we build a variant of G′_{n}, where we do not admit all isomorphisms of the normal bundle of ℝ_{+}^{n}, but only those that fix the first axis. Thus an action of 𝔖_{n−1} appears, instead of 𝔖_{n} in the case of G′_{n}.

Let σ ∈ 𝔖_{n}, and ψ_{σ} defined as above. Let’s identify the subgroup of 𝔖_{n} of permutations for which 1 is invariant with 𝔖_{n−1}.

Let Δ_{1} = {x ∈ Y_{n}, x_{1} ≠ 0 ou x = 0} and Δ′ = {x ∈ Y_{n}, x_{1} = 0}. Let’s define

H′_{n} = ⋃_{σ∈𝔖_{n−1}} ψ_{σ}(Δ_{1} × ℝ_{+}^{∗}^{n}),

H′′_{n} = ⋃_{σ∈𝔖_{n−1}} ψ_{σ}(Δ′ × ℝ_{+}^{∗}^{n}),

H_{n} = H′_{n} ∪ H′′_{n} = ⋃_{σ∈𝔖_{n−1}} ψ_{σ}(Y_{n} × ℝ_{+}^{∗}^{n}).

By definition of G′_{n}, H_{n} is an open sub-groupoid of G′_{n}, and one gets a morphism from C^{∗}(H_{n}) to C^{∗}(G′_{n}). Moreover, H′_{n} (resp. H′′_{n}) is its restriction to Δ_{1} (resp. Δ′), which is saturated.

More precisely, H′_{n} = G(ℝ_{+}) × ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1}. Indeed, ψ_{σ}(t, λ) is given by (λ_{1}t_{1}, t_{1}, α_{σ,t,λ}) if t_{1} ≠ 0 and by (0, 0, λ_{1}, α_{σ,1,λ}) if t_{1} = 0.

One thus obtains the following exact sequence, induced by that of G(ℝ_{+}):

0 → K ⊗ C^{∗}(ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1}) → C^{∗}(H′_{n}) → C^{∗}(ℝ_{+}^{∗} × ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1}) → 0.

The boundary map in K-theory induced by this exact sequence is consequently the Bott isomorphism

τ_{n} : K_{∗}(C^{∗}(ℝ_{+}^{∗} × ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1})) → K_{1−∗}(C^{∗}(ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−1})).

In addition, H′′_{n} = ℝ_{+}^{∗} × G′_{n−1}. Let t ∈ Δ′, and λ ∈ ℝ_{+}^{∗}^{n} : t = (0, t′), and λ = (λ_{1}, λ′). The element ψ_{σ}(t, λ) is given by (λ′t′, t′, α_{σ,t,λ}). But the normal space to the boundary at t necessarily factors through Δ_{1}, and Δ_{1} is invariant by σ, thus α_{σ,t,λ} is given by λ_{1} and by α_{σ,t′,λ′}. Finally, ψ_{σ}(t, λ) is given by (λ_{1}, ψ_{σ}(t′, λ′)).

One gets the following exact sequence, from that of G′_{n−1} :

0 → K ⊗ C^{∗}(ℝ_{+}^{∗}^{n−1} ⋊ 𝔖_{n−2}) → C^{∗}(ℝ_{+}^{∗}^{×}G′_{n−1}) → C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n−1}) → 0,

whose boundary map in K-theory is τ_{n−1}^{−1} ∘ ∂_{n−1} ∘ τ_{n}.

This induces the following diagram of exact sequences:

0 ⟶ C^{∗}(Y′_{n}×Y′_{n}×ℝ_{+}^{∗}^{n−1}⋊𝔖_{n−1}) ⟶^{j_{n}} C^{∗}(G′_{n}) ⟶^{q_{n}} C_{0}(ℝ_{+}^{∗}^{n})⋊𝔖_{n} ⟶ 0
↑  ↑  ↑ k_{n}
0 → \begin{matrix}C^{∗}(Δ^{∘}_{1}×Δ^{∘}_{1}×ℝ_{+}^{∗}^{n−1}⋊𝔖_{n−1})⊕ \\ C^{∗}(Y′_{n−1}×Y′_{n−1}×ℝ_{+}^{∗}^{n−1}⋊𝔖_{n−2}) \end{matrix} → C^{∗}(H_{n}) → C^{∗}(ℝ_{+}^{∗}^{n}⋊𝔖_{n−1}) → 0

At the K-theory level, one has the equality:

∂_{n} ∘ k_{n}_{∗} = τ_{n} + k_{n−1}_{∗} ∘ τ_{n−1}^{−1} ∘ ∂_{n−1} ∘ τ_{n},

which implies (composing by τ_{n}^{−1}) that Ker ∂_{n−1} ⊂ Im ∂_{n}.

4. K-theory of C_{0}(ℝ^{n}) ⋊ 𝔖_{n}

In this section we aim to compute the K-theory groups K_{∗}(C_{0}(ℝ^{n}) ⋊ 𝔖_{n}) (where 𝔖_{n} acts on ℝ^{n} by the standard representation), since by Fourier transform, C^{∗}(ℝ_{+}^{∗}^{n} ⋊ 𝔖_{n}) = C^{∗}(ℝ_{+}^{∗}^{n}) ⋊ 𝔖_{n} = C_{0}(ℝ^{n}) ⋊ 𝔖_{n}. Let us mention that lemma 4.1 and proposition 4.1 were explained to us by Kroum Tzanev.

We try to address this problem by using the equivariant cohomology of P. Baum and A. Connes ([1]), whose definition we recall in the case of a finite group. The K-theory groups of C_{0}(ℝ_{+}^{∗}^{n})⋊𝔖_{n} can be deduced from this using Chern’s isomorphism for discrete groups ([1], theorem 1.19).

Let Γ be a finite group acting by homeomorphisms on a topological space X . We write X̂ = {(x, g) ∈ X × Γ ; xg = x}, on which Γ acts by (x, g′)g = (xg, g^{−1}g′g) .

Definition 4.1. One defines ([1] definition 1.5) H^{0}(X, Γ) and H^{1}(X, Γ) by

H^{0}(X, Γ) = ∏_{j∈ℕ}H_{c}^{2j}(X̂/Γ; ℂ) and H^{1}(X, Γ) = ∏_{j∈ℕ}H_{c}^{2j+1}(X̂/Γ; ℂ)

(with H_{c}^{∗}(X̂/Γ; ℂ) the compactly supported Čech cohomology groups).

Lemma 4.1. Let < Γ > be the set of conjugation classes of Γ and g_{1}, ⋯, g_{p} be a system of representants. We denote by C_{g_{i}} the centralizer of g_{i} in Γ and X^{g_{i}} = {x ∈ X; xg_{i} = x} . Then:

H^{∗}(X, Γ) = ⨁_{g_{i}∈<Γ>} H_{c}^{∗}(X^{g_{i}}/C_{g_{i}}; ℂ) .

Proof. This lemma is induced by the homeomorphism between X̂/Γ and the disjoint union ⋃_{g_{i}∈<Γ>} X^{g_{i}}/C_{g_{i}} .

We define, moreover, for any 1 ≤ i ≤ p : X_{g_{i}} = ⋃_{g∈Γ} X^{gg_{i}g^{−1}} . We get a

decomposition of X̂ as a disjoint union ⋃_{g_{i}∈<Γ>} X_{g_{i}} of subsets which are

invariant under the action of Γ . The natural map X^{g_{i}} ⊂ X_{g_{i}} → X_{g_{i}}/Γ induces a homeomorphism from X^{g_{i}}/C_{g_{i}} to X_{g_{i}}/Γ .

As C_{g_{i}} ⊂ Γ this application is well defined.

Moreover X^{g_{i}} ⊂ X_{g_{i}} → X_{g_{i}}/Γ is surjective: if z′ ∈ X_{g_{i}}, ∃h ∈ Γ, z′h ∈ X^{hg_{i}h^{−1}}. Thus z′hg_{i} = z′h, and z′h ∈ X^{g_{i}}.

Last, it is injective because if (y, g_{i}) and (z, g_{i}) ∈ X_{g_{i}} and (y, g_{i})g = (z, g_{i}) then g^{−1}g_{i}g = g_{i} i.e. g ∈ C_{g_{i}} .

Let p_{n} be the number of partitions of n of the shape n = λ_{1}+⋯+λ_{2k} with 1 ≤ λ_{1} < ⋯ < λ_{2k} and i_{n} the number of partitions of n of the shape n = λ_{1} + ⋯ + λ_{2k+1} with 1 ≤ λ_{1} < ⋯ < λ_{2k+1} .

The main result of this section is:

Proposition 4.1. H^{0}(ℝ^{n}, 𝔖_{n}) = ℂ^{p_{n}} and H^{1}(ℝ^{n}, 𝔖_{n}) = ℂ^{i_{n}} .

Proof. We recall first ([3]) that the conjugation classes of 𝔖_{n} are in oneto-one correspondence with Young tableaux.

Let σ ∈ 𝔖_{n} and T be the Young tableau which represents the conjugation class of σ. We denote by λ_{1} < λ_{2} < ⋯ < λ_{m} the lengths of the lines of T and by n_{k} the number of lines of length λ_{k} in T. The total number of lines in T is then ℓ = n_{1} + ⋯ + n_{m} and n = n_{1}λ_{1} + ⋯ + n_{m}λ_{m}.

We will prove that in this case

(ℝ^{n})^{σ} /C_{σ} ≅ ℝ^{n_{1}}/𝔖_{n_{1}} × ⋯ × ℝ^{n_{m}}/𝔖_{n_{m}}

Recall that if we consider 𝔖_{n} as acting on a set of n boxes, the Young tableau of σ is obtained by sorting the orbits of its action by decreasing length, which corresponds to the decomposition of σ into a product of disjoint circular permutations. With our notation λ_{1} is the length of the shortest orbit (or cycle) and there are n_{1} orbits of this length.

One checks also that C_{σ} ≅ ∏_{k=1}^{m} (ℤ/λ_{k}ℤ)^{n_{k}} ⋊ 𝔖_{n_{k}} where each (ℤ/λ_{k}ℤ) acts by circular permutation on one of the n_{k} orbits of length λ_{k} and 𝔖_{n_{k}} acts by interchanging the n_{k} orbits of length λ_{k} (this is done by choosing in non canonical manner some origin in every λ_{k}-orbit).

But an element of ℝ^{n} is σ-invariant if its coordinates belonging to the same orbit are equal. Thus the action of (ℤ/λ_{k}ℤ)^{n_{k}} on (ℝ^{n})^{σ} is trivial; (ℝ^{n})^{σ} identifies with ℝ^{ℓ} and the action of C_{σ} factors through the natural action of 𝔖_{n_{1}} ×⋯×𝔖_{n_{m}} ⊂ 𝔖_{ℓ} on ℝ^{ℓ} . One thus obtains that (ℝ^{n})^{σ} /C_{σ} is homeomorphic to ℝ^{l}/𝔖_{n_{1}} ×⋯×𝔖_{n_{m}} and consequently that (ℝ^{n})^{σ} /C_{σ} ≅ ℝ^{n_{1}}/𝔖_{n_{1}} × ⋯ × ℝ^{n_{m}}/𝔖_{n_{m}} .

An element of ℝ^{n_{k}}/𝔖_{n_{k}} is determined by an ordered n_{k}-tuple of reals x_{1} ≤ ⋯ ≤ x_{n_{k}} and the application

(x_{1} ≤ ⋯ ≤ x_{n_{k}}) ↦ (x_{1}, x_{2} − x_{1}, ⋯, x_{n_{k}} − x_{n_{k}−1})

gives an homeomorphism between ℝ^{n_{k}}/𝔖_{n_{k}} and ℝ× (ℝ_{+})^{n_{k}−1}. So the cohomology of ℝ^{n_{k}}/𝔖_{n_{k}} is zero as soon as n_{k} ≠ 1 and finally by Künneth’s formula one gets H_{c}^{∗} ((ℝ^{n})^{σ} /C_{σ}) = 0 except if n_{1} = ⋯ = n_{m} = 1. In this case (ℝ^{n})^{σ} /C_{σ} ≅ ℝ^{ℓ}, so H_{c}^{∗} ((ℝ^{n})^{σ} /C_{σ}) is null in any degree distinct of ℓ, with H_{c}^{ℓ} ((ℝ^{n})^{σ} /C_{σ}) = ℂ. The result comes out by applying lemma 4.1.

Corollary 4.1. K_{0}(C_{0}(ℝ^{n}) ⋊ 𝔖_{n}) = ℤ^{p_{n}} and K_{1}(C_{0}(ℝ^{n}) ⋊ 𝔖_{n}) = ℤ^{i_{n}}.

Proof. By a result of Karoubi ([4]) K_{∗}(C_{0}(ℝ^{n})⋊𝔖_{n}) = K_{𝔖_{n}}^{∗}(ℝ^{n}) is finitely generated and torsion free, so the result is an obvious consequence of the Chern isomorphism for discrete groups [1], theorem 1.19: K_{𝔖_{n}}^{∗}(ℝ^{n})⊗ℂ = H^{∗}(ℝ^{n}, 𝔖_{n}) .

Let q_{n} = i_{n} + p_{n} . The generating function of the sequence (q_{n})_{n∈ℕ} is

∑_{n=0}^{∞} q_{n}x^{k} = ∏_{n=1}^{∞}(1 + x^{n}) , and admits the following asymptotic equivalent:

\frac{e^{π\sqrt{\frac{n}{3}}}}{4.3^{\frac{1}{4}} n^{\frac{3}{4}}} . If q′_{n} = i_{n} − p_{n} , the generating function of the sequence (q′_{n})_{n∈ℕ}

is ∑_{n=0}^{∞} q′_{n}x^{k} = ∏_{n=1}^{∞}(1 − x^{n}) = ∑_{k=0}^{∞}(−1)^{k}x^{\frac{3k^{2}+k}{2}} .

5. K-theory of C^{∗}(G(X))

Proposition 5.1. K_{0}(C^{∗}(G_{n})) = ℤ^{ℓ_{n}} and K_{1}(C^{∗}(G_{n})) = ℤ^{m_{n}}, where ℓ_{n} and m_{n} satisfy the following equalities, with i_{n} and p_{n} defined as above:

ℓ_{n} = p_{n} − m_{n−1}, m_{n} = i_{n} − ℓ_{n−1}.

Proof. This is a direct consequence of proposition 3.1.

Let’s give a few examples of K_{∗}(C^{∗}(G_{n})):

\begin{matrix}K_{0}(C^{∗}(G_{0})) = ℤ^{1} & K_{1}(C^{∗}(G_{0})) = ℤ^{0} \\ K_{0}(C^{∗}(G_{1})) = ℤ^{0} & K_{1}(C^{∗}(G_{1})) = ℤ^{0} \\ K_{0}(C^{∗}(G_{2})) = ℤ^{0} & K_{1}(C^{∗}(G_{2})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{3})) = ℤ^{0} & K_{1}(C^{∗}(G_{3})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{4})) = ℤ^{0} & K_{1}(C^{∗}(G_{4})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{5})) = ℤ^{1} & K_{1}(C^{∗}(G_{5})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{6})) = ℤ^{1} & K_{1}(C^{∗}(G_{6})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{7})) = ℤ^{2} & K_{1}(C^{∗}(G_{7})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{8})) = ℤ^{2} & K_{1}(C^{∗}(G_{8})) = ℤ^{1} \\ K_{0}(C^{∗}(G_{9})) = ℤ^{3} & K_{1}(C^{∗}(G_{9})) = ℤ^{2} \\ K_{0}(C^{∗}(G_{10})) = ℤ^{3} & K_{1}(C^{∗}(G_{10})) = ℤ^{2} \\ K_{0}(C^{∗}(G_{20})) = ℤ^{17} & K_{1}(C^{∗}(G_{20})) = ℤ^{18} \\ K_{0}(C^{∗}(G_{50})) = ℤ^{966} & K_{1}(C^{∗}(G_{50})) = ℤ^{967} \\ K_{0}(C^{∗}(G_{100})) = ℤ^{115826} & K_{1}(C^{∗}(G_{100})) = ℤ^{115825} \\ K_{0}(C^{∗}(G_{200})) = ℤ^{125443897} & K_{1}(C^{∗}(G_{200})) = ℤ^{125443898} \\ K_{0}(C^{∗}(G_{300})) = ℤ^{29434172078} & K_{1}(C^{∗}(G_{300})) = ℤ^{29434172078} \end{matrix}

References

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Laboratoire Emile Picard, Université Paul Sabatier (UFR MIG), 118 route de Narbonne, F-31062 Toulouse CEDEX 4

 E-mail address: legall@picard.ups-tlse.fr
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