Travaux mathématiques, Volume 17 (2007), 45–55, \begin{matrix}c \\ ◯ \end{matrix} Université du Luxembourg

An approach toward a finite-dimensional definition of twisted K-theory

by Kiyonori Gomi

Abstract

This is an expository account of the following result: we can construct a group by means of twisted ℤ_{2}-graded vectorial bundles which is isomorphic to K-theory twisted by any degree three integral cohomology class.

1 Introduction

Topological K-theory admits a twisting by a degree three integral cohomology class. The resulting K-theory, known as twisted K-theory [2], has its origin in the works of Donovan-Karoubi [8] and Rosenberg [15], and has applications to D-brane charges [5, 12, 17], Verlinde algebras [9] and so on.

As is well-known, ordinary K-theory has definitions by means of:

(1) vector bundles;

(2) the C^{∗}-algebra of continuous functions; and

(3) the space of Fredholm operators. Twisted K-theory is usually defined by twisting the definitions (2) or (3). For a definition parallel to (1), there are the notions of twisted vector bundles ([13, 14, 16], see also [5, 8, 12]), and of bundle gerbe K-modules [4]. However, the definitions by means of these geometric objects are only valid for twisted K-theory whose “twisting”, a third integral cohomology class, is of finite order: otherwise, there is no non-trivial such geometric object in finite-dimensions.

Toward a finite-dimensional definition of twisted K-theory valid for degree three integral cohomology classes of infinite order, we explain in this article Furuta’s notion of generalized vector bundles [10], which we call vectorial bundles. We also explain a notion of finite dimensional approximation of Fredholm operators, which provides a linear version of the finite dimensional approximation of the monopole equations [11]. We can use these notions to construct a group and an isomorphism from K-theory twisted by any degree three integral cohomology class. The proof of the result is only outlined. The detailed treatment will be provided elsewhere.

A possible application of the result above is to generalize the notions of 2vector bundles [3, 6]. A 2-vector bundle of rank 1 due to Brylinski [6] is a stack which reproduces the category of twisted vector bundles. Replacing the category of vector bundles by that of vectorial bundles, we can directly generalize the 2vector bundles in [6]. Similarly, we can also generalize the 2-vector bundles due to Baas, Dundas and Rognes [3], which they studied in an approach to geometric realization of elliptic cohomology. It seems interesting to apply their study of 2vector bundles to the generalization of 2-vector bundles made of vectorial bundles.

Acknowledgments. I am grateful to M. Furuta for helpful discussions concerning this work. I am also indebted to T. Moriyama and A. Henriques for useful suggestions. I thank the organizers of School on Poisson geometry and related topics for the invitation to give a talk.

2 Twisted K-theory and twisted vector bundles

In this section, we review the definition of twisted K-theory by means of the space of Fredholm operators, following [2]. We also review the notion of twisted vector bundles [4, 5, 8, 12, 13, 14, 16].

Unless otherwise mentioned, X is a compact manifold through this article.

2.1 Twisted K-theory

The twisted K-theory we consider in this article is associated to a degree three integral cohomology class. To give the precise definition, we represent the class by a projective unitary bundle. Let H be a separable Hilbert space of infinite dimension, and P U(H) the projective unitary group P U(H) = U(H)/U(1).

Definition 2.1. For a principal P U(H)-bundle over X, we define the twisted Kgroup K_{P} (X) to be the fiberwise homotopy classes of sections of the associated bundle P ×_{Ad} F(H) over X, where P U(H) acts on the space F(H) of Fredholm operators on H by adjoint.

In the above definition, the topologies on P U(H) and F(H) are understood to come from the operator norm. Notice that we can also use the compact-open topology in the sense of [2].

If P is a trivial bundle, then K_{P} (X, P) is exactly the homotopy classes of continuous functions X → F(H). Thus, in this case, we recover the K-group of X by the well-known fact that F(H) is a classifying space of K-theory [1].

P U(H)-bundles over X are classified by H^{3}(X, ℤ): since U(H) is contractible by Kuiper’s theorem, P U(H) is homotopy equivalent to the Eilenberg-MacLane space K(ℤ, 2), so that the classifying space BP U(H) is homotopy equivalent to K(ℤ, 3). If P and P′ are isomorphic P U(H)-bundles, then the twisted K-groups K_{P} (X) and K_{P′}(X) are also (non-canonically) isomorphic. So we often speak of “twisted K-theory twisted by a class in H^{3}(X, ℤ)”.

An approach toward a finite-dimensional definition of twisted K-theory 47

We will call the cohomology class corresponding to P the Dixmier-Douady class, and denote it by δ(P) ∈ H^{3}(X, ℤ). For later convenience, we recall here the construction of δ(P): take an open cover U = {U_{α}} of X so that:

• there are local sections s_{α} : U_{α} → P |_{U_{α}};

• there are lifts g_{αβ} : U_{αβ} → U(H) of the transition functions \overline{g}_{αβ} : U_{αβ} →

P U(H). Here we write U_{αβ} for the overlap U_{α}∩U_{β}, and the transition function is defined by the relation s_{α}\overline{g}_{αβ} = s_{β}. Because of the cocycle condition for \{\overline{g}_{αβ}\}, we can find a map c_{αβγ} : U_{αβγ} → U(1) such that g_{αβ}g_{βγ} = c_{αβγ}g_{αγ}. These maps comprise a C̆ech 2-cocycle (c_{αβγ}) ∈ Ž^{2}(U, \underline{U(1)}) with its coefficients in the sheaf of germs of U(1)- valued functions, which represents δ(P) through the isomorphism Ȟ^{2}(X, \underline{U(1)}) ≅ H^{3}(X, ℤ).

2.2 Twisted vector bundles

For a P U(H)-bundle P over X, a twisted vector bundle consists essentially of the data (U, E_{α}, ϕ_{αβ}):

• an open cover U = {U_{α}} of X;

• vector bundles E_{α} over U_{α};

• isomorphisms of vector bundles ϕ_{αβ} : E_{β}|_{U_{αβ}} → E_{α}|_{U_{αβ}} over U_{αβ} satisfying the “twisted cocycle condition” on U_{αβγ}:

ϕ_{αβ}ϕ_{βγ} = c_{αβγ}ϕ_{αγ},

where c_{αβγ} is as in the previous subsection. In the rigorous definition of twisted vector bundles, we have to include the choices of the local sections s_{α} and the lifts g_{αβ}. Though it is crucial to specify these choices in considering isomorphism classes of twisted vector bundles, we omit them for simplicity.

The isomorphism classes of twisted vector bundles Vect_{P} (X) gives rise to a semi-group by the direct sum of local vector bundles. Let K(Vect_{P} (X)) denote the group given by applying the Grothendieck construction to Vect_{P} (X). Then the following fact is known. (See [5, 8, 12, 13, 16].)

Proposition 2.2. For a P U(H)-bundle whose Dixmier-Douady class δ(P) is of finite order, there exists an isomorphism:

K_{P} (X) ⟶ K(Vect_{P} (X)).

Instead of twisted vector bundles, we can use bundle gerbe K-modules to obtain an equivalent result [4, 7].

The rank of a twisted vector bundle is a multiple of the order of δ(P). This can be seen readily as follows. Suppose that a twisted vector bundle has a finite rank r. Taking the determinant of the twisted cocycle condition, we have:

detϕ_{αβ}detϕ_{βγ} = c_{αβγ}^{r}detϕ_{αγ}.

Hence (c_{αβγ}^{r}) ∈ Ž^{2}(U, \underline{U(1)}) is a coboundary and rδ(P) = 0.

Because of the above remark, there are no no-trivial twisted vector bundles in the case where δ(P) is infinite order. So we cannot use twisted vector bundles of finite dimensions to realize K_{P} (X) generally. In spite of this fact, collections of locally defined vector bundles seem to have the potential in defining K_{P} (X) by means of finite-dimensional objects. An approach is to use the usual technique proving the isomorphism K(X) ≅ [X, F(H)]. In this approach, however, some complications prevent us from transparent management, in particular, in giving equivalence relation. The usage of Furuta’s generalized vector bundle provides a more efficient approach, which we explain in the next section.

3 Furuta’s generalized vector bundle

In this section, we explain a generalization of the notion of vector bundles introduced by M. Furuta [10]. We call the generalized vector bundles vectorial bundles for short. This notion is closely related to a finite-dimensional approximation of Fredholm operators. Applying these notions, we approach to our problem of defining twisted K-theory finite-dimensionally.

3.1 Approximation of a Fredholm operator

We begin with the simplest situation. A ℤ_{2}-graded vectorial bundle over a single point is a pair (E, h) consisting of:

• a ℤ_{2}-graded Hermitian vector space E = E^{0} ⊕ E^{1} of finite rank; and

• a Hermitian map h : E → E of degree 1.

By using a ℤ_{2}-graded vectorial bundle over a point, we can approximate a single Fredholm operator as follows. Let A : H → H be a Fredholm operator. For simplicity, we assume the kernel or cokernel of A is non-trivial We define the ℤ_{2}-graded Hilbert space Ĥ by Ĥ = H⊕H, and the self-adjoint Fredholm operator

Â : Ĥ → Ĥ of degree 1 by Â = ( \begin{matrix}0 & A^{∗} \\ A & 0 \end{matrix} ). By the assumption, the spectrum

σ(Â^{2}) of the non-negative operator Â^{2} contains 0. Since Â^{2} is also Fredholm, 0 ∈ σ(Â^{2}) is a discrete spectrum. Hence there is a positive number μ such that:

• μ ∉ σ(Â^{2});

• the subset σ(Â^{2}) ∩ [0, μ) consists of a finite number of eigenvalues; • the eigenspace Ker(Â^{2} − λ) is finite dimensional for λ ∈ σ(Â^{2}) ∩ [0, μ). Let 0 = λ_{1} < λ_{2} < ⋯ < λ_{n} < μ be the distinct eigenvalues in σ(Â^{2}) ∩ [0, μ).

Then we have the following orthogonal decomposition of Ĥ:

Ĥ = (Ĥ, Â)_{λ_{1}} ⊕ (Ĥ, Â)_{λ_{2}} ⊕ ⋯ ⊕ (Ĥ, Â)_{λ_{n}} ⊕ Ĥ′,

where (Ĥ, Â)_{λ} = Ker(Â^{2} − λ) is the eigenspace of Â^{2} with its eigenvalue λ, and Ĥ′ is the orthogonal complement. Notice that Â preserves each eigenspace as well as the orthogonal complement. More precisely, Â restricts to the trivial map on (Ĥ, Â)_{λ_{1}} ≅ KerA ⊕ kerA^{∗}, while Â induces isomorphisms on (H, Â)_{λ_{i}} for i > 1 and Ĥ′.

Now, cutting off the infinite-dimensional part Ĥ′, we define E and h by E = ⨁_{i}(Ĥ, Â)_{λ_{i}} and h = Â|_{E}. The pair (E, h) is nothing but a ℤ_{2}-graded vectorial bundle over a point.

As a finite-dimensional approximation of a single Fredholm operator, we obtained a ℤ_{2}-graded vectorial bundle over a single point. As will be explained in Subsection 3.3, a similar approximation is possible for a family of Fredholm operators parameterized by X. The resulting object is a ℤ_{2}-graded vectorial bundle over X.

3.2 Definition of vectorial bundle

We now introduce ℤ_{2}-graded vectorial bundles:

Definition 3.1. Let (U, (E_{α}, h_{α}), ϕ_{αβ}) be the following data:

• an open cover U = {U_{α}} of X;

• ℤ_{2}-graded Hermitian vector bundles E_{α} over U_{α};

• Hermitian maps h_{α} : E_{α} → E_{α} of degree 1;

• vector bundle maps ϕ_{αβ} : E_{β}|_{U_{αβ}} → E_{α}|_{U_{αβ}} of degree 0 over U_{αβ} such that h_{α}ϕ_{αβ} = ϕ_{αβ}h_{β}.

A ℤ_{2}-graded vectorial bundle over X is defined to be data (U, (E_{α}, h_{α}), ϕ_{αβ}) satisfying the following conditions:

\begin{matrix}ϕ_{αα} & ≐ & 1 & on U_{α}, \\ ϕ_{αβ}ϕ_{βγ} & ≐ & ϕ_{αγ} & on U_{αβγ}. \end{matrix}

In the above definition, the symbol ≐ stands for an equivalence relation. The meaning of the first condition ϕ_{αα} ≐ 1 is as follows.

For any point x ∈ U_{α}, there are a neighborhood V ⊂ U_{α} of x and a positive number μ such that: for all y ∈ V and v ∈ (E_{α}, h_{α})_{y,<μ} we have ϕ_{αα}(v) = v.

Here (E_{α}, h_{α})_{y,<μ} is the subspace in the fiber of E_{α} at y given by the direct sum of eigenspaces of (h_{α})_{y}^{2} whose eigenvalues are less than μ:

(E_{α}, h_{α})_{y,<μ} = ⨁_{λ<μ} Ker ((h_{α})_{y}^{2} − λ) = ⨁_{λ<μ} {v ∈ (E_{α})_{y}| (h_{α})_{y}^{2}v = λv} .

The meaning of the second condition ϕ_{αβ}ϕ_{βγ} ≐ ϕ_{αγ} is now obvious.

Definition 3.2. Let 𝔼 = (U, (E_{α}, h_{α}), ϕ_{αβ}) and 𝔼′ = (U, (E′_{α}, h′_{α}), ϕ′_{αβ}) be ℤ_{2}- graded vectorial bundles over X.

(a) A set (f_{α}) of vector bundle maps f_{α} : E_{α} → E′_{α} of degree 0 such that f_{α}h_{α} = h′_{α}f_{α} on U_{α} is said to be a homomorphism from 𝔼 to 𝔼′, if we have f_{α}ϕ_{αβ} ≐ ϕ′_{αβ}f_{β} on U_{αβ}.

(b) A homomorphism (f_{α}) : 𝔼 → 𝔼′ is said to be an isomorphism, if there exists a homomorphism (f′_{α}) : 𝔼′ → 𝔼 such that f_{α}f′_{α} ≐ 1 and f′_{α}f_{α} ≐ 1 on U_{α}.

In the above definition of homomorphism, 𝔼 and 𝔼′ share the same open cover. In the case where they have different open covers U and U′ respectively, it suffices to take a common refinement of U and U′.

Definition 3.3. A homotopy between ℤ_{2}-graded vectorial bundles 𝔼 and 𝔼′ over X is defined to be a ℤ_{2}-graded vectorial bundle 𝔼̃ over X × [0, 1] such that 𝔼 and 𝔼′ are isomorphic to 𝔼̃|_{X×{0}} and 𝔼̃|_{X×{1}}, respectively.

We write KF (X) for the set of homotopy classes of isomorphism classes of ℤ_{2}-graded vectorial bundles. The set KF (X) gives rise to a group by means of the direct sum of vector bundles given locally.

A ℤ_{2}-graded (ordinary) vector bundle E gives an example of a ℤ_{2}-graded vectorial bundle by setting U = {X} and h = 0. This construction induces a well-defined homomorphism K(X) → KF (X). In [10], Furuta proved:

Proposition 3.4. The homomorphism K(X) → KF (X) is an isomorphism.

3.3 Approximation of a family of Fredholm operators

As a family version of the construction in Subsection 3.1, we can show:

Lemma 3.5. Let A = {A_{x}} : X → F(H) be a continuous map. For any point p ∈ X, there are a neighborhood U_{p} of p and a positive number μ_{p} such that the following family of vector spaces gives rise to a vector bundle over U_{p}:

⋃_{x∈U_{p}}(Ĥ, Â_{x})_{<μ_{p}} = ⋃_{x∈U_{p}} ⨁_{λ<μ_{p}} Ker(Â_{x}^{2} − λ).

A key to this lemma is that eigenvalues of A_{x} is continuous in x.

By means of the lemma, the family of Fredholm operators A : X → F(H) yields a ℤ_{2}-graded vectorial bundle ({U_{p}}_{p∈X}, (E_{U_{p}}, h_{U_{p}}), ϕ_{U_{p}U_{q}}), where the ℤ_{2}- graded vector bundle E_{U_{p}} is that in Lemma 3.5, the Hermitian map h_{U_{p}} is given by restricting the Fredholm operator: h_{U_{p}}|_{x} = Â_{x}|_{E_{Up}}, and the map of vector bundles ϕ_{U_{p}U_{q}} : E_{U_{q}} → E_{U_{p}} is the following composition of the natural inclusion and the orthogonal projection:

⋃_{x∈U_{p}∩U_{q}}(Ĥ, Â_{x})_{<μ_{q}} ⟶ (U_{p} ∩ U_{q}) × Ĥ ⟶ ⋃_{x∈U_{p}∩U_{q}}(Ĥ, Â_{x})_{<μ_{p}}.

The construction above induces a well-defined homomorphism

α : [X, F(H)] ⟶ KF (X).

This homomorphism is compatible with the isomorphism ind : [X, F(H)] → K(X) in [1]. Namely, the following diagram is commutative:

[X, F(H)]  [X, F(H)]
ind↓  ↓^{α}
K(X) ⟶ KF (X).

The compatibility follows from the fact that one can realize any vector bundle E → X as E = ⋃_{x∈X} KerÂ_{x} by taking A : X → F(H) such that σ(Â_{x}^{2}) = {0, 1}. (See the proof of the surjectivity of ind in [1].)

3.4 Twisted vectorial bundle and twisted K-theory

We now apply vectorial bundles and finite dimensional approximations explained so far to twisted K-theory.

Recall that twisted vector bundles are defined by “twisting” the ordinary cocycle condition for vector bundles. In a similar way, for a P U(H)-bundle P, we define a twisted ℤ_{2}-graded vectorial bundle by replacing the “cocycle condition” ϕ_{αβ}ϕ_{βγ} ≐ ϕ_{αγ} in Definition 3.1 by the “twisted cocycle condition”:

ϕ_{αβ}ϕ_{βγ} ≐ c_{αβγ}ϕ_{αγ}.

A twisted ℤ_{2}-graded vectorial bundle can be constructed from a section 𝔸 : X → P ×_{Ad} F(H). The section gives a set of maps {A_{p} : W_{p} → F(H)}_{p∈X} such that A_{p} = g_{pq}A_{q}g_{pq}^{−1}, where W_{p} is an open set containing p and g_{pq} : W_{p} ∩ W_{q} → U(H) is a lift of transition function of P. Now, we use Lemma 3.5 to define a Hermitian vector bundle over U_{p} ⊂ W_{p} by E_{U_{p}} = ⋃_{x∈U_{p}}(Ĥ, (Â_{p})_{x})_{<μ_{p}}. The map A_{p} also defines a Hermitian map h_{U_{p}} on E_{U_{p}} by restriction. If we define ϕ_{U_{p}U_{q}} : E_{U_{q}} → E_{U_{p}} by the following composition:

⋃_{x∈U_{pq}}(Ĥ, (Â_{q})_{x})_{<μ_{q}} → U_{pq} × Ĥ ⟶^{id×g_{pq}} U_{pq} × Ĥ → ⋃_{x∈U_{pq}}(Ĥ, (Â_{p})_{x})_{<μ_{p}},

then ({U_{p}}, (E_{U_{p}}, h_{U_{p}}), ϕ_{U_{p}U_{q}}) is a twisted ℤ_{2}-graded vectorial bundle.

Introducing isomorphisms and homotopies in a similar way, we obtain the group KF_{P} (X) of homotopy classes of isomorphism classes of twisted ℤ_{2}-graded vectorial bundles. The above construction of twisted vectorial bundles induces the well-defined homomorphism

α : K_{P} (X) ⟶ KF_{P} (X).

Since this map generalizes α : [X, F(H)] → KF (X), it is reasonable to expect that α gives rise to an isomorphism. In fact, we have:

Theorem 3.6. For any P U(H)-bundle P over a compact manifold X, the homomorphism α : K_{P} (X) ⟶ KF_{P} (X) is bijective.

We sketch the proof of this result in the next subsection.

3.5 Sketch of the proof of Theorem 3.6

The fundamental idea to prove Theorem 3.6 is to construct a kind of generalized cohomology theory on CW complexes by means of KF_{P} (X).

As is known [2, 7], the twisted K-group K_{P} (X) fits into a certain generalized cohomology theory {K_{P}^{−n}(X, Y)}_{n∈ℤ}. In particular, for a CW pair (X, Y) equipped with a P U(H)-bundle P → X, we have the long exact sequence:

⋯ → K_{P |_{Y}}^{−n−1}(Y) →^{δ_{−n−1}} K_{P}^{−n}(X, Y) → K_{P}^{−n}(X) → K_{P |_{Y}}^{−n} (Y) →^{δ_{−n}} · · · .

Note that we can identify K_{P}^{1} (X, Y) with K_{P}^{−1}(X, Y) = K_{P ×I}(X × I, Y × I ∪ X × ∂I), and δ_{0} : K_{P |_{Y}}^{0} (Y) → K_{P}^{1} (X, Y) with the composition of the following maps:

K_{P |_{Y}}^{0} (Y) ⟶^{β} K_{P |_{Y} ×D^{2}}^{0}(Y × D^{2}, Y × S^{1}) = K_{P |_{Y}}^{−2} (Y) ⟶^{δ_{−2}} K_{P}^{−1}(X, Y).

Here β induces the Bott periodicity, and is given by “multiplying” a map T : D^{2} → F(H) representing the generator of K(D^{2}, S^{1}).

To construct a similar cohomology theory, we define KF_{P} (X, Y) by using twisted ℤ_{2}-graded vectorial bundles on X whose support do not intersect Y. (The support of a twisted ℤ_{2}-graded vectorial bundle 𝔼 = (U, (E_{α}, h_{α}), ϕ_{αβ}) on X is the closure of the points x ∈ X such that (h_{α})_{x} is not invertible for an α.) For n ≥ 0, we put:

KF_{P}^{−n}(X, Y) = KF_{P ×I^{n}}(X × I^{n}, Y × I^{n} ∪ X × ∂I^{n}).

Then K_{P}^{−n}(X, Y) satisfies the (suitably modified) homotopy axiom and the excision axiom in the Eilenberg-Steenrod axioms. In a way parallel to the method in [1], we can also introduce a natural map δ_{−n} : KF_{P |_{Y}}^{−n} (Y) → KF_{P}^{−n+1}(X, Y), and obtain the long exact sequence for a pair:

⋯ → KF_{P}^{−1}(X) → KF_{P |_{Y}}^{−1} (Y) →^{δ_{−1}} KF_{P}^{0}(X, Y) → KF_{P}^{0}(X) → KF_{P |_{Y}}^{0} (Y).

To extend this sequence, we put KF_{P}^{1}(X, Y) = KF_{P}^{−1}(X, Y) and define δ_{0} : KF_{P |_{Y}}^{0} (Y) → KF_{P}^{1}(X, Y) to be the composition of:

KF_{P |_{Y}}^{0} (Y) ⟶^{β} KF_{P |_{Y} ×D^{2}}^{0}(Y × D^{2}, Y × S^{1}) = KF_{P |_{Y}}^{−2} (Y) ⟶^{δ_{−2}} KF_{P}^{−1}(X, Y),

where β is given by tensoring a vector bundle representing the generator of K(D^{2}, S^{1}). Then the composition of KF_{P}^{0}(X) → KF_{P |_{Y}}^{0} (Y) → KF_{P}^{1}(X, Y) is trivial. (This sequence is not yet shown to be exact at this stage.)

The spaces X × I^{n} and Y × I^{n} ∪ X × ∂I^{n} in the definition of KF_{P}^{−n}(X, Y) are used in that of K_{P}^{−n}(X, Y). Hence the finite-dimensional approximation induces the natural homomorphism α_{−n} : K_{P}^{−n}(X, Y) → KF_{P}^{−n}(X, Y) for n ≥ −1. We can readily see that δ_{−n} (n ≥ 1) commutes with α_{−n}, since δ_{−n} is essentially defined by an inclusion map of spaces. If X is compact, then β commutes with α_{−n}, so that δ_{0} does. The key to this fact is that the compactness allows us to choose a map T : D^{2} → F(H) realizing the generator of K(D^{2}, S^{1}) in a way appropriate for the finite-dimensional approximation.

Now, for a finite CW complex X and a P U(H)-bundle P → X, we can prove the bijectivity of α_{−n} : K_{P}^{−n}(X) → KF_{P}^{−n}(X) (n ≥ 0) by the induction on the number of cells in X. Notice that, if P is trivial, then an argument by using Proposition 3.4 implies the bijectivity of α_{−n} (n ≥ 0). Thus, α_{−n} (n ≥ 0) is bijective in the case where X is a point and has only one cell. If the number of the cells in X is r > 1, then we can express X as X = Y ∪ e^{q}, where Y ⊂ X is a subcomplex with (r − 1) cells, and e^{q} ⊂ X is a cell of dimension q. We here make use of the commutative diagram (n ≥ 0):

K_{P |_{Y}}^{−n−1}(Y) → K_{P}^{−n}(X, Y) → K_{P}^{−n}(X) → K_{P |_{Y}}^{−n} (Y) → K_{P}^{−n+1}(X, Y)
↓ ↓ ↓ ↓ ↓
KF_{P |_{Y}}^{−n−1}(Y) → KF_{P}^{−n}(X, Y) → KF_{P}^{−n}(X) → KF_{P |_{Y}}^{−n} (Y) → KF_{P}^{−n+1}(X, Y).

We can assume that the first and the forth columns are bijective in the induction. The excision axiom implies that K_{P}^{−n}(X, Y) ≅ K_{P}^{−n}(D^{q}, S^{q}) and KF_{P}^{−n}(X, Y) ≅ K_{P}^{−n}(D^{q}, S^{q}). Since any P U(H)-bundle over D^{q} is trivial, the second and fifth columns are also bijective. Thus, so is the third column. (The exactness at KF_{P |_{Y}}^{−n} (Y) is not necessary in the five-term lemma.)

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Kiyonori Gomi Graduate school of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-Ku, Tokyo, 153-8914 Japan. e-mail: kgomi@ms.u-tokyo.ac.jp 56