Homotopy theory of bundles with fiber matrix algebra

A.V. Ershov^{∗}

Abstract In the present paper we consider a special class of locally trivial bundles with fiber a matrix algebra. On the set of such bundles over a finite CW -complex we define a relevant equivalence relation. The obtained stable theory gives us a geometric description of the H-space structure BSU_{⊗} on BSU related to the tensor product of virtual SU-bundles of virtual dimension 1.

Contents

1 Main definitions 2 1.1 Category 𝔉𝔄𝔅(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Classifying spaces 4 2.1 Canonical bundles over Gr_{k, l}, k, l > 1 . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Homotopy groups of Gr_{k, l} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 The universal property of Gr_{k, l} . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The stable theory 10 3.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 The group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 One interesting property of a bundle A_{k}, which is a part of FAB (A_{k}, μ, M̃_{kl}) 13 3.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Relation between ÃB^{1} and K̃SU-theory 19

5 Appendix 1: a GL-version 22

6 Appendix 2: A remark on Thom spaces 23

Introduction

In the present paper we consider a special type of locally trivial bundles with fiber matrix algebra M_{k}(ℂ), k > 1 that are equipped with an embedding into a trivial bundle with fiber M_{kl}(ℂ) for some l coprime with k. More precisely, using such bundles, we construct a homotopy functor ÃB^{1} to the category of Abelian group. The construction can be viewed as a counterpart of the one for the usual topological K-functor by means of vector bundles. In particular, the role of classifying spaces in our case is played by matrix counterparts Gr_{k, l} of the usual Grassmann manifolds. Those Gr_{k, l} are homogeneous spaces parameterizing all subalgebras in M_{kl}(ℂ) that are isomorphic to M_{k}(ℂ) (for different pairs {k, l}, (k, l) = 1). Furthermore, we define a stable equivalence relation on the set of considered bundles such that the classifying space for its equivalence classes is just the direct limit of the matrix Grassmannians lim_{\overrightarrow{(k,l)=1}} Gr_{k, l}. This equivalence relation can be treated as a counterpart of

∗Partially supported by RFFI Grant 02-01-00572.

the usual stable equivalence relation on vector bundles considered in K-theory. Let us remark that the above condition (k, l) = 1 allows us to avoid the localization of classifying spaces under the stabilization. Finally, we show that the obtained homotopy functor ÃB^{1} is equivalent to the multiplicative group of the usual { \atop KSU}-functor (i.e. to the group ξ ∗ η = ξ + η + ξη, ξ, η ∈ { \atop KSU}(X) for a finite CW -complex X). This gives us a geometric description of the H-space structure BSU_{⊗} on BSU in terms of considered type of bundles. Acknowledgments I would like to thank the Max Planck Institute for Mathematics in Bonn for their kind support and generous hospitality while this work partially has been completed. I am grateful to E.V. Troitsky for constant attention to this work and allround support. A number of related questions were discussed with V.M. Manuilov and A.S. Mishchenko and I would like to thank them too.

1 Main definitions

1.1 Category 𝔉𝔄𝔅(X)

Let X be a finite CW -complex. By M̃_{n} denote the trivial bundle (over X) with fiber M_{n}(ℂ), where M_{n}(ℂ) is the algebra of all n × n matrices over ℂ.

Definition 1. Let A_{k} (k > 1) be a locally trivial bundle over X with fiber M_{k}(ℂ). Suppose there is a bundle map μ

\xymatrix{A_{k} \ar[rr]^{μ} \ar[dr] &  & M̃_{kl} \ar[dl] \\  & X}

such that for any point x ∈ X it embeds the fiber (A_{k})_{x} ≅ M_{k}(ℂ) into the fiber (M̃_{kl})_{x} ≅ M_{kl}(ℂ) as a central simple subalgebra. Then the triple (A_{k}, μ, M̃_{kl}) is called an algebra bundle (abbrev. AB) over X. Moreover, if the positive integers k, l are coprime (i.e. their greatest common divisor (k, l) = 1), then the triple (A_{k}, μ, M̃_{kl}) is called to be a floating algebra bundle (abbrev. FAB) over X.

Remark 2. Let A be a central simple algebra over a field 𝕂, B ⊂ A a central simple subalgebra in A. It is well known that the centralizer Z_{A}(B) of B in A is a central simple subalgebra in A again, moreover, the equality A = B⊗_{𝕂}Z_{A}(B) holds. Taking centralizers for all fibers of the subbundle A_{k} ⊂ M̃_{kl} in the corresponding fibers of the trivial bundle M̃_{kl}, we get the complementary subbundle B_{l} with fiber M_{l}(ℂ) together with its embedding ν: B_{l} ↪ M_{kl}(ℂ). Thus, we have the complementary subbundle ν(B_{l}) ⊂ M̃_{kl}, where B_{l} is the locally trivial bundle with fiber M_{l}(ℂ), and ν is its evident embedding into M̃_{kl}. Moreover, A_{k} ⊗ B_{l} = M̃_{kl}.

Conversely, to a given pair (A_{k}, B_{l}) consisting of M_{k}(ℂ)-bundle A_{k} and M_{l}(ℂ)-bundle B_{l} over X such that A_{k} ⊗ B_{l} = M̃_{kl} we can construct a unique triple (A_{k}, μ, M̃_{kl}), where μ is the embedding A_{k} ↪ A_{k} ⊗ B_{l}, a ↦ a ⊗ 1_{B_{l}}. Note also that the operation of taking centralizer is involutive, i.e Z_{A}(Z_{A}(B)) = B.

Definition 3. A morphism from a (F)AB (A_{k}, μ, M̃_{kl}) to a (F)AB (C_{m}, ν, M̃_{mn}) over X is a pair (f, g) of bundle maps f: A_{k} ↪ C_{m}, g: M̃_{kl} ↪ M̃_{mn} such that

• they are fiberwise homomorphisms of algebras (i.e. they are embeddings in fact);

• the square diagram

\xymatrix{M̃_{kl} \ar[r]^{g} & M̃_{mn} \\ A_{k} \ar[u]^{μ} \ar[r]^{f} & C_{m} \ar[u]_{ν}}

commutes;

• let B_{l} ⊂ M̃_{kl}, D_{n} ⊂ M̃_{mn} be the complementary subbundles (see the remark above) for A_{k}, C_{m}, respectively, then g maps B_{l} into D_{n}.

Note that a morphism (f, g): (A_{k}, μ, M̃_{kl}) → (C_{m}, ν, M̃_{mn}) exists only if k|m, l|n.

In particular, an isomorphism between (F)ABs (A_{k}, μ, M̃_{kl}) and (C_{k}, ν, M̃_{kl}) is a pair of bundle maps f : A_{k} → C_{k}, g : M̃_{kl} → M̃_{kl} which are fiberwise isomorphisms of algebras such that the following diagram

\xymatrix{M̃_{kl} \ar[r]^{g} & M̃_{kl} \\ A_{k} \ar[u]^{μ} \ar[r]^{f} & C_{k} \ar[u]_{ν}}

commutes.

Clearly, ABs (resp. FABs) over X with morphisms just defined form a category which we will denote by 𝔄𝔅(X) (resp. 𝔉𝔄𝔅(X)).

For a continuous map φ: X → Y we have the functor φ^{∗}: (𝔉)𝔄𝔅(Y) → (𝔉)𝔄𝔅(X). Remark 4. Since Aut(M_{n}(ℂ)) ≅ Aut(ℂP^{n−1}) ≅ PGL_{n}(ℂ), there is an equivalent theory of locally trivial bundles which can be developed by replacing matrix algebras M_{n}(ℂ) with the corresponding projective spaces ℂP^{n−1}. For example, let us give a counterpart of Definition 1.

Let ℂ̃P^{kl−1} be the trivial bundle over X with fiber ℂP^{kl−1}.

Definition 5. Let P^{k−1}, Q^{l−1} be locally trivial bundles over X with fibers ℂP^{k−1}, ℂP^{l−1}, respectively. Suppose there is a bundle map

\xymatrix{P^{k−1}×_{X}Q^{l−1} \ar[rr]^{λ} \ar@{-->}[dr] &  & ℂ̃P^{kl−1} \ar[dl] \\  & X}

(here P^{k−1}×_{X}Q^{l−1} is the fibered product of bundles over X) such that for any point x ∈ X

its restriction λ |_{x}: (P^{k−1}×_{X}Q^{l−1})_{x} ≅ ℂP^{k−1} × ℂP^{l−1} → (ℂ̃P^{kl−1})_{x} ≅ ℂP^{kl−1} is Segre’s

embedding (in other words, under some appropriate choice of homogeneous coordinates on the projective spaces it is the map

ℂP^{k−1} × ℂP^{l−1} → ℂP^{kl−1},
([x_{0} : …: x_{k−1}], [y_{0} : …: y_{l−1}]) ↦ [x_{0}y_{0} : …: x_{i}y_{j} : …: x_{k−1}y_{l−1}]).

Then P^{k−1}×_{X}Q^{l−1} is called a bundle of Segre’s product, and in the case (k, l) = 1 a floating bundle of Segre’s product (abbrev. FBSP).

For example, using FBSPs, we can construct a counterpart of Thom’s spaces (see Appendix 2).

2 Classifying spaces

2.1 Canonical bundles over Gr_{k, l}, k, l > 1

For any pair k, l > 1 there is the space Gr_{k, l}, parameterizing k-subalgebras (i.e. those isomorphic to M_{k}(ℂ)) in the fixed matrix algebra M_{kl}(ℂ). As a homogeneous space it can be represented as follows:

Gr_{k, l} = PGL_{kl}(ℂ)/PGL_{k}(ℂ) ⊗ PGL_{l}(ℂ),

where by PGL_{k}(ℂ)⊗PGL_{l}(ℂ) we denote the image of the embedding PGL_{k}(ℂ)×PGL_{l}(ℂ) → PGL_{kl}(ℂ) induced by the Kronecker product of matrices.

By M̃_{kl} denote the trivial bundle Gr_{k, l} ×M_{kl}(ℂ). There is a canonical (F)AB (A_{k, l}, μ, M̃_{kl}) over Gr_{k, l} which can be defined as follows: the fiber (A_{k, l})_{x} over x ∈ Gr_{k, l} is the k-subalgebra in M_{kl}(ℂ) = (M̃_{kl})_{x} corresponding to this point.

By Fr_{k, l} denote the homogeneous space PGL_{kl}(ℂ)/E_{k} ⊗ PGL_{l}(ℂ). The space Gr_{k, l} is the base of the following principal PGL_{k}(ℂ)-bundle:

\begin{aligned}PGL_{k}(ℂ) & ↪^{...⊗E_{l}} & Fr_{k, l} \\ & & ↓ \\ & & Gr_{k, l} . \end{aligned}  (6)

Now we want to show that (6) can be treated as a frame fibration for the M_{k}(ℂ)-bundle A_{k, l} over Gr_{k, l}.

More precisely, an ordered collection of n^{2} linearly independent matrices {α_{i, j}: 1 ≤ i, j ≤ n} in M_{n}(ℂ) such that α_{i, j}α_{k, l} = δ_{jk}α_{i, l}, 1 ≤ i, j, k, l ≤ n is called to be an n-frame in M_{n}(ℂ). In particular, the collection of matrices {e_{i, j}: 1 ≤ i, j ≤ n}, where e_{i, j} = E_{ij} are the matrix units, is an n-frame. The map e_{i, j} ↦ α_{i, j}, 1 ≤ i, j ≤ n can be extended to the automorphism of M_{n}(ℂ) which is identity on the center ℂE_{n} ⊂ M_{n}(ℂ). Now applying Noether-Scolem’s theorem we see that there is an element g ∈ GL_{n}(ℂ) such that α_{i, j} = ge_{i, j}g^{−1}, 1 ≤ i, j ≤ n. Therefore, the group PGL_{n}(ℂ) acts transitive on the set of n-frames. Moreover, the stabilizer in PGL_{n}(ℂ) of the n-frame {e_{i, j}: 1 ≤ i, j ≤ n} is trivial. This implies that the set of n-frames in M_{n}(ℂ) can be identified with the group space of PGL_{n}(ℂ). It follows easily that bundle (6) is the bundle of k-frames associated with the M_{k}(ℂ)-bundle A_{k, l}.

Similarly, we can define a k-frame in the matrix algebra M_{kl}(ℂ) as an ordered collection of k^{2} linearly independent matrices {α_{i, j}: 1 ≤ i, j ≤ k} in M_{kl}(ℂ) such that α_{i, j}α_{r, s} = δ_{jr}α_{i, s}, 1 ≤ i, j, r, s ≤ k. The space Fr_{k, l} can be treated as the space of k-frames in the matrix algebra M_{kl}(ℂ). Indeed, it follows from Noether-Scolem’s theorem that the group PGL_{kl}(ℂ) = Aut(M_{kl}(ℂ)) acts transitive on the set of such frames and the stabilizer of the k-frame {e_{i, j} = E_{ij} ⊗ E_{l}: 1 ≤ i, j ≤ k} (here E_{ij} is a matrix unit in M_{k}(ℂ) and E_{l} is the unit l × l matrix) is just the subgroup E_{k} ⊗ PGL_{l}(ℂ) ⊂ PGL_{kl}(ℂ). Remark 6. The noncompact spaces Gr_{k, l} and Fr_{k, l} can be replaced by homotopy equivalent compact ones Gr_{k, l}^{U}, Fr_{k, l}^{U}, respectively. More precisely, let M_{k}(ℂ) ⊗ ℂE_{l} ⊂ M_{kl}(ℂ) be the “standard” k-subalgebra. A subalgebra A ≅ M_{k}(ℂ) in M_{kl}(ℂ) is called unitary if there is an element g ∈ U(kl) ⊂ GL_{kl}(ℂ) such that g(M_{k}(ℂ) ⊗ ℂE_{l})g^{−1} = A. In other words, A is unitary if it conjugates with the standard k-subalgebra by a unitary transformation. The set of unitary subalgebras in M_{kl}(ℂ) is parameterized by the subspace

Gr_{k, l}^{U} := PU(kl)/ PU(k) ⊗ PU(l) ⊂ Gr_{k, l},

where PU(n) is the projective unitary group, i.e. the quotient group U(n)/{αE_{n} | α ∈ ℂ^{∗}, |α| = 1}. It is clear that Gr_{k, l}^{U} is compact and homotopy equivalent to Gr_{k, l} . The principle bundle

PU(k) ↪^{···⊗E_{l}} Fr_{k, l}^{U} := PU(kl)/E_{k} ⊗ PU(l)
↓
Gr_{k, l}^{U}

(7)

(compare with bundle (6)) can be considered as the bundle of k-frames that are unitary with respect to the Hermite scalar product tr(X\overline{Y}^{t}) in M_{kl}(ℂ). There is a unitary counterpart (A_{k, l}^{U}, μ^{U}, M̃_{kl}^{U}) over Gr_{k, l}^{U} of the canonical (F)AB over Gr_{k, l}, where A_{k, l}^{U} is the M_{k}(ℂ)-bundle associated with (7). Because of the homotopy-equivalences Gr_{k, l}^{U} ≃ Gr_{k, l}, Fr_{k, l}^{U} ≃ Fr_{k, l}, we will not make a difference between Gr_{k, l}^{U} and Gr_{k, l}, Fr_{k, l}^{U} and Fr_{k, l} below.

2.2 Homotopy groups of Gr_{k, l}

Let k, l, m, n be integers greater than 1 such that k|m, l|n. Clearly, a homomorphism of the matrix algebras M_{kl}(ℂ) → M_{mn}(ℂ) induces the embedding

Gr_{k, l} ↪ Gr_{m, n} .  (8)

The aim of this subsection is to compute the homotopy groups of the spaces Gr_{k, l} and to study their behavior under the maps (8). We show that (8) induces an isomorphism π_{r}(Gr_{k, l}) ≅ π_{r}(Gr_{m, n}), r ≤ 2 min{k, l} under the condition (m, n) = 1 (see Corollary 10). This is the reason why the condition (k, l) = 1 is imposed. In the general case the localization of the homotopy groups occurs.

Proposition 7. The space Gr_{k, l} has the following homotopy groups in dimensions ≤ 2 min{k, l}:

π_{2}(Gr_{k, l}) ≅ ℤ/(k, l)ℤ; π_{2r}(Gr_{k, l}) ≅ ℤ, if 2 ≤ r ≤ min{k, l};
π_{2r−1}(Gr_{k, l}) ≅ ℤ/(k, l)ℤ, if 1 ≤ r ≤ min{k, l}.

Remark 8. The homotopy groups π_{r}(Gr_{k, l}), r ≤ 2 min{k, l} will be called “stable“ below. Proof. Let us consider the homotopy sequence of the fibration:

PU(k) ⊗ PU(l) ↪ PU(kl)
↓
Gr_{k, l} .

Suppose 2 ≤ r ≤ min{k, l}, then we have:

0 → π_{2r}(Gr_{k, l}) →^{1} ℤ ⊕ ℤ →^{2} ℤ → π_{2r−1}(Gr_{k, l}) → 0.

It is clear from the description of the embedding PU(k) ⊗ PU(l) ↪ PU(kl) (recall that it is induced by the Kronecker product of matrices) that map 2 is following:

ℤ ⊕ ℤ → ℤ, (α, β) ↦ lα + kβ.

Now in the case r ≥ 2 the required result follows from the exactness of the homotopy sequence. Moreover, we obtain the description of map 1:

ℤ → ℤ ⊕ ℤ, γ ↦ (\frac{k}{(l, k)}γ, −\frac{l}{(l, k)}γ).

In the case r = 1 we have:

0 → π_{2}(Gr_{k, l}) → ℤ/kℤ ⊕ ℤ/lℤ → ℤ/klℤ → π_{1}(Gr_{k, l}) → 0.

The exactness of this sequence yields π_{2}(Gr_{k, l}) ≅ ℤ/(k, l)ℤ, π_{1}(Gr_{k, l}) ≅ ℤ/(k, l)ℤ. □ Now we study the behavior of the stable homotopy groups under maps (8).

Proposition 9. If 2 ≤ r ≤ min{k, l}, then the homomorphism of the stable homotopy groups π_{2r}(Gr_{k, l}) → π_{2r}(Gr_{m, n}) is the following map:

ℤ → ℤ, γ ↦ \frac{(m, n)}{(k, l)}γ.

If r = 1, then the homomorphism π_{2}(Gr_{k, l}) → π_{2}(Gr_{m, n}) is the monomorphism:

ℤ/(k, l)ℤ ↪ ℤ/(m, n)ℤ.

The image of π_{2r−1}(Gr_{k, l}) ≅ ℤ/(k, l)ℤ in the group π_{2r−1}(Gr_{m, n}) ≅ ℤ/(m, n)ℤ under the map π_{2r−1}(Gr_{k, l}) → π_{2r−1}(Gr_{m, n}) is the subgroup generated by [\frac{mn}{kl}] := \frac{mn}{kl} mod(m, n) (in particular, the order #\{im(π_{2r−1}(Gr_{k, l}))\} is equal to the order of [\frac{mn}{kl}] in the group ℤ/(m, n)ℤ).

Proof. Consider the following commutative diagram:

\xymatrix{ & PU(m) ⊗ PU(n) \ar[r] & PU(mn) \ar[d] \\ PU(k) ⊗ PU(l) \ar@{-->}[ur] \ar[r] & PU(kl) \ar[d] \ar@{-->}[ur] & _{Gr_{m, n}} \\  & Gr_{k, l} . \ar[ur]}  (10)

Let us describe all the maps in (10). The left-hand arrow: (A, B) ↦ (E_{m/k} ⊗ A, B ⊗ E_{n/l}), where matrices A ∈ U(k), B ∈ U(l) represent elements of PU(k) and PU(l), respectively. The right-hand arrow: C ↦ E_{m/k} ⊗ (C ⊗ E_{n/l}) = (E_{m/k} ⊗ C) ⊗ E_{n/l}, C ∈ U(kl). The two horizontal arrows are induced by the Kronecker product of matrices: (A, B) ↦ A ⊗ B. The commutativity of the diagram follows from the identity (E_{m/k} ⊗ A) ⊗ (B ⊗ E_{n/l}) = E_{m/k} ⊗ ((A ⊗ B) ⊗ E_{n/l}) = (E_{m/k} ⊗ (A ⊗ B)) ⊗ E_{n/l} for all A ∈ U(k), B ∈ U(l).

Diagram (10) induces the morphism of the homotopy sequences of fibrations. If 2 ≤ r ≤ min{k, l}, then we have the diagram:

\xymatrix{0 \ar[r] & π_{2r}(Gr_{m, n}) \ar[r]^{1′} & ℤ ⊕ ℤ \ar[r]^{2′} & ℤ \ar[r]^{3′} & π_{2r−1}(Gr_{m, n}) \ar[r] & 0 \\ 0 \ar[r] & π_{2r}(Gr_{k, l}) \ar[r]_{1} \ar[u]^{4} & ℤ ⊕ ℤ \ar[r]_{2} \ar[u]^{5} & ℤ \ar[r]_{3} \ar[u]^{6} & π_{2r−1}(Gr_{k, l}) \ar[r] \ar[u]^{7} & 0.}  (11)

The description of monomorphisms 1, 1′ :

ℤ → ℤ ⊕ ℤ, γ ↦ (\frac{k}{(l, k)}γ, −\frac{l}{(l, k)}γ),

ℤ → ℤ ⊕ ℤ, ν ↦ (\frac{m}{(n, m)}ν, −\frac{n}{(n, m)}ν),

and 2, 2′ :

(α, β) ↦ lα + kβ, (λ, μ) ↦ nλ + mμ

can be extracted from the proof of Proposition 7. The epimorphisms 3 and 3′ are the reductions modulo (k, l) and modulo (m, n) respectively. The description of the diagonal arrows in (10) yields the following description of homomorphisms 5, 6 in (11):

(α, β) ↦ (\frac{m}{k}α, \frac{n}{l}β), τ ↦ \frac{mn}{kl}τ,

respectively. Now in the case r ≥ 2 the required result follows from the commutativity of diagram (11).

Now suppose r = 1. In this case diagram (10) yields us the diagram:

[figure: 0; π_{2}(Gr_{m, n}); ℤ/mℤ ⊕ ℤ/nℤ; 0; π_{2}(Gr_{k, l}); 4; ℤ/kℤ ⊕ ℤ/lℤ; 5; ℤ/mnℤ; π_{1}(Gr_{m, n}); 0; ℤ/klℤ; 6; π_{1}(Gr_{k, l}); 0.]  (12)

The commutativity of (12) and the description of the diagonal arrows in (10) imply that arrows 4, 5, 6 are monomorphisms (here we use the evident fact that the map PU(k) →^{...⊗E_{l}} PU(kl) induces the monomorphism of the fundamental groups

π_{1}(PU(k)) ↪ π_{1}(PU(kl)), ℤ/kℤ ↪ ℤ/klℤ). □

Corollary 10. Suppose (k, l) = (m, n) = 1. Then the map Gr_{k, l} → Gr_{m, n} induces isomorphisms π_{r}(Gr_{k, l}) →^{≅} π_{r}(Gr_{m, n}) of homotopy groups, where r ≤ 2 min{k, l}. Thus, the spaces Gr_{k, l} and Gr_{m, n} are homotopy equivalent in the stable dimensions (i.e. up to dimension r = 2 min{k, l}).

Now let us describe the (stable) homotopy groups of total spaces of frame fibrations (see (7)) and their behavior under maps.

The following Proposition follows easily from the exactness of the homotopy sequence of fibration (7).

Proposition 11.

π_{2r}(Fr_{k, l}) = 0, π_{2r−1}(Fr_{k, l}) = ℤ/kℤ, 1 ≤ r ≤ l.

Remark 12. Suppose (m, n) = 1 (⇒ (k, l) = 1). Consider the following morphism of fiber bundles

\xymatrix{ & PU(n) \ar[r]^{E_{m}⊗...} & PU(mn) \ar[d] \\ PU(l) \ar@{-->}[ur]^{...⊗E_{\frac{n}{l}}} \ar[r]^{E_{k}⊗...} & PU(kl) \ar[d] \ar[ur]^{E_{\frac{m}{k}} ⊗...⊗E_{\frac{n}{l}}} & _{Fr_{m, n}} \\  & _{Fr_{k, l}} \ar[ur]^{α}}

and the corresponding morphism of homotopy sequences. Clearly, the order of element \frac{mn}{kl} mod m ∈ ℤ/mℤ equals k. It implies that α induces monomorphisms π_{2r−1}(Fr_{k, l}) ↪ π_{2r−1}(Fr_{m, n}), ℤ/kℤ ↪ ℤ/mℤ for all r ≤ l.

2.3 The universal property of Gr_{k, l}

Let Ψ_{k, l}(X) be the set of isomorphism classes of FABs of the form (A_{k}, μ, M̃_{kl}) over X (⇒ (k, l) = 1).

Proposition 13. The assignment:

[X, Gr_{k, l}] → Ψ_{k, l}(X), φ ↦ φ^{∗}(A_{k, l}, μ, M̃_{kl})

is a bijection for X, dim X ≤ 2 min{k, l}.

Proof. Let φ_{0}, φ_{1} be two classifying maps X → Gr_{k, l} for (A_{k}, μ, M̃_{kl}). We must prove that there is a map Φ: X × I → Gr_{k, l} such that Φ|_{X×{0}} = φ_{0}, Φ|_{X×{1}} = φ_{1}. Suppose such a Φ exists, then Φ^{∗}(A_{k, l}, μ, M̃_{kl}) ≅ π^{∗}(A_{k}, μ, M̃_{kl}), where π: X × I → X is the projection.

We construct Φ by induction on dimension r of the skeleton of the relative CW -complex (X × I, X × {0} ⊔ X × {1}).

Suppose we have already constructed a map Φ^{(r)} : (X × I, X × {0} ⊔ X × {1})^{(r)} → Gr_{k, l} with the required properties (Φ^{(r)}|_{X×{0}} = φ_{0}, Φ^{(r)}|_{X×{1}} = φ_{1}) and with an isomorphism Φ^{(r)∗}(A_{k, l}, μ, M̃_{kl}) ≅ π^{∗}(A_{k}, μ, M̃_{kl})|_{(X×I, X×{0}⊔X×{1})^{(r)}}.

Let e_{r+1} be a relative cell in X × I (it has the form e_{r} × I, where e_{r} is a cell in X). If r = 2s + 1, then using π_{2s+1}(Gr_{k, l}) = 0 (see Proposition 7), we can extend Φ^{(r)} from

the boundary ∂e_{r+1} = S^{2s+1} to the whole cell e_{r+1} and therefore to the r + 1-skeleton.

If r = 2s, then we construct Φ^{(r+1)} in the following way. The restriction of π^{∗}(A_{k}, μ, M̃_{kl}) to a cell e_{r+1} is a trivial FAB (see Definition 14). Thus, we have the maps:

(i) π^{∗}(A_{k}, μ, M̃_{kl})|_{e_{r+1}} →^{≅} (e_{r+1} × M_{k}(ℂ), τ, e_{r+1} × M_{kl}(ℂ));

(ii) Φ^{(r)} : ∂e_{r+1} → Gr_{k, l} satisfying the conditions Φ^{(r)} |_{e_{r}×{0}}= φ_{0}, Φ^{(r)} |_{e_{r}×{1}}= φ_{1};

(iii) Φ^{(r)∗}(A_{k, l}, μ, M̃_{kl})|_{∂e_{r+1}} ≅ (∂e_{r+1} × M_{k}(ℂ), τ, ∂e_{r+1} × M_{kl}(ℂ)).

\xymatrix{∂e_{r+1} \ar[r]^{Ψ^{(r)}} \ar[dr]_{Φ^{(r)}} & _{Fr_{k, l}} \ar[d] \\  & _{Gr_{k, l}}}  (14)

Thus, we have a lifting Ψ^{(r)} of Φ^{(r)} (see (14)) which to any point x ∈ ∂e_{r+1} assigns a frame over this point (here we use isomorphism (iii)). Because of r = 2s and π_{2s}(Fr_{k, l}) = 0 (see Proposition 11), the lifting Ψ^{(r)} can be extended to the whole r + 1-cell e_{r+1} and therefore Φ^{(r)} can be extended to the (r + 1)-skeleton (X × I, X × {0} ⊔ X × {1})^{(r+1)}. □

3 The stable theory

3.1 Stabilization

Suppose (km, ln) = 1. Define the product ∘ of two FABs (A_{k}, μ, M̃_{kl}), (B_{m}, ν, M̃_{mn}) over X as

(A_{k}, μ, M̃_{kl}) ∘ (B_{m}, ν, M̃_{mn}) = (A_{k} ⊗ B_{m}, μ ⊗ ν, M̃_{kl} ⊗ M̃_{mn})

(notice that M̃_{kl} ⊗ M̃_{mn} = M̃_{klmn}).

Definition 14. A FAB of the form (M̃_{k}, τ, M̃_{kl}) is called to be trivial if τ: M̃_{k} → M̃_{kl} is the following map:

X × M_{k}(ℂ) → X × M_{kl}(ℂ), (x, T) ↦ (x, T ⊗ E_{l})

(under some choice of trivializations on M̃_{k} and M̃_{kl}) for any point x ∈ X, where E_{l} is the unit l × l matrix and T ⊗ E_{l} denotes the Kronecker product of matrices. In other words, the bundle M̃_{k} is embedded into M̃_{kl} as a fixed subalgebra.

Definition 15. Two FABs (A_{k}, μ, M̃_{kl}) and (B_{m}, ν, M̃_{mn}) over X are said to be stable equivalent, if there is a sequence of pairs {t_{i}, u_{i}} ∈ ℕ^{2}, 1 ≤ i ≤ s such that

• {t_{1}, u_{1}} = {k, l}, {t_{s}, u_{s}} = {m, n};

• (t_{i}t_{i+1}, u_{i}u_{i+1}) = 1 if s > 1, 1 ≤ i ≤ s − 1,

and a sequence of FABs (A_{t_{i}}, μ_{i}, M̃_{t_{i}u_{i}}) over X such that

• (A_{t_{1}}, μ_{1}, M̃_{t_{1}u_{1}}) = (A_{k}, μ, M̃_{kl}), (A_{t_{s}}, μ_{s}, M̃_{t_{s}u_{s}}) = (B_{m}, ν, M̃_{mn});

• (A_{t_{i}}, μ_{i}, M̃_{t_{i}u_{i}}) ∘ (M̃_{t_{i+1}}, τ, M̃_{t_{i+1}u_{i+1}}) ≅ (A_{t_{i+1}}, μ_{i+1}, M̃_{t_{i+1}u_{i+1}}) ∘ (M̃_{t_{i}}, τ, M̃_{t_{i}u_{i}}), where 1 ≤ i ≤ s − 1 and (M̃_{t_{i}}, τ, M̃_{t_{i}u_{i}}) are trivial FABs.

By ÃB^{1}(X) denote the set of stable equivalence classes of FABs over X.

The following theorem justifies the previous definition.

Theorem 16. 1) For all sequences of pairs of positive integers {k_{j}, l_{j}}_{j∈ℕ} such that

(i) k_{j}, l_{j} → ∞; (ii) k_{j}|k_{j+1}, l_{j}|l_{j+1}; (iii) (k_{j}, l_{j}) = 1 ∀j,

the corresponding direct limits lim_{\overrightarrow{j}} Gr_{k_{j}, l_{j}} are homotopy equivalent. This unique homotopy

type we denote by lim_{\overrightarrow{(k,l)=1}} Gr_{k, l}.

2) The space lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} is a classifying space for stable equivalence classes of FABs over a

finite CW -complex X. In other words, the functor X ↦ ÃB^{1}(X) from the homotopy category of finite CW -complexes to the category 𝔖𝔢𝔱 is represented by the space lim_{\overrightarrow{(k,l)=1}} Gr_{k, l}.

Proof. 1) First, suppose (k, l) = 1 = (m, n), (km, ln) = 1, then the common stable parts of the spaces Gr_{k, l} and Gr_{m, n} are homotopy equivalent. Indeed, according to Corollary 10 the maps λ, κ :

\xymatrix{ & _{Gr_{km, ln}} \\ _{Gr_{k, l}} \ar@{-->}[ur]^{λ} &  & _{Gr_{m, n}} \ar@{-->}[ul]_{κ}}  (15)

induce an isomorphism of homotopy groups.

Secondly, suppose (km, ln) > 1. Let us take sufficiently large t, u such that (t, u) = 1 and (t, l) = (u, k) = (t, n) = (u, m) = 1 (hence (kt, lu) = 1 = (tm, nu)). Now, using the diagram

\xymatrix{ & _{Gr_{kt, lu}} &  & _{Gr_{mt, nu}} \\ _{Gr_{k, l}} \ar[ur] &  & _{Gr_{t, u}} \ar[ul] \ar@{-->}[ur] &  & Gr_{m, n}, \ar[ul]}  (16)

we complete the proof of the first part of the theorem.

2) The proof of this part is based on Proposition 13. More precisely, suppose dim X ≤ 2 min{k, l, m, n}. Let (A_{k}, μ, M̃_{kl}), (B_{m}, ν, M̃_{mn}) be FABs over X, let φ_{A}: X → Gr_{k, l}, φ_{B}: X → Gr_{m, n} be their classifying maps. Let us remark that according to Proposition 13 the classifying maps φ_{A}, φ_{B} exist and unique up to homotopy. Suppose the composite maps

X →^{φ_{A}} Gr_{k, l} ↪ lim_{\overrightarrow{(t,u)=1}} Gr_{t, u},
X →^{φ_{B}} Gr_{m, n} ↪ lim_{\overrightarrow{(t,u)=1}} Gr_{t, u}

are homotopic to each other. Under the condition (km, ln) = 1 we use diagram (15). Since dim X ≤ 2 min{k, l, m, n}, we see that already the maps λ ∘ φ_{A}, κ ∘ φ_{B} are homotopic to each other. Note that

λ^{∗}(A_{km, ln}, μ, M̃_{klmn}) ≅ (A_{k, l}, μ, M̃_{kl}) ∘ (M̃_{m}, τ, M̃_{mn}),
κ^{∗}(A_{km, ln}, μ, M̃_{klmn}) ≅ (A_{m, n}, μ, M̃_{mn}) ∘ (M̃_{k}, τ, M̃_{kl}),

where (M̃_{k}, τ, M̃_{kl}), (M̃_{m}, τ, M̃_{mn}) are trivial FABs over Gr_{m, n} and Gr_{k, l} respectively. Therefore λ ∘ φ_{A} and κ ∘ φ_{B} are classifying maps for (A_{k}, μ, M̃_{kl}) ∘ (M̃_{m}, τ, M̃_{mn}) and (B_{m}, ν, M̃_{mn}) ∘ (M̃_{k}, τ, M̃_{kl}) respectively. Hence (A_{k}, μ, M̃_{kl}) ∘ (M̃_{m}, τ, M̃_{mn}) ≅ (B_{m}, ν, M̃_{mn}) ∘ (M̃_{k}, τ, M̃_{kl}), i.e. (A_{k}, μ, M̃_{kl}) ∼ (B_{m}, ν, M̃_{mn}).

Conversely, suppose (A_{k}, μ, M̃_{kl}) ∼ (B_{m}, ν, M̃_{mn}). Since dim X ≤ 2 min{k, l, m, n}, we have (A_{k}, μ, M̃_{kl}) ∘ (M̃_{m}, τ, M̃_{mn}) ≅ (B_{m}, ν, M̃_{mn}) ∘ (M̃_{k}, τ, M̃_{kl}) (recall that we have assumed that (km, ln) = 1). Then it follows from Proposition 13 that the compositions λ ∘ φ_{A} : X → Gr_{km, ln} and κ ∘ φ_{B} : X → Gr_{km, ln} are homotopic to each other.

The required assertion in the case (kl, mn) > 1 can be obtained similarly, but instead of (15) we should use diagram (16). □

3.2 The group structure

For a FAB (A_{k}, μ, M̃_{kl}) over X by [(A_{k}, μ, M̃_{kl})] we denote its stable equivalence class. Define the product ⋄ of two classes [(A_{k}, μ, M̃_{kl})], [(B_{m}, ν, M̃_{mn})] as

[(A_{k}, μ, M̃_{kl})] ⋄ [(B_{m}, ν, M̃_{mn})] = [(A_{k}, μ, M̃_{kl}) ∘ (B_{m}, ν, M̃_{mn})].

Clearly, this product is well defined if (km, ln) = 1. The following lemma allows us to reject this restriction.

Lemma 17. For any pair {k, l} such that (i) (k, l) = 1, (ii) 2 min{k, l} ≥ dim X, any stable equivalence class of FABs over X has a representative of the form (A_{k}, μ, M̃_{kl}).

Proof easily follows from Theorem 16. □

Clearly, the product ⋄ is associative, commutative, and has identity element [(M̃_{k}, τ, M̃_{kl})], where (M̃_{k}, τ, M̃_{kl}) is an arbitrary trivial FAB. Moreover, for any class [(A_{k}, μ, M̃_{kl})] there exists the inverse element. In order to find it, let us recall the following fact. The centralizer Z_{P} (Q) of a central simple subalgebra Q in a central simple algebra P (over some field 𝕂) is a central simple subalgebra again, moreover, the equality P = Q⊗_{𝕂}Z_{P} (Q) holds. Therefore, taking the centralizers for all fibers of the subbundle A_{k} in M̃_{kl}, we obtain the complementary subbundle B_{l} with fiber M_{l}(ℂ) together with its embedding ν: B_{l} ↪ M_{kl}(ℂ) into the trivial bundle. Moreover, A_{k} ⊗ B_{l} = M̃_{kl}. It is not hard to prove that [(B_{l}, ν, M̃_{kl})] is the inverse element for [(A_{k}, μ, M̃_{kl})]. Thus, the functor X ↦ ÃB^{1}(X) takes values in the category of Abelian groups 𝔄𝔟.

The proof of the following proposition is clear.

Proposition 18. There is the structure of H-space on lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} such that the H-space

lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} represents the functor X ↦ ÃB^{1}(X) to the category of Abelian groups 𝔄𝔟. In

other words, the functors X ↦ ÃB^{1}(X) and [X, lim_{\overrightarrow{(k,l)=1}} Gr_{k, l}] are naturally equivalent as

homotopy functors to the category 𝔄𝔟.

Remark 19. Note that for any pairs {k, l}, {m, n} such that (km, ln) = 1 we have the maps

Gr_{k, l} × Gr_{m, n} → Gr_{km, ln}

induced by the tensor product of matrix algebras M_{kl}(ℂ) × M_{mn}(ℂ) →^{⊗} M_{kl}(ℂ)⊗_{ℂ}M_{mn}(ℂ) ≅ M_{klmn}(ℂ). Clearly, the multiplication in the H-space lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} is determined by such maps

on the finite subspaces Gr_{k, l} ⊂ lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} .

Furthermore, for any pair {k, l} we have map Gr_{k, l} → Gr_{l, k} which assigns to any ksubalgebra in M_{kl}(ℂ) its centralizer and is a classifying map for the complementary bundle (B_{l, k}, ν, M̃_{kl}) for the canonical bundle (A_{k, l}, μ, M̃_{kl}) over Gr_{k, l}. These maps induce the inversion map in the H-space lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} .

Now we simplify our notation: the H-space lim_{\overrightarrow{(k,l)=1}} Gr_{k, l} we denote simply by Gr.

3.3 One interesting property of a bundle A_{k}, which is a part of FAB (A_{k}, μ, M̃_{kl})

Definition 20. Let (A_{k}, μ, M̃_{kl}) be an AB over X. The locally trivial Aut(M_{k}(ℂ)) ≅ PGL_{k}(ℂ)-bundle A_{k} is said to be a core of the AB (A_{k}, μ, M̃_{kl}).

Lemma 21. Suppose A_{k} is the core of a FAB (A_{k}, μ, M̃_{kl}), then the structural group of A_{k} can be reduced from Aut M_{k}(ℂ) ≅ PGL_{k}(ℂ) to SL_{k}(ℂ), i.e. actually to SU(k).

Proof. We have the covering

μ_{n} ↪ SU(n)
↓
PU(n),

where μ_{n} is the group of nth degree roots of unity. Suppose (k, l) = 1, then the embedding SU(k) × SU(l) ↪^{⊗} SU(kl) induces an isomorphism of the center of SU(k) × SU(l) (which is isomorphic to μ_{k} × μ_{l} = μ_{kl}) to the center of SU(kl). Therefore we have the diagram

[figure: PU(k); PU(kl)/E_{k}⊗PU(l); SU(k); SU(kl)/E_{k}⊗SU(l); PU(kl)/ PU(k)⊗PU(l); SU(kl)/ SU(k)⊗SU(l),; φ]

where φ is an isomorphism.

Another way to prove the lemma is based on the observation that π_{2}(Gr_{k, l}) = 0 if (k, l) = 1. Hence the PU(k)-cocycle determining the PU(k)-bundle A_{k, l} over Gr_{k, l} can be lifted to a SU(k)-cocycle. □

Corollary 22. The core A_{k} of arbitrary FAB (A_{k}, μ, M̃_{kl}) has the form End(ξ_{k}) for some vector SU(k)-bundle ξ_{k}. The bundle ξ_{k} is determined by A_{k} uniquely up to isomorphism.

Proof. Let λ̄_{k}: Gr_{k, l} → BPU(k) be a classifying map for A_{k, l} as a PU(k)-bundle. We must prove the uniqueness of lifting λ_{k} of λ̄_{k}:

\xymatrix{K(μ_{k}; 1) \ar[r] & BSU(k) \ar[d] \\ _{Gr_{k, l}} \ar@{-->}[ur]^{λ_{k}} \ar[r]_{λ̄_{k}} & BPU(k).}  (19)

It follows from (19) that the obstruction to a homotopy between two liftings λ_{k}, λ′_{k} of λ̄_{k} belongs to the group

H^{1}(Gr_{k, l}; π_{1}(K(μ_{k}; 1))) = H^{1}(Gr_{k, l}; μ_{k}) = 0. □

It can be proved that any locally trivial M_{k}(ℂ)-bundle A_{k} over X is the core of some AB. The following lemma shows that, in contrast to the general case, cores of FABs have some specific property.

Lemma 23. Let X be a finite CW -complex. Suppose dim X ≤ 2 min{k, m}; then the following conditions are equivalent:

(i) A_{k} is the core of some FAB over X;

(ii) for arbitrary m such that 2m ≥ dim X there is a bundle B_{m} with fiber M_{m}(ℂ) such that A_{k} ⊗ M̃_{m} ≅ B_{m} ⊗ M̃_{k};

(iii) A_{k} ⊗ M̃_{m} ≅ B_{m} ⊗ M̃_{k} for some locally trivial bundle B_{m} with fiber M_{m}(ℂ) such that (k, m) = 1.

Moreover, for any pair of bundles (A_{k}, B_{m}) such that (k, m) = 1 and A_{k} ⊗ M̃_{m} ≅ B_{m} ⊗ M̃_{k} there exists a unique stable equivalence class of FABs over X which has (for sufficiently large n, (km, n) = 1) FABs of the form (A_{k}, μ, M̃_{kn}), (B_{m}, ν, M̃_{mn}) as representatives (for some embeddings μ, ν).

Proof. The implication (i)⇒(ii) follows from Lemma 17, the implication (ii)⇒(iii) is trivial. Thus, we have to prove the implication (iii)⇒(i).

Let λ̄_{k}: Gr_{k, n} → BPU(k) be a classifying map for the core A_{k, n} of the canonical FAB over Gr_{k, n} as a PU(k)-bundle. The map λ̄_{k} induces the morphism

\xymatrix{ & PU(k) \ar[r] & ∗ \ar[d] \\ PU(k) \ar@{-->}[ur] \ar[r]^{···⊗E_{n}} & _{Fr_{k, n}} \ar[ur] \ar[d] & BPU(k) \\  & _{Gr_{k, n}} \ar[ur]^{λ̄_{k}}}

from the k-frame bundle Fr_{k, n} ⟶^{PU(k)} Gr_{k, n} to the universal principle PU(k)-bundle over BPU(k). It follows from the corresponding morphism of the homotopy sequences that λ̄_{k∗} : π_{2r}(Gr_{k, n}) = ℤ → π_{2r}(BPU(k)) = ℤ, 2 ≤ r ≤ min{k, n} is the homomorphism which takes 1 to k (if r = 1, then π_{2}(Gr_{k, n}) = 0 and π_{2}(BPU(k)) = π_{1}(PU(k)) = μ_{k} is the group of kth degree roots of unity).

The map λ̄_{k} can be considered as a fibration:

\xymatrix{_{Fr_{k, n}} \ar[r] & _{Gr_{k, n}} \ar[d]^{λ̄_{k}} \\ X \ar[ur]_{φ̃_{A_{k}}} \ar[r]_{φ_{A_{k}}} & BPU(k).}  (20)

Clearly, if φ_{A_{k}}: X → BPU(k) is a classifying map for the bundle A_{k} (as a PU(k)-bundle), then A_{k} is the core of some FAB iff φ_{A_{k}} has a lifting φ̃_{A_{k}}: X → Gr_{k, n} (see (20)). Furthermore, a fiber of λ̄_{k} is the space Fr_{k, n} of k-frames in M_{kn}(ℂ). Recall that

π_{2r}(Fr_{k, n}) = 0, π_{2r−1}(Fr_{k, n}) = ℤ/kℤ, 1 ≤ r ≤ n

(see Proposition 11).

Now consider the diagram

\xymatrix{ &  & BPU(km) \\ BPU(k) \ar[urr]^{ψ} & _{Gr_{k, n}} \ar[l]_{λ̄_{k}} \ar[r]^{γ_{k}} & _{Gr_{km, n^{2}}} \ar[u]^{λ̄_{km}} & _{Gr_{m, n}} \ar[l]_{γ_{m}} \ar[r]^{λ̄_{m}} & BPU(m) \ar[ull]_{χ} \\  &  & X \ar[ull]_{φ_{k}} \ar[urr]^{φ_{m}}}  (21)

(where m is a positive integer such that (km, n) = (k, m) = 1 and dim X ≤ 2 min{k, m, n}). The maps γ_{k}, γ_{m} in (21) are induced by matrix algebras homomorphisms (they are homotopy equivalences in the stable dimensions, see Corollary 10), ψ and χ are induced by the maps PU(k) ⟶^{...⊗E_{m}} PU(km), PU(m) ⟶^{...⊗E_{k}} PU(km), respectively, φ_{k} and φ_{m} are the classifying maps for the bundles A_{k}, B_{m} as in (iii) of lemma’s statement.

It follows from our assumption (iii) that ψ ∘ φ_{k} ≃ χ ∘ φ_{m}. Moreover, it is clear that ψ ∘ λ̄_{k} ≃ λ̄_{km} ∘ γ_{k}, χ ∘ λ̄_{m} ≃ λ̄_{km} ∘ γ_{m}. In order to verify the implication (iii) ⇒ (i), we must construct a common lifting of the maps φ_{k}, φ_{m} to Gr and prove its uniqueness up to homotopy.

We construct the common lifting of φ_{k}, φ_{m} by induction on the dimension of the cellular skeleton of X. Suppose we have already constructed liftings φ̃_{k}^{2r}: X^{(2r)} → Gr, φ̃_{m}^{2r}: X^{(2r)} → Gr of φ_{k}, φ_{m} to the 2r-dimensional skeleton X^{(2r)} which are homotopic, i.e. φ̃_{k}^{2r} ≃ φ̃_{m}^{2r}. Let (φ̃_{k}^{2r+1})′, (φ̃_{m}^{2r+1})′ be arbitrary liftings of φ_{k}, φ_{m} which extend φ̃_{k}^{2r}, φ̃_{m}^{2r} to the (2r+1)-skeleton X^{(2r+1)}. They can be considered as liftings of the map ψφ_{k} ≃ χφ_{m} to X^{(2r+1)}. Consider the distinguishing cochain d((φ̃_{k}^{2r+1})′, (φ̃_{m}^{2r+1})′) ∈ C^{2r+1}(X; π_{2r+1}(F_{km, n^{2}})) = C^{2r+1}(X; ℤ/kmℤ). The maps ψ, χ induce the monomorphisms of the cochain groups

ψ_{∗}: C^{2r+1}(X; ℤ/kℤ) → C^{2r+1}(X; ℤ/kmℤ),
χ_{∗}: C^{2r+1}(X; ℤ/mℤ) → C^{2r+1}(X; ℤ/kmℤ).

Moreover, since (k, m) = 1 by our assumption, we have

C^{2r+1}(X; ℤ/kmℤ) = C^{2r+1}(X; ℤ/kℤ) ⊕ C^{2r+1}(X; ℤ/mℤ).

Note also that the distinguishing cochains for liftings of φ_{k}, φ_{m} belong to C^{2r+1}(X; ℤ/kℤ) and C^{2r+1}(X; ℤ/mℤ), respectively. It is well-known from the obstruction theory that for an arbitrary cochain d_{1} ∈ C^{2r+1}(X; ℤ/kℤ) and (φ̃_{k}^{2r+1})′ as above there exists an extension φ̃_{k}^{2r+1} of the lifting φ̃_{k}^{2r} of φ_{k} such that d(φ̃_{k}^{2r+1}, (φ̃_{k}^{2r+1})′) = −d_{1}; similarly, for (φ̃_{m}^{2r+1})′ and d_{2} ∈ C^{2r+1}(X; ℤ/mℤ).

Suppose d((φ̃_{k}^{2r+1})′, (φ̃_{m}^{2r+1})′) = d_{1} − d_{2}, where d_{1} ∈ C^{2r+1}(X; ℤ/kℤ), d_{2} ∈ C^{2r+1}(X; ℤ/mℤ). Then

d(φ̃_{k}^{2r+1}, (φ̃_{k}^{2r+1})′) + d((φ̃_{k}^{2r+1})′, (φ̃_{m}^{2r+1})′) + d((φ̃_{m}^{2r+1})′, φ̃_{m}^{2r+1}) =
d(φ̃_{k}^{2r+1}, φ̃_{m}^{2r+1}) = −d_{1} + d_{1} − d_{2} + d_{2} = 0.

In other words, we can replace (φ̃_{k}^{2r+1})′ by another lifting of φ_{k} to the (2r + 1)-skeleton and (φ̃_{m}^{2r+1})′ by another lifting of φ_{m} to the (2r + 1)-skeleton such that the liftings extend the corresponding liftings φ_{k}^{2r} φ_{m}^{2r} and such that the obtained liftings are homotopic as maps from X^{(2r+1)} to Gr . Clearly, the liftings φ̃_{k}^{2r+1} φ̃_{m}^{2r+1} which satisfy this condition are unique up to homotopy.

Now extend φ̃_{k}^{2r+1}, φ̃_{m}^{2r+1} to the (2r + 2)-skeleton X^{(2r+2)}. We have well-known formula δd(f, g) = c(f) − c(g) in the obstruction theory, where c(f) and c(g) are the obstruction cochains for lifting of f and g, respectively, and δ is the coboundary map. Hence 0 = δd(φ̃_{k}^{2r+1}, φ̃_{m}^{2r+1}) = c(φ̃_{m}^{2r+1}) − c(φ̃_{k}^{2r+1}). But φ̃_{m}^{2r+1} is a lifting of φ_{m}, therefore c(φ̃_{k}^{2r+1}) ∈ C^{2r+1}(X; ℤ/kℤ); similarly, c(φ̃_{k}^{2r+1}) ∈ C^{2r+1}(X; ℤ/mℤ). Thus, c(φ̃_{m}^{2r+1}) = c(φ̃_{k}^{2r+1}) = 0. □

3.4 Localization

Let X be a finite CW -complex, k ≥ 2 a fixed integer. The set of isomorphism classes of bundles of the form A_{k^{m}} (for arbitrary m ∈ ℕ) over X with fiber M_{k^{m}}(ℂ) is a monoid with respect to the operation ⊗ (with the identity element M_{k^{0}}(ℂ) ≅ ℂ).

Let us consider the following equivalence relation

A_{k^{m}} ∼ B_{k^{n}} ⟺ ∃r, s ∈ ℕ such that A_{k^{m}} ⊗ M̃_{k^{r}} ≅ B_{k^{n}} ⊗ M̃_{k^{s}}  (22)

(⇒ m + r = n + s). The set of equivalence classes [A_{k^{m}}] is a group with respect to the operation induced by ⊗. This group we denote by ÃB^{k}(X).

Let us consider the direct limit lim_{\overrightarrow{n}} BPU(k^{n}) with respect to the maps induced by the group monomorphisms

\begin{matrix}PU(k^{n}) & ↪ & PU(k^{n+1}) \\ A & ↦ & A ⊗ E_{k} \end{matrix}

(here the symbol ⊗ denotes the Kronecker product of matrices). Clearly, the functor X ↦ ÃB^{k}(X) is represented by the space BPU(k^{∞}) := lim_{\overrightarrow{n}} BPU(k^{n}).

According to Lemma 17, for any stable equivalence class of FABs over X there is a representative of the form (A_{k^{m}}, μ, M̃_{(kl)^{m}}) (for some l coprime with k). Therefore, for any k we can define the natural transformation ÃB^{1}(X) → ÃB^{k}(X) by letting

[(A_{k^{m}}, μ, M̃_{(kl)^{m}})] ↦ [A_{k^{m}}].

More precisely, consider the diagram

[figure: _{Gr_{k^{2}, l^{2}}}; λ̄_{k^{2}}; BPU(k^{2}); _{Gr_{k, l}}; λ̄_{k}; BPU(k),]  (23)

where λ̄_{k} (resp. λ̄_{k^{2}}) is a classifying map for the bundle A_{k, l} as a PU(k)-bundle (resp. for A_{k^{2}, l^{2}} as a PU(k^{2})-bundle). The diagram determines the map of classifying spaces λ̄_{k^{∞}} := lim_{\overrightarrow{n}} λ̄_{k^{n}}: Gr → BPU(k^{∞}) such that λ̄_{k^{∞}∗}: [X; Gr] → [X; BPU(k^{∞})] is the natural

transformation ÃB^{1}(X) → ÃB^{k}(X) as above. Note that the kernel of the homomorphism λ̄_{k^{∞}∗}: ÃB^{1}(X) → ÃB^{k}(X) is the k-torsion subgroup in ÃB^{1}(X). Remark 24. Consider the following equivalence relation on the set of bundles over X with fiber a matrix algebra:

A_{k} ∼ B_{l} ⇔ ∃m, n ∈ ℕ such that A_{k} ⊗ M̃_{m} ≃ B_{l} ⊗ M̃_{n}.  (24)

Let ÃB(X) be the group of equivalence classes of the relation with respect to the operation induced by the tensor product of bundles. Then the functor X ↦ ÃB(X) is represented by the H-space lim_{\overrightarrow{k}} BPU(k) (it is the direct limit over all k ∈ ℕ with respect to the maps induced by PU(k) ↪ PU(m), A ↦ A⊗E\frac{m}{k} for any k|m). Note that π_{2}(lim_{\overrightarrow{k}} BPU(k)) = ℚ/ℤ (more precisely, it is the group μ_{∞} of finite degree roots of unity), π_{2r}(lim_{\overrightarrow{k}} BPU(k)) = ℚ, if r ≥ 2, and π_{2r+1}(lim_{\overrightarrow{k}} BPU(k)) = 0 ∀r ∈ ℕ. We have the natural transformation of functors

ÃB^{1} → ÃB, (A_{m}, μ, M̃_{mn}) ↦ [A_{m}] which can be passed through ÃB^{k} for any k > 1. Clearly, ker{ÃB^{1}(X) → ÃB(X)} is just the torsion subgroup in ÃB^{1}(X). Remark 25. Note that the equivalence relations (22) and (24) do not preserve the dimension of bundles. Hence, ÃB^{k} and ÃB are counterparts of the reduced K-theory. In order to obtain the nonreduced theory, for example, instead of (22) we should consider the following equivalence relation:

A_{k^{m}} ∼ B_{k^{n}} ⇔ m = n and there is l such that A_{k^{m}} ⊗ M̃_{k^{l}} ≅ B_{k^{m}} ⊗ M̃_{k^{l}},

and repeat the above procedure for relation (24). Denote the corresponding functors by AB^{k}, AB, respectively. The group AB(pt) can easily be described: it is the symmetrization of the (multiplicative) monoid ℕ^{∗}. Thus, AB(pt) ≅ ℚ_{+}^{∗}, where ℚ_{+}^{∗} is the group of positive rational numbers with respect to the multiplication. The group AB^{k}(pt) can be identified with the subgroup ℤ ↪ ℚ_{+}^{∗}, 1 ↦ k.

Theorem 26. There is a homotopy equivalence Gr ≃ BSU .

Proof. Let p, q be two prime numbers, p ≠ q. Clearly, Gr_{p^{k}, q^{k}} ≃ Gr_{q^{k}, p^{k}} . Consider the following homotopy commutative diagrams

[figure: _{Gr_{p^{2}, q^{2}}}; λ_{p^{2}}; BSU(p^{2}); ψ_{p^{2}}; _{Gr_{p, q}}; φ; λ_{p}; BSU(p),; ψ_{p}; _{Gr_{p^{2}, q^{2}}}; λ_{q^{2}}; BSU(q^{2}); ψ_{q^{2}}; _{Gr_{p, q}}; φ; λ_{q}; BSU(q),; ψ_{q}]

where λ_{p^{k}}, λ_{q^{k}} are classifying maps for the cores A_{p^{k}, q^{k}}, A_{q^{k}, p^{k}} of canonical FABs over Gr_{p^{k}, q^{k}} and Gr_{q^{k}, p^{k}}, respectively (see Lemma 21; notice that λ̄_{k} in (23) is the composition of λ_{k} with the map BSU(k) → BPU(k) induced by the natural epimorphism SU(k) → PU(k) with the kernel μ_{k}), the map φ is induced by a homomorphism of matrix algebras M_{pq}(ℂ) ↪ M_{(pq)^{2}}(ℂ), and the map ψ_{p^{k}} (resp. ψ_{q^{k}}) is induced by the homomorphism

SU(p^{k}) ↪^{...⊗E_{p}} SU(p^{k+1}) (resp. SU(q^{k}) ↪^{...⊗E_{q}} SU(q^{k+1}))

of Lie groups.

Note that the map λ_{p^{k}}: Gr_{p^{k}, q^{k}} → BSU(p^{k}) is a fibration with fiber Fr_{p^{k}, q^{k}} (compare with diagram (20)). A simple computations with homotopy groups show that the map

λ_{p^{∞}} := lim_{\overrightarrow{k}} λ_{p^{k}}: lim_{\overrightarrow{k}} Gr_{p^{k}, q^{k}} → lim_{\overrightarrow{k}}(BSU(p^{k}), ψ_{p^{k}})

is just the localization at p, similarly for q. In particular, BSU[\frac{1}{p}] (resp. BSU[\frac{1}{q}]) is a ℤ[\frac{1}{p}]-local space (resp. a ℤ[\frac{1}{q}]-local space). Thus, we have the fibered square

\xymatrix{lim_{\overrightarrow{k}} Gr_{p^{k}, q^{k}} \ar[r]^{λ_{q^{∞}}} \ar[d]_{λ_{p^{∞}}} & BSU[\frac{1}{q}] \ar[d]^{g} \\ BSU[\frac{1}{p}] \ar[r]_{f} & BSU[\frac{1}{pq}],}

where the maps f, g correspond to the homomorphisms ℤ[\frac{1}{p}] → ℤ[\frac{1}{pq}] and ℤ[\frac{1}{q}] → ℤ[\frac{1}{pq}] respectively. It is known [1] that the space

lim_{\overrightarrow{k}} Gr_{p^{k}, q^{k}} = BSU[\frac{1}{p}] ×_{BSU[ \frac{1}{pq}]}^{h} BSU[\frac{1}{q}]

is determined by the maps f, g uniquely up to a homotopy equivalence. Now, applying Theorem 16, item 1), we complete the proof. □

In the next section we construct a natural bijection between the sets ÃB^{1}(X) and { \atop KSU}(X). This gives another way to prove the homotopy equivalence Gr ≃ BSU (here { \atop KSU} is the reduced K-functor constructed by means of SU-bundles, recall that it is represented by the space BSU).

Now let us consider the following situation, in which FABs appear naturally. As usually, let X be a finite CW -complex, {k, l} a pair of natural numbers such that 2 min{k, l} ≥ dim X and (k, l) = 1. Let A_{kl} be a locally trivial bundle over X with fiber M_{kl}(ℂ). Suppose for simplicity that the structural group of A_{kl} is reducible to the special linear group SL_{kl}(ℂ) (i.e. actually to SU(kl)). Then it can be proved that there are bundles B_{k}, C_{l} over X with fibers M_{k}(ℂ), M_{l}(ℂ), respectively, such that their structural groups can be reduced to the corresponding special linear groups and A_{kl} ≅ B_{k} ⊗ C_{l}. Moreover, such pairs (B_{k}, C_{l}) are in one-to-one correspondence with stable equivalence classes of FABs over X.

In order to prove the described results, consider the fibration

Gr_{k, l} ↪^{κ} BSU(k) × BSU(l)
↓ ϑ
BSU(kl),

(25) where ϑ is induced by the tensor product of the universal SU-bundles over classifying spaces. It follows from the obstruction theory that an arbitrary map X → BSU(kl) can be lifted to the total space of the considered fibration. Thus, we have A_{kl} = B_{k} ⊗ C_{l} for some bundles B_{k}, C_{l}. Furthermore, the second part of the assertion follows from the exactness of the sequence

0 → [X; Gr_{k, l}] →^{κ_{∗}} [X; BSU(k) × BSU(l)] →^{ϑ_{∗}} [X; BSU(kl)] → 0

(let us remark that the injectivity of κ_{∗} follows from the obvious fact that a FAB (A_{k}, μ, M̃_{kl}) is trivial iff its core A_{k} and its complementary subbundle B_{l} (i.e. the subbundle B_{l} ⊂ M̃_{kl} such that A_{k} ⊗ B_{l} = M̃_{kl}, see page 12) are both trivial as locally trivial bundles with fibers M_{k}(ℂ) and M_{l}(ℂ), respectively). Remark 27. Consider also the sequence of fibrations (25):

Gr_{k^{n}, l^{n}} ↪^{κ_{k^{n}, l^{n}}} BSU(k^{n}) × BSU(l^{n})
↓ ϑ_{k^{n}, l^{n}}
BSU((kl)^{n})

(clearly, κ_{p, q} = λ_{p} × λ_{q} in the notation of the previous theorem) and its direct limit as n → ∞ :

Gr ↪^{κ_{k^{∞}, l^{∞}}} BSU(k^{∞}) × BSU(l^{∞}) ⟶^{ϑ_{k^{∞}, l^{∞}}} BSU((kl)^{∞}).

It can easily be checked that this fibration corresponds to the exact sequence of groups

0 → ℤ → ℤ[\frac{1}{k}] × ℤ[\frac{1}{l}] → ℤ[\frac{1}{kl}] → 0.

4 Relation between ÃB^{1} and K̃SU-theory

Let { \atop KSU}(X) be the reduced K-functor constructed by means of SU-bundles over X. Let k, m be a pair of positive integers such that (k, m) = 1. Without loss of generality we can assume that m > k. Suppose dim X ≤ 2k. Let ξ_{k} be a k-dimensional vector SU(k)-bundle over X. Let us consider the pair

ξ_{k} ⊗ [k] − [k(k − 1)], (ξ_{k} ⊕ [m − k]) ⊗ [m] − [m(m − 1)]  (27)

of virtual bundles of virtual dimensions k, m, respectively, where [n] is the trivial ndimensional bundle over X. By ξ_{k}^{•}, (ξ_{k} ⊕ [m − k])^{•} denote geometric representatives of stable equivalence classes (27) (since dim X ≤ 2 min{k, m}, the geometric representatives exist and are unique up to isomorphism).

Proposition 28. (End ξ_{k}^{•}) ⊗ M̃_{m} ≅ (End(ξ_{k} ⊕ [m − k])^{•}) ⊗ M̃_{k}.

Proof. It can easily be checked that ξ_{k}^{•} ⊗ [m] ≅ (ξ_{k} ⊕ [m − k])^{•} ⊗ [k]. □

Now we see that the pair (A_{k}, B_{m}) = (End(ξ_{k}^{•}), End(ξ_{k} ⊕[m−k])^{•}) satisfies the condition of the second part of Lemma 23, and we conclude that every pair ξ_{k}^{•}, (ξ_{k}⊕[m−k])^{•}, (k, m) = 1 determines a unique stable equivalence class of FABs over X. Moreover, one can easily verify that this assignment depends only on the stable equivalence class [ξ_{k}] ∈ { \atop KSU}(X) of the initial bundle ξ_{k}. Therefore, the corresponding natural transformation of functors

ϕ_{X}: { \atop KSU}(X) → ÃB^{1}(X)

is well-defined. Moreover, it is a bijection:

Proposition 29. For any finite CW -complex X the map ϕ_{X}: { \atop KSU}(X) → ÃB^{1}(X) is a natural bijection. Consequently, the spaces Gr and BSU are homotopy-equivalent.

Proof. In order to prove the theorem, we produce the inverse map ψ_{X}: ÃB^{1}(X) → { \atop KSU}(X) for ϕ_{X}. Let (ξ_{k}, ξ_{m}) be a pair consisting of SU(k) and SU(m)-bundle ((k, m) = 1) such that ξ_{k} ⊗ [m] ≅ ξ_{m} ⊗ [k]. Let us take l, n such that kl + mn = 1. Then in the group KSU(X) the identity ξ_{k} = klξ_{k} +mnξ_{k} = klξ_{k} +knξ_{m} = k(lξ_{k} +nξ_{m}) holds. Similarly, ξ_{m} = m(lξ_{k} +nξ_{m}). Thus, we have ξ_{k} = kη, ξ_{m} = mη for the virtual SU-bundle η (η = lξ_{k} + nξ_{m}) of virtual dimension 1. Clearly, the assignment (A_{k}, B_{m}) ↦ η (where A_{k} = End ξ_{k}, B_{m} = End ξ_{m}) is the required map ψ_{X} satisfying ψ_{X} ∘ ϕ_{X} = id_{{ \atop KSU}(X)}, ϕ_{X} ∘ ψ_{X} = id_{ÃB^{1}(X)} . □

Let us remember that BSU_{⊗} is the space BSU with the structure of H-space related to the tensor product of virtual SU-bundles of virtual dimension 1.

Theorem 30. The H-spaces Gr and BSU_{⊗} are isomorphic to each other.

Proof. Let ξ be a virtual SU-bundle over X of virtual dimension 1; ξ_{k}, ξ_{m} the geometric representatives of dimension k, m of the classes kξ, mξ, respectively (the numbers k, m are assumed to be sufficiently large). Given a virtual SU-bundle η of virtual dimension 1, the bundles η_{l}, η_{n} are defined in the same way. Clearly, ξ_{k} ⊗ η_{l} is a geometric representative of dimension kl of klξ ⊗ η ∈ KSU(X); the same relation holds between ξ_{m} ⊗ η_{n} and mnξ ⊗ η. Clearly, End(ξ_{k} ⊗η_{l})⊗M̃_{mn} ≅ End(ξ_{m} ⊗η_{n})⊗M̃_{kl}. Suppose (kl, mn) = 1; then, according to Lemma 23, the pair (End(ξ_{k} ⊗ η_{l}), End(ξ_{m} ⊗ η_{n})) determines the stable equivalence class of FABs over X. We claim that this class is equal to the product of the classes corresponding to the pairs (End ξ_{k}, End ξ_{m}) and (End η_{l}, End η_{n}). Indeed, this also follows from Lemma 23. □

We see that the group ÃB^{1}(X) is isomorphic to the multiplicative group of the ring { \atop KSU}(X), i.e. to the group of elements of { \atop KSU}(X) with respect to the operation ξ ∗ η :=

ξ + η + ξη (ξ, η ∈ { \atop KSU}(X)).

Note that we have obtained a geometric description of the H-space structure on BSU_{⊗} . For example, the construction of the inverse stable equivalence class [(B_{m}, ν, M̃_{mn})] for a given one [(A_{k}, μ, M̃_{kl})] is closely connected with taking centralizer for a subalgebra in a fixed matrix algebra.

Let us remark that the H-space BSU_{⊗} is an infinite loop space [2], so is Gr.

Now, using the described relation between ÃB^{1} and { \atop KSU}, we study (stable) characteristic classes of FABs. In particular, we obtain a formula which expresses characteristic classes of the product of stable equivalence classes of FABs in terms of the characteristic classes of factors (i.e. a counterpart of the Whitney formula).

Definition 31. Let (A_{k}, μ, M̃_{kl}) be a FAB over X (since we study the stable theory, without loss of generality we can assume that k, l are sufficiently large). Let ξ_{k} be an SU-bundle over X such that ϕ_{X}[ξ_{k}] = [(A_{k}, μ, M̃_{kl})] (in particular, A_{k} ≅ End ξ_{k}^{•}), where [ξ_{k}] ∈ { \atop KSU}(X) is the stable equivalence class of ξ_{k} and ϕ_{X} as in Proposition 29. Then, by definition, Chern class c̃_{i}((A_{k}, μ, M̃_{kl})) of (A_{k}, μ, M̃_{kl}) is the usual i-th Chern class c_{i}(ξ_{k}) of ξ_{k}. In the same way we define Newton classes (corresponding to the Newton polynomials) s̃_{i}((A_{k}, μ, M̃_{kl})), i ≥ 1. Furthermore, by definition, put s̃_{0}(A_{k}, μ, M̃_{kl}) = 1 for any FAB (this convention is justified in Remark 32 below).

Below we will write simple c̃_{i}(A_{k}) instead of c̃_{i}((A_{k}, μ, M̃_{kl})), similarly for Newton classes. It follows immediately from the definition that the introduced characteristic classes are

well defined on the stable equivalence classes of FABs. Clearly, H^{∗}(Gr ; ℤ) = ℤ[c̃_{2}, c̃_{3}, …]. Remark 32. Let ξ_{k}^{•} be an SU(k)-bundle such that End ξ_{k}^{•} ≅ A_{k}. Then for integer l > 1 such that (k, l) = 1 and 2l ≥ dim X there exists an SU(l)-bundle η_{l}^{•} such that ξ_{k}^{•} ⊗ [l] ≅ η_{l}^{•} ⊗ [k] (compare with Lemma 23). Then for the usual Newton class s_{i} (recall that s_{0}(ξ_{k}^{•}) = dim ξ_{k}^{•} = k) we have s_{i}(ξ_{k}^{•} ⊗ [l]) = s_{i}(η_{l}^{•} ⊗ [k]), i.e. ls_{i}(ξ_{k}^{•}) = ks_{i}(η_{l}^{•}). Now we see that the assignment

ξ_{k}^{•} ↦ \frac{s_{i}(ξ_{k}^{•})}{k} = s̃_{i}(End ξ_{k}^{•}) ∈ H^{2i}(X; ℤ), i ≥ 0,  (28)

determines a well-defined characteristic class on the stable equivalence classes of FABs (which coincides with the Newton class s̃_{i} defined above). Note that the cohomology H^{∗}(Gr ; ℤ) are torsion-free, this allows us to divide s_{i}(ξ_{k}^{•}) in (28) by k ∈ ℤ. Remark 33. It follows from Definition 31 that the relation between the classes c̃_{i} and s̃_{i} is the same as the one between the usual Chern and Newton classes:

s_{k} − s_{k−1}c_{1} + ⋯ + (−1)^{k}kc_{k} = 0,  k ≥ 1.  (29)

Now we want to deduce the counterpart for FABs of Whitney’s formula. Let (A_{k}, μ, M̃_{kl}) (B_{m}, ν, M̃_{mn}) be FABs over X, dim X ≤ 2 min{k, l, m, n}. Suppose ξ_{k}^{•}, η_{m}^{•} are SU-bundles such that A_{k} ≅ End ξ_{k}^{•}, B_{m} ≅ End η_{m}^{•}. Then there are SU-bundles ξ_{k}, η_{m} such that the bundles ξ_{k}^{•}, η_{m}^{•} are geometric representatives of the equivalence classes (ξ_{k} ⊗ [k] − [k(k − 1)]), (η_{m} ⊗[m]−[m(m−1)]), respectively. By (k, s_{2}, s_{3}, …), (m, s′_{2}, s′_{3}, …) we denote the Newton classes of ξ_{k}, η_{m}, respectively. Then the bundles ξ_{k}^{•}, η_{m}^{•} have the following Newton classes: (k, ks_{2}, ks_{3}, …) and (m, ms′_{2}, ms′_{3}, …) respectively. Recall that for any vector bundles ξ, η there is the relation

s_{r}(ξ ⊗ η) = ∑_{{i+j=r \atop i,j≥0}} \frac{r!}{i!j!}s_{i}(ξ)s_{j}(η)  (30)

(recall that s_{0}(ξ) = dim ξ) between Newton classes of ξ, η and the ones of their tensor product. From the other hand, we have:

ξ_{k}^{•} ⊗ η_{m}^{•} = ζ_{km}^{•} = ζ_{km} ⊗ [km] − [km(km − 1)]

for some SU(km)-bundle ζ_{km}.

Set s′′_{r} := s_{r}(ζ_{km}). Using (30), we get:

s′′_{r} = s_{r} + ∑_{{i+j=r \atop i,j≥1}} \frac{r!}{i!j!}s_{i}s′_{j} + s′_{r}, r ≥ 1

(note that cohomology groups H^{∗}(Gr ; ℤ) = H^{∗}(BSU; ℤ) have no torsion). Hence,

s̃_{r}(A_{k} ⊗ B_{m}) = s̃_{r}(A_{k}) + ∑_{{i+j=r \atop i,j≥1}} \frac{r!}{i!j!}s̃_{i}(A_{k})s̃_{j}(B_{m}) + s̃_{r}(B_{m})

= ∑_{{i+j=r \atop i,j≥0}} \frac{r!}{i!j!}s̃_{i}(A_{k})s̃_{j}(B_{m}), r ≥ 0,

because s̃_{0}(A_{k}) = 1 = s̃_{0}(B_{m}) by our convention.

Now, using Newton’s formulas (29) and Remark 33, one can obtain similar formulas for Chern classes. In particular,

c̃_{2}(A_{k} ⊗ B_{m}) = c̃_{2}(A_{k}) + c̃_{2}(B_{m})
c̃_{3}(A_{k} ⊗ B_{m}) = c̃_{3}(A_{k}) + c̃_{3}(B_{m})
c̃_{4}(A_{k} ⊗ B_{m}) = c̃_{4}(A_{k}) − 5c̃_{2}(A_{k})c̃_{2}(B_{m}) + c̃_{4}(B_{m})
c̃_{5}(A_{k} ⊗ B_{m}) = c̃_{5}(A_{k}) − 11c̃_{3}(A_{k})c̃_{2}(B_{m}) − 11c̃_{2}(A_{k})c̃_{3}(B_{m}) + c̃_{5}(B_{m}),

etc. Using the previous formulas, one can also obtain the expressions for Chern classes of the inverse FAB for a given one.

5 Appendix 1: a GL-version

In this appendix we introduce the “GL-version” of the previous results which is related to the H-space structure BU_{⊗} .

Consider the canonical map BU(k) → BPU(k) induced by the group homomorphism U(k) → PU(k). By Ĝr_{k, l} denote the total space of the Fr_{k, l}-fibration induced by the fibration Gr_{k, l} ⟶^{Fr_{k, l}} BPU(k) and the map BU(k) → BPU(k) (as ever, the integers k, l are assumed to be coprime).

It follows easily that there is a ℂP^{∞}-fibration Ĝr_{k, l} → Gr_{k, l}. Moreover, there is the diagram of fibrations:

Ĝr_{k, l}  ⟶^{Fr_{k, l}} BU(k)
ℂP^{∞} ↓  ↓ ℂP^{∞}
Gr_{k, l}  ⟶^{Fr_{k, l}} BPU(k)

(the spaces over/next to the arrows are the corresponding fibers).

Consider the following morphism of U(k)-fibrations:

\xymatrix{ & U(k) \ar[r] & ∗ \ar[d] \\ U(k) \ar[ur]^{=} \ar[r] & _{Fr_{k, l}} \ar[d] \ar[ur] & BU(k) \\  & Ĝr_{k, l}, \ar@{-->}[ur]^{λ̂_{k, l}}}

where λ̂_{k, l} is a classifying map for the canonical U(k)-bundle over Ĝr_{k, l} and by ∗ we denote a contractible space. A simple computation with homotopy sequences of the fibrations shows that λ̂_{k, l∗}: π_{2r}(Ĝr_{k, l}) → π_{2r}(BU(k)), r ≤ min{k, l} is just the monomorphism ℤ → ℤ, 1 ↦ k · 1 (note that the odd-dimensional stable homotopy groups of both spaces are equal to 0). This implies that the direct limit map λ̂_{k^{∞}}: Ĝr → BU(k^{∞}) is just the localization at k (in the sense that k is invertible; in particular, BU(k^{∞}) is a ℤ[\frac{1}{k}]-local space), where

Ĝr := lim_{\overrightarrow{(k,l)=1}} Ĝr_{k, l}.

The space Ĝr is an H-space with respect to the multiplication induced by the tensor product of bundles. It can be proved that Ĝr ≅ BU_{⊗} as H-spaces. Let us also recall that there are isomorphisms of H-spaces

BU_{⊗} ≅ BSU_{⊗} ×ℂP^{∞}

and Gr ≅ BSU_{⊗}, hence Ĝr ≅ Gr ×ℂP^{∞}. In particular, the H-space Ĝr represents the functor of “multiplicative group” of the ring K̃_{ℂ}, i.e. the functor X ↦ K̃_{ℂ}(X), where K̃_{ℂ}(X) is considered as a group with respect to the operation ξ ∗ η := ξ + η + ξη, ξ, η ∈ K̃_{ℂ}(X) (here K̃_{ℂ} is the reduced complex K-functor).

6 Appendix 2: A remark on Thom spaces

In algebraic topology for any stable type of bundles there is the corresponding bordism (and cobordism) theory. Therefore, it is natural to ask the following question: what is the bordism theory corresponding to FABs? In this appendix we define an analog of Thom spaces in this hypothetical theory.

In order to construct the analog of Thom spaces, we shall use the interpretation of floating bundles in term of bundles on projective spaces given in Remark 4.

First let us consider the embedding of pairs

\begin{matrix}(ℂP^{k} × ℂP^{l}, ℂP^{k} × ℂP^{l−1} ∪ ℂP^{k−1} × ℂP^{l}) ↪ (ℂP^{kl}, ℂP^{kl−1}), \\ ([x_{0} : …: x_{k−1} : x_{k}], [y_{0} : …: y_{l−1} : y_{l}]) ↦ [x_{0}y_{0} : …: x_{i}y_{j} : …: x_{k−1}y_{l−1} : x_{k}y_{l}] \end{matrix} (37)

(note that the subspaces ℂP^{k−1} ⊂ ℂP^{k} and ℂP^{l−1} ⊂ ℂP^{l} are defined by the equations x_{k} = 0 and y_{l} = 0, respectively). Using the well-known formula X × Y/(X × B ∪ A × Y) ≅ (X/A) ∧ (Y/B), we get the map

S^{2k} ∧ S^{2l} → S^{2kl},  (38)

which corresponds (37).

Recall that the definition of a floating bundle of Segre’s product (abbrev. FBSP) was given in Definition 5. Clearly, the space Gr_{k, l} parameterizes embeddings ℂP^{k−1} × ℂP^{l−1} ↪ ℂP^{kl−1} that under some choice of homogeneous coordinates are equivalent to Segre’s embedding ([x_{0} : …: x_{k−1}], [y_{0} : …: y_{l−1}]) ↦ [x_{0}y_{0} : …: x_{i}y_{j} : …: x_{k−1}y_{l−1}].

Clearly, there is the canonical FBSP P^{k−1} ×_{Gr_{k, l}}Q^{l−1} over Gr_{k, l} equipped with the canonical

embedding λ: P^{k−1} ×_{Gr_{k, l}}Q^{l−1} ↪ Gr_{k, l} ×ℂP^{kl−1} into the trivial bundle (for any x ∈ Gr_{k, l} its

restriction λ |_{x} to the fiber over x coincides with the embedding ℂP^{k−1} × ℂP^{l−1} ↪ ℂP^{kl−1} parameterized by x).

It follows from Lemma 21 that there are SU(k) and SU(l)-bundles 𝔗_{k}, 𝔖_{l} over Gr_{k, l} such that P^{k−1} = ℙ(𝔗_{k}), Q^{l−1} = ℙ(𝔖_{l}), where ℙ(ξ_{m}) is the projectivization of a vector bundle ξ_{m}. Clearly, 𝔗_{k} ⊗ 𝔖_{l} is a trivial ℂ^{kl}-bundle [kl] over Gr_{k, l} .

Now consider the following commutative diagram of the embeddings:

ℙ(𝔗_{k} ⊕ [1]) ×_{Gr_{k, l}}ℙ(𝔖_{l}) ∪ ℙ(𝔗_{k}) ×_{Gr_{k, l}}ℙ(𝔖_{l} ⊕ [1]) ⟶ ℙ(𝔗_{k} ⊕ [1]) ×_{Gr_{k, l}}ℙ(𝔖_{l} ⊕ [1])
↓  ↓
ℙ(𝔗_{k} ⊗ 𝔖_{l}) = ℙ([kl])  ⟶  ℙ((𝔗_{k} ⊗ 𝔖_{l}) ⊕ [1])

and the corresponding map of the pairs

(ℙ(𝔗_{k} ⊕ [1]) ×_{Gr_{k, l}}ℙ(𝔖_{l} ⊕ [1]), ℙ(𝔗_{k} ⊕ [1]) ×_{Gr_{k, l}}ℙ(𝔖_{l}) ∪ ℙ(𝔗_{k}) ×_{Gr_{k, l}}ℙ(𝔖_{l} ⊕ [1]))

↪ (ℙ((𝔗_{k} ⊗ 𝔖_{l}) ⊕ [1]), ℙ(𝔗_{k} ⊗ 𝔖_{l}))

(compare with (37)), where by [n] we denote a trivial ℂ^{n}-bundle. Thus, we have the canonical map of Thom spaces

T(𝔗_{k}) ∧_{Gr_{k, l}}T(𝔖_{l}) ↪ T([kl]),

(compare with (38)), where T([kl]) is the Thom space of the trivial bundle [kl] = 𝔗_{k} ⊗ 𝔖_{l} over Gr_{k, l}, T(𝔗_{k}) ∧_{Gr_{k, l}}T(𝔖_{l}) is the smash product over Gr_{k, l} of the Thom spaces T(𝔗_{k}) and T(𝔖_{l}).

Finally, let us remark that the natural stabilization on such Thom spaces does not coincide with the usual suspension. This is closely related to the stabilization of FABs: we stabilize our bundles by taking the tensor product with trivial bundles.

References

[1] M.M. Postnikov Localization of topological spaces. (Russian) Uspehi Mat. Nauk 32 (1977), no. 6(198), 117–181, 287. {English translation: Russian Math. Surveys 32 (1977), no. 6, 121–184.}

[2] G.B. Segal Categories and cohomology theories. Topology 13 (1974), 293–312.

[3] A.V. Ershov On the K-theory of fibre bundles for matrix algebras. (Russian) Uspehi Mat. Nauk 55 (2000), no. 2(332), 137–138. {English translation: Russian Math. Surveys 55 (2000), no. 2, 336–337.}

[4] A.V. Ershov On homotopy properties of bundles with a structural group of automorphisms of matrix algebras. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1999, no.6, 56–61. {English translation: Moscow Univ. Math. Bull. 54 (1999), no. 6, 43–46.}