A generalization of the topological Brauer group

A.V. Ershov^{∗}

Abstract In the present paper we introduce some generalization of the topological Brauer

group H_{tors}^{3}(X; ℤ). We hope that this construction can be transferred to algebraic geometry and would provide a basis for introducing new birational invariants of varieties.

Contents

1 Fans 𝕄_{k, l} 3 1.1 The definition of 𝕄_{k, l} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The topology on 𝕄_{k, l} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The algebraic structure on 𝕄_{k, l} . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 The space of k-frames in M_{kl}(ℂ) . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The action ν_{k, l}: 𝔊_{k, l} × Fr_{k, l} → Fr_{k, l} . . . . . . . . . . . . . . . . . . . . . . . 7

2 𝕄_{k, l}-bundles 8 2.1 An exact sequence of functors . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Relation to the frame spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 A stable theory of 𝕄_{k, l}-bundles 13 3.1 A weak homotopy equivalence 𝔊_{k, l^{∞}} ≃ Fr_{k, l^{∞}} . . . . . . . . . . . . . . . . . 13 3.2 The tensor product of fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 A stable equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 A GL-version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 0 → ℤ → ℚ → ℚ/ℤ → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 A generalization of the topological Brauer group 24 4.1 More about H-spaces B GL(ℂ) and B PGL(ℂ) . . . . . . . . . . . . . . . . . 24 4.2 Reminder: the classical Brauer group . . . . . . . . . . . . . . . . . . . . . . 25 4.3 The generalized Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Appendix 1: The functor represented by the H-space Gr 29

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

5.1 Category 𝔉𝔄𝔅(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.4 The group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.5 One interesting property of a bundle A_{k} which is a part of FAB (A_{k}, μ, M̃_{kl}) 33 5.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.7 Relation between ÃB^{1} and K̃SU-theory . . . . . . . . . . . . . . . . . . . . . 34 5.8 A GL-version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Appendix 2: The space Fr_{k^{∞}} is a loop space 35

Introduction

The aim of this paper is to introduce a homotopy functor GBr which can be viewed as a generalization of the “topological” Brauer group Br.

Suppose X is a finite CW -complex. The topological Brauer group Br(X) can be defined as the group of equivalence classes of locally-trivial bundles A_{k} over X with fibers M_{k}(ℂ) (for arbitrary k ∈ ℕ) with respect to the following equivalence relation:

A_{k} ∼ B_{l} ⇔ A_{k} ⊗ End(ξ_{m}) ≅ B_{l} ⊗ End(η_{n})

for some complex vector bundles ξ_{m}, η_{n} of rank m, n respectively (in particular, km = ln). Roughly speaking, we can say that Br(X) is the group of obstructions to the lifting of locally trivial bundles A_{k} over X with fiber the matrix algebra M_{k}(ℂ) to bundles of the form End(ξ_{k}) for some locally trivial ℂ^{k}-bundle ξ_{k} over X.

In the case of our generalized Brauer group GBr, instead of bundles with fibers M_{k}(ℂ), k ∈ ℕ, we consider bundles 𝔄_{k, l} whose fibers are “fans” 𝕄_{k, l}, k, l ∈ ℕ, (k, l) = 1. Similar to the usual Brauer group, we define GBr(X) as the group of obstructions for the lifting of bundles 𝔄_{k, l} to bundles of the form F_{k, l}^{X}(End(ξ_{k})), where ξ_{k} is some locally trivial ℂ^{k}-bundle over X. Here F_{k, l}^{X}: 𝔄𝔅_{k}(X) → 𝔉𝔞𝔫_{k, l}(X) is a functor from the category of M_{k}(ℂ)-bundles to the category of 𝕄_{k, l}-bundles over X; the functor is induced by the map M_{k}(ℂ) ↪ 𝕄_{k, l} (see Subsection 2.1). Compared to the classical case, a whole new step is added to the procedure of the lifting. At first, we have to lift locally trivial bundles 𝔄_{k, l} with fibers 𝕄_{k, l} to bundles of the form F_{k, l}^{X}(A_{k}) for some A_{k}. The second step essentially coincides with the lifting of bundles A_{k} to bundles of the form End(ξ_{k}) for some locally trivial bundle ξ_{k}, i.e. with the classical case.

According to a classical theorem of J.-P. Serre, there is an isomorphism Br(X) ≅ H_{tors}^{3}(X; ℤ) [1]. In other words, all obstructions for the lifting of locally-trivial bundles A_{k} over X with fiber M_{k}(ℂ) to bundles of the form End(ξ_{k}) are independent of the higherdimensional integer cohomologies of dimensions greater than 3. In contrast to the classical case, the generalized Brauer group GBr(X) actually depends on the higher-dimensional cohomologies of X.

On the other hand, just as in the classical group Br(X), any element of GBr(X) has finite order (moreover, the order of the equivalence class represented by a fan bundle 𝔄_{k, l} with fiber 𝕄_{k, l} is a power of k). Acknowledgments I would like to thank the Max Planck Institute for Mathematics in Bonn for their kind support and generous hospitality while this work was being completed. Regretfully, I no longer have a chance to express my gratitude to A.N. Tyurin , who was the first to listen to a preliminary version of this work and whose kind support was very important to me. I am grateful to E.V. Troitsky for constant attention to this work and allround support. My special thanks to E.L. Pervova for very helpfull discussions concerning many aspects of the present research. A number of related questions were discussed with M.R. Bakuradze, Yu.B. Rudyak, and V.E. Voskressenski at Max Planck Institute and I would like to thank them too.

1 Fans 𝕄_{k, l}

1.1 The definition of 𝕄_{k, l}

Let us fix two coprime integers k, l. By M_{n}(ℂ) we denote the algebra of all n × n matrices over ℂ. Let 𝔸_{k, l} be the set of all central simple subalgebras in M_{kl}(ℂ) that are isomorphic to M_{k}(ℂ), and for any α ∈ 𝔸_{k, l} let M_{k, α} ⊂ M_{kl}(ℂ) be the corresponding subalgebra.

Definition 1. The set of matrices

𝕄_{k, l} := ⋃_{α∈𝔸_{k, l}}M_{k, α} ⊂ M_{kl}(ℂ)

is called a fan of k-subalgebras in M_{kl}(ℂ).

A fan 𝕄_{k, l} will be equipped with some algebraic and topological structures but first, let us give a few other (equivalent) descriptions of 𝕄_{k, l}.

Noether-Scolem’s theorem claims that all subalgebras in M_{kl}(ℂ) isomorphic to M_{k}(ℂ) are conjugate to each other. In particular, any one of them is conjugate to the so-called “standard subalgebra” M_{k}(ℂ)⊗_{ℂ}ℂE_{l} ⊂ M_{kl}(ℂ), where E_{l} is the unit l × l matrix. Hence

𝕄_{k, l} = ⋃_{g∈GL_{kl}(ℂ)}g(M_{k}(ℂ)⊗_{ℂ}ℂE_{l})g^{−1} ⊂ M_{kl}(ℂ).

In order to obtain yet another description of 𝕄_{k, l}, we consider the matrix Grassmannian Gr_{k, l} := PGL_{kl}(ℂ)/ PGL_{k}(ℂ)⊗PGL_{l}(ℂ) (it is a homogeneous space and there is a canonical bijection of sets Gr_{k, l} ≅ 𝔸_{k, l}, see Appendix 1). Let A_{k, l} → Gr_{k, l} be the tautological bundle over Gr_{k, l} with fiber M_{k}(ℂ) (see Appendix 1). There is a canonical embedding μ of A_{k, l} into the trivial bundle Gr_{k, l} ×M_{kl}(ℂ):

\xymatrix{A_{k, l} \ar[dr] \ar[rr]^{μ} &  & Gr_{k, l} ×M_{kl}(ℂ) \ar[dl]^{π_{1}} \\  & Gr_{k, l},}

where for any x ∈ Gr_{k, l} the fiber (A_{k, l})_{x} is embedded by μ into the fiber (Gr_{k, l} ×M_{kl}(ℂ))_{x} = M_{kl}(ℂ) as the central simple subalgebra corresponding to x. Consider the composite map

A_{k, l} →^{μ} Gr_{k, l} ×M_{kl}(ℂ) →^{π_{2}} M_{kl}(ℂ).

Clearly, 𝕄_{k, l} = im(π_{2} ∘ μ) ⊂ M_{kl}(ℂ). Remark 2. Notice the following property of fans. Consider the fans 𝕄_{k, l} and 𝕄_{l, k} as contained in the same algebra M_{kl}(ℂ). Then their intersection 𝕄_{k, l} ∩ 𝕄_{l, k} is just the center ℂE_{kl} of M_{kl}(ℂ) (the reason being that k, l are assumed to be coprime). Indeed, this result can be proved by using Jordan’s normal form of matrices and the assumption (k, l) = 1.

1.2 The topology on 𝕄_{k, l}

Any fan 𝕄_{k, l} is a subset in the Euclidean space M_{kl}(ℂ). By O denote the induced topology on 𝕄_{k, l}. It follows immediately from the definition that O is the weakest topology on 𝕄_{k, l} such that the embedding 𝕄_{k, l} ↪ M_{kl}(ℂ) is continuous. Furthermore, for any α ∈ 𝔸_{k, l} the embedding M_{k, α} ↪ 𝕄_{k, l} is a continuous map.

1.3 The algebraic structure on 𝕄_{k, l}

The set 𝕄_{k, l} is not a ℂ-algebra, moreover, not even a vector space, in any natural sense. But for elements in the same k-subalgebra in 𝕄_{k, l} we have partial operations “ · ” and “+” induced by the usual multiplication and addition.

Since all k-subalgebras in 𝕄_{k, l} contain the identity element, their intersection includes the set of scalar matrices ℂE_{kl}. In particular, the vector space structures over the center ℂE_{kl} ≅ ℂ on distinct k-subalgebras are concordant.

Proposition 3. The intersection of all k-subalgebras in 𝕄_{k, l} is the center ℂE_{kl} ⊂ 𝕄_{k, l}, where E_{kl} is the unit kl × kl-matrix. In the previous notation,

⋂_{α∈𝔸_{k, l}}M_{k, α} = ℂE_{kl}.

Proof. Clearly, ∩_{α}M_{k, α} is a subalgebra in M_{kl}(ℂ); moreover, it is invariant with respect to all automorphisms of the algebra M_{kl}(ℂ). The required assertion follows from the well-known decomposition of the adjoint representation Ad of GL_{n}(ℂ) into irreducible ones. □

Definition 4. An isomorphism of fans ϑ: 𝕄_{k, l} → 𝕄_{k, l} is a homeomorphism of the topological spaces {𝕄_{k, l}, O} such that for any k-subalgebra M_{k, α} ⊂ 𝕄_{k, l} the restriction ϑ |_{M_{k, α}} is a ℂ-algebras isomorphism with some k-subalgebra M_{k, β} ⊂ 𝕄_{k, l} (α, β ∈ 𝔸_{k, l}), i.e. for any α ∈ 𝔸_{k, l} the corresponding diagram

𝕄_{k, l} →^{ϑ} 𝕄_{k, l}
⋃  ⋃
M_{k, α} →^{≅} M_{k, β}

commutes (here β = ϑ̄(α), where ϑ̄ is the induced map on 𝔸_{k, l}).

By 𝔊_{k, l} denote the full group of automorphisms Aut(𝕄_{k, l}). We shall consider 𝔊_{k, l} as a topological group with respect to the compact-open topology related to the action 𝔊_{k, l} × 𝕄_{k, l} → 𝕄_{k, l}. In other words, a subbase of the open sets of 𝔊_{k, l} is formed by sets W (C, U) := {g ∈ 𝔊_{k, l} | g·C ⊂ U} for all open U ⊂ 𝕄_{k, l} and compact C ⊂ 𝕄_{k, l}. Since 𝕄_{k, l} is a Hausdorff space, so is 𝔊_{k, l}. Furthermore, since 𝕄_{k, l} is regular and locally compact, then the natural maps, the multiplication 𝔊_{k, l} ×𝔊_{k, l} → 𝔊_{k, l} and the action 𝔊_{k, l} ×𝕄_{k, l} → 𝕄_{k, l}, (g, x) ↦ gx, are continuous. Remark 5. Note that the continuity of the reversal map 𝔊_{k, l} → 𝔊_{k, l}, g ↦ g^{−1} is not obvious in our case. One way to eliminate this defect is to use a modification of the compact-open topology given by Arens (see [2], p.20).

Acting on M_{kl}(ℂ), the group PGL_{kl}(ℂ) preserves the subset 𝕄_{k, l} ⊂ M_{kl}(ℂ). Hence we have a group homomorphism

κ_{k, l}: PGL_{kl}(ℂ) → 𝔊_{k, l}.

Proposition 6. The homomorphism κ_{k, l} is injective. Moreover, it is a continuous map.

Proof. Obviously, Z_{PGL_{kl}(ℂ)}(M_{k}(ℂ)⊗_{ℂ}ℂE_{l}) = E_{k} ⊗ PGL_{l}(ℂ) ⊂ PGL_{kl}(ℂ), where by Z we denote a centralizer and by E_{k} ⊗ PGL_{l}(ℂ) we denote the image of the embedding

PGL_{l}(ℂ) ∋ A ↦ E_{k} ⊗ A ∈ PGL_{kl}(ℂ),

induced by the Kronecker product of matrices. It is sufficient to prove that

⋂_{g∈GL_{kl}(ℂ)}g(E_{k} ⊗ PGL_{l}(ℂ))g^{−1} = {e}.

We have already proven that

⋂_{g∈GL_{kl}(ℂ)}g(ℂE_{k}⊗_{ℂ}M_{l}(ℂ))g^{−1} = ℂE_{kl}.

Hence

GL_{kl}(ℂ) ∩ ⋂_{g∈GL_{kl}(ℂ)}g(ℂE_{k}⊗_{ℂ}M_{l}(ℂ))g^{−1} = GL_{kl}(ℂ) ∩ ℂE_{kl} = ℂ^{∗}E_{kl}.

Let us factorize by ℂ^{∗}E_{kl}. The right-hand side is clearly equal to

(GL_{kl}(ℂ) ∩ ℂE_{kl})/ℂ^{∗}E_{kl} = ℂ^{∗}E_{kl}/ℂ^{∗}E_{kl} = {e}.

The left-hand side is actually equal to

⋂_{g∈GL_{kl}(ℂ)}g(E_{k}⊗ GL_{l}(ℂ))g^{−1},

therefore when factorized by ℂ^{∗}E_{kl}, it gives us

{e} = ( ⋂_{g∈GL_{kl}(ℂ)}g(E_{k}⊗ GL_{l}(ℂ))g^{−1})/ℂ^{∗}E_{kl} = ⋂_{g∈GL_{kl}(ℂ)}g(E_{k}⊗ GL_{l}(ℂ))g^{−1}/ℂ^{∗}E_{kl}

= ⋂_{g∈GL_{kl}(ℂ)}g(E_{k}⊗ GL_{l}(ℂ)/ℂ^{∗}E_{kl})g^{−1} = ⋂_{g∈GL_{kl}(ℂ)}g(E_{k} ⊗ PGL_{l}(ℂ))g^{−1}.

Thus, we get the desired equality.

Recall that the standard topology on PGL_{kl}(ℂ) coincides with the compact-open topology related to the action PGL_{kl}(ℂ) × M_{kl}(ℂ) → M_{kl}(ℂ). Take a compact subset C ⊂ 𝕄_{k, l}, and an open one U ⊂ 𝕄_{k, l}. Put W′(C, U) := {g ∈ PGL_{kl}(ℂ) | g · C ⊂ U} ⊂ PGL_{kl}(ℂ). Take an open subset V ∈ M_{kl}(ℂ) such that U = V ∩ 𝕄_{k, l}. Consider C as a subset in M_{kl}(ℂ). Then the subset W′′(C, V) ⊂ PGL_{kl}(ℂ) (related to the action PGL_{kl}(ℂ) × M_{kl}(ℂ) → M_{kl}(ℂ)) is open with respect to the standard topology on PGL_{kl}(ℂ). From the other hand, since 𝕄_{k, l} is a PGL_{kl}(ℂ)-invariant subset in M_{kl}(ℂ), we have W′(C, U) = W′′(C, V). Clearly, W′(C, U) = W (C, U)∩PGL_{kl}(ℂ), where 𝔊_{k, l} ⊃ W (C, U) := {g ∈ 𝔊_{k, l} | g ·C ⊂ U}. Finally note that subsets of the form W (C, U) form a subbase of the compact-open topology in 𝔊_{k, l}. □

Thus, the group 𝔊_{k, l} contains PGL_{kl}(ℂ) as a subgroup, which consists of all automorphisms of 𝕄_{k, l} that can be extended to automorphisms of the matrix algebra M_{kl}(ℂ).

Consider an embedding M_{k}(ℂ) → 𝕄_{k, l}, which maps M_{k}(ℂ) isomorphically onto a ksubalgebra in 𝕄_{k, l}.

Proposition 7. The embedding M_{k}(ℂ) → 𝕄_{k, l} is functorial.

Proof. Any automorphism θ of a matrix k-subalgebra A_{k} ⊂ M_{kl}(ℂ) can be uniquely extended to the automorphism θ̃ of the whole algebra M_{kl}(ℂ) such that θ̃ |_{Z(A_{k})}= id_{Z(A_{k})}, where Z(A_{k}) ≅ M_{l}(ℂ) is the centralizer of the subalgebra A_{k} in M_{kl}(ℂ). Actually, the assignment θ ↦ θ̃ corresponds to the group monomorphism PGL_{k}(ℂ) ⟶^{...⊗E_{l}} PGL_{kl}(ℂ). Thus, we obtain the automorphism ϑ = κ_{k, l}(θ̃) of the fan 𝕄_{k, l}, where κ_{k, l}: PGL_{kl}(ℂ) → 𝔊_{k, l} is the group monomorphism as above. □ Remark 8. Let us remark that the group monomorphism PGL_{k}(ℂ) ↪ 𝔊_{k, l} corresponded to the embedding M_{k}(ℂ) → 𝕄_{k, l}, is continuous. Indeed, it is the composite homomorphism PGL_{k}(ℂ) ↪^{...⊗E_{l}} PGL_{kl}(ℂ) ↪^{κ_{k, l}} 𝔊_{k, l} and therefore the required assertion follows from the previous proposition.

1.4 The space of k-frames in M_{kl}(ℂ)

Definition 9. A k-frame in M_{kl}(ℂ) is an ordered collection of k^{2} linearly independent matrices {α_{i, j} | 1 ≤ i, j ≤ k} such that α_{i, j}α_{m, n} = δ_{jm}α_{i, n} for all 1 ≤ i, j, m, n ≤ k, where δ_{jm} is the Kronecker delta-symbol.

Clearly, every such frame is a basis in a certain k-subalgebra in M_{kl}(ℂ). We have the “standard” k-frame {e_{i, j} | 1 ≤ i, j ≤ k}, where e_{i, j} := E_{ij} ⊗ E_{l} is the Kronecker product of a “matrix unit” E_{ij} of order k with the unit l × l matrix E_{l}. It is a frame in the subalgebra M_{k}(ℂ)⊗_{ℂ}ℂE_{l} ⊂ M_{kl}(ℂ). Applying Noether-Scolem’s theorem, we see that all k-frames in M_{kl}(ℂ) are conjugate to each other. Thus, the set of k-frames in M_{kl}(ℂ) is the homogeneous space Fr_{k, l} = PGL_{kl}(ℂ)/E_{k} ⊗PGL_{l}(ℂ). It is the total space of the following principal bundle

\xymatrix{PGL_{k}(ℂ) \ar[r]^{...⊗E_{l}} & _{Fr_{k, l}} \ar[d] \\  & _{Gr_{k, l}}}

over the matrix Grassmannian Gr_{k, l} . The fiber of the bundle over x ∈ Gr_{k, l} consists of all k-frames contained in the k-subalgebra M_{k, x} ⊂ M_{kl}(ℂ) parameterized by x. The tautological M_{k}(ℂ)-bundle A_{k, l} over Gr_{k, l} is associated with this principal bundle.

1.5 The action ν_{k, l}: 𝔊_{k, l} × Fr_{k, l} → Fr_{k, l}

The group 𝔊_{k, l} acts upon 𝕄_{k, l} and therefore upon the space of k-frames Fr_{k, l}. We shall consider Fr_{k, l} as a topological space with respect to the standard topology of the homogeneous space PGL_{kl}(ℂ)/E_{k} ⊗ PGL_{l}(ℂ).

Proposition 10. The natural action ν_{k, l}: 𝔊_{k, l} × Fr_{k, l} → Fr_{k, l} is continuous.

Proof. We have k^{2} continuous maps α̃_{m, n}: Fr_{k, l} → 𝕄_{k, l}, 1 ≤ m, n ≤ k which to any k-frame {α_{i, j} | 1 ≤ i, j ≤ k} assign its m, n-component α_{m, n} ∈ 𝕄_{k, l}. Clearly, the product map α̃ := α̃_{1, 1} × … × α̃_{k, k}: Fr_{k, l} → 𝕄_{k, l}^{×k^{2}} is an embedding. Obviously, the topology on Fr_{k, l} coincides with the induced topology related to the embedding Fr_{k, l} α̃ ; ↪ 𝕄_{k, l}^{×k^{2}} ⊂ M_{kl}(ℂ)^{×k^{2}}. Now the required assertion follows from the commutative diagram

\begin{matrix}𝔊_{k, l} × Fr_{k, l} & ⟶^{ν_{k, l}} & Fr_{k, l} \\ id_{𝔊_{k, l}} ×α̃ ↓ & & ↓ α̃ \\ 𝔊_{k, l} × 𝕄_{k, l}^{×k^{2}} & ⟶ & 𝕄_{k, l}^{×k^{2}}, \end{matrix}  (2)

where 𝔊_{k, l} × 𝕄_{k, l}^{×k^{2}} → 𝕄_{k, l}^{×k^{2}} is the diagonal action. □ Remark 11. Note that the action ν_{k, l}: 𝔊_{k, l} ×Fr_{k, l} → Fr_{k, l} preserves the structure of the fiber bundle Fr_{k, l} ⟶^{PGL_{k}(ℂ)} Gr_{k, l} . Since the topology on Gr_{k, l} coincides with the quotient topology, the corresponding action 𝔊_{k, l}×Gr_{k, l} → Gr_{k, l} is continuous. Furthermore, naturally acting on the direct product Gr_{k, l} ×𝕄_{k, l}, the group 𝔊_{k, l} preserves the tautological M_{k}(ℂ)-subbundle A_{k, l} ⊂ Gr_{k, l} ×𝕄_{k, l}. Since the standard topology on A_{k, l} coincides with the topology induced by the natural embedding A_{k, l} ↪ Gr_{k, l} ×𝕄_{k, l}, the induced action 𝔊_{k, l} × A_{k, l} → A_{k, l} is also continuous.

Suppose St(x) ⊂ 𝔊_{k, l} is the stabilizer of x ∈ Fr_{k, l}, then St(y) = g St(x)g^{−1} for y = gx.

Proposition 12.

⋂_{y∈Fr_{k, l}} St(y) = ⋂_{g∈𝔊_{k, l}}g St(x)g^{−1} = {e} ⊂ 𝔊_{k, l}.

In other words, the action ν_{k, l}: 𝔊_{k, l} × Fr_{k, l} → Fr_{k, l} is effective.

Proof. Suppose h ∈ ∩_{g∈𝔊_{k, l}}g St(x)g^{−1}, then h keeps all points of Fr_{k, l} fixed ⇒ h keeps all k-frames in 𝕄_{k, l} fixed ⇒ for any k-subalgebra A_{k} ⊂ 𝕄_{k, l} the restriction h |_{A_{k}} is the identity homomorphism id_{A_{k}} ⇒ h is the identity map of the fan 𝕄_{k, l} ⇒ h = e. □

Thus, the group 𝔊_{k, l} is a group of homeomorphisms of Fr_{k, l} .

Proposition 13. The action map ν_{k, l} equips Fr_{k, l} with the structure of homogeneous space over the group 𝔊_{k, l}.

Proof easily follows from the observation (which is a consequence of Noether-Scolem’s theorem) that even the subgroup PGL_{kl}(ℂ) ⊂ 𝔊_{k, l} acts transitively on Fr_{k, l} . □ Remark 14. Consider the standard Hermitian metric on M_{kl}(ℂ) given by ⟨A, B⟩ = tr(A\overline{B}^{t}). We say that a k-subalgebra M_{k, α} ⊂ M_{kl}(ℂ), α ∈ 𝔸_{k, l}, is unitary if there is a unitary transformation in Aut(M_{kl}(ℂ)) which takes the standard subalgebra M_{k}(ℂ)⊗_{ℂ}ℂE_{l} onto the subalgebra M_{k, α}. In other words, M_{k, α} = g(M_{k}(ℂ)⊗_{ℂ}ℂE_{l})g^{−1} for some g ∈ U(kl) ⊂ GL_{kl}(ℂ). Let 𝔸_{k, l}^{U} be the set of unitary k-subalgebras in M_{kl}(ℂ). By definition, put

𝕄_{k, l}^{U} := ⋃_{α∈𝔸_{k, l}^{U}}M_{k, α} = ⋃_{g∈U(kl)}g(M_{k}(ℂ)⊗_{ℂ}ℂE_{l})g^{−1} ⊂ 𝕄_{k, l}.

Automorphisms of 𝕄_{k, l}^{U} can be defined in the same way as for 𝕄_{k, l}. By 𝔊_{k, l}^{U} denote the group Aut(𝕄_{k, l}^{U}). In this “unitary” version we also replace the spaces Gr_{k, l} and Fr_{k, l} by the homotopy equivalent compact ones Gr_{k, l}^{U} := PU(kl)/ PU(k) ⊗ PU(l) and Fr_{k, l}^{U} := PU(kl)/E_{k} ⊗ PU(l) respectively. Note that Gr_{k, l}^{U} and Fr_{k, l}^{U} are just the Grassmannian of unitary k-subalgebras in M_{kl}(ℂ) and the space of unitary k-frames in 𝕄_{k, l}^{U} respectively. We endow 𝔊_{k, l}^{U} with the compact-open topology, using the action 𝔊_{k, l}^{U} × 𝕄_{k, l}^{U} → 𝕄_{k, l}^{U}. Note that this “unitary” version of the stable theory (see below) coincides with the “noncompact” one. In particular, the topological groups 𝔊_{k, l^{∞}} and 𝔊_{k, l^{∞}}^{U} are homotopy equivalent to each other.

2 𝕄_{k, l}-bundles

2.1 An exact sequence of functors

Fix a pair of coprime integers k, l. Let X be a finite CW -complex. By 𝔉𝔄𝔅_{k, l}(X), 𝔄𝔅_{k}(X), 𝔉𝔞𝔫_{k, l}(X) we denote the categories of floating algebra bundles of the form (A_{k}, μ, M̃_{kl}) (see Appendix 1), locally trivial bundles with fiber M_{k}(ℂ), and locally trivial bundles with fiber 𝕄_{k, l} over X, respectively. We stress that any 𝕄_{k, l}-bundle is assumed to be a locally trivial bundle with the structural group 𝔊_{k, l}.

We have the forgetful functor

T_{k, l}^{X}: 𝔉𝔄𝔅_{k, l}(X) → 𝔄𝔅_{k}(X), (A_{k}, μ, M̃_{kl}) ↦ A_{k},

and the functor

F_{k, l}^{X}: 𝔄𝔅_{k}(X) → 𝔉𝔞𝔫_{k, l}(X)

defined by the embedding of M_{k}(ℂ) as the “standard” subalgebra M_{k}(ℂ)⊗_{ℂ}ℂE_{l} ⊂ 𝕄_{k, l}. We also have the functor

R_{k, l}^{X}: 𝔄𝔅_{kl}(X) → 𝔉𝔞𝔫_{k, l}(X)

defined by the assignment M_{kl}(ℂ) ↦ 𝕄_{k, l}. In other words, to every bundle A_{kl} over X with fiber M_{kl}(ℂ), the functor R_{k, l}^{X} assigns the corresponding 𝕄_{k, l}-bundle. Remark 15. Note that functors F_{k, l}^{X} and R_{k, l}^{X} correspond to the continuous monomorphisms of the structural groups PGL_{k}(ℂ) → 𝔊_{k, l} (see Proposition 7) and PGL_{kl}(ℂ) ⟶^{κ_{k, l}} 𝔊_{k, l}, respectively.

Given an M_{k}(ℂ)-bundle A_{k} over X, let Γ(X; A_{k}) be the Banach algebra of its global sections. Let GL_{l}(A_{k}) be the group GL_{l}(Γ(X; A_{k})) of all invertible l × l matrices with coefficients in Γ(X; A_{k}). In other words, GL_{l}(A_{k}) = GL_{1}(M_{l}(Γ(X; A_{k}))) = GL_{1}(Γ(X; A_{k} ⊗ M̃_{l})), where M̃_{l} is a trivial M_{l}(ℂ)-bundle over X. We have the group monomorphism

GL_{1}(Γ(X; ℂ)) ↪ GL_{l}(A_{k}), α ↦ αE_{kl},

where GL_{1}(Γ(X; ℂ)) = Γ(X; ℂ^{∗}) and E_{kl} is the unit matrix. By PGL_{l}(A_{k}) we denote the corresponding quotient group.

Proposition 16. Let A_{k}, B_{kl}, B′_{kl} be locally trivial bundles over X with fibers M_{k}(ℂ), M_{kl}(ℂ), M_{kl}(ℂ), respectively. Put 𝔄_{k, l} := R_{k, l}^{X}(B_{kl}), 𝔄′_{k, l} := R_{k, l}^{X}(B′_{kl}). Suppose we have two embeddings μ: A_{k} ↪ B_{kl}, μ′: A_{k} ↪ B′_{kl} such that for any point x ∈ X their restrictions μ_{x}, μ′_{x} take the fiber (A_{k})_{x} into (B_{kl})_{x}, (B′_{kl})_{x}, as a central simple subalgebra. Then the corresponding fan bundles 𝔄_{k, l}, 𝔄′_{k, l} are isomorphic to each other. Furthermore, they both have the form ⋃_{g∈GL_{l}(A_{k})}g(A_{k}⊗_{ℂ}ℂE_{l})g^{−1} (i.e. the corresponding sections form a “fan

of subbundles” ⋃_{g∈GL_{l}(A_{k})}g(Γ(X; A_{k}) ⊗_{Γ(X; ℂ)}Γ(X; ℂ)E_{l})g^{−1}). Moreover, any isomorphism α of

the subbundles 𝔄_{k, l} ⊃ μ(A_{k}) →^{α} μ′(A_{k}) ⊂ 𝔄′_{k, l} can be extended to a canonical PGL_{l}(A_{k})- equivariant isomorphism α̃: 𝔄_{k, l} → 𝔄′_{k, l}.

Proof. First, let us prove that the Gr_{k, l}-bundle Gr_{k, l}(B_{kl}) associated with B_{kl} is trivial as a fiber space with structural group PGL_{kl}(ℂ). Indeed, it follows easily from our assumptions that there is an M_{l}(ℂ)-bundle C_{l} such that the embedding A_{k} ↪ B_{kl} has the form A_{k} ↪ A_{k} ⊗C_{l} = B_{kl} (here we use the fact that every locally trivial bundle with fiber matrix algebra has a canonical unit section). By Prin(B_{kl}), Prin(A_{k}), Prin(C_{l}) denote the principal bundles corresponding to the bundles B_{kl}, A_{k}, C_{l}, respectively. We have the embedding Prin(A_{k}) ⊗ Prin(C_{l}) ↪ Prin(B_{kl}) corresponding to the morphism of the bundles A_{k}×C_{l} →^{⊗} B_{kl}. We have free actions of groups PGL_{kl}(ℂ), PGL_{k}(ℂ), PGL_{l}(ℂ) upon Prin(B_{kl}), Prin(A_{k}), Prin(C_{l}), respectively. Moreover, the following diagram

PGL_{kl}(ℂ) × Prin(B_{kl})  ⟶  Prin(B_{kl})
↑  ↑
(PGL_{k}(ℂ) ⊗ PGL_{l}(ℂ)) × (Prin(A_{k}) ⊗ Prin(C_{l})) ⟶ Prin(A_{k}) ⊗ Prin(C_{l})

commutes, i.e. the action is compatible with the embedding of the subbundles. Let Prin(B_{kl})/(Prin(A_{k}) ⊗ Prin(C_{l})) be the homogeneous space of cosets {g · (Prin(A_{k}) ⊗ Prin(C_{l})) | g ∈ PGL_{kl}(ℂ)}. Consider the PGL_{kl}(ℂ)-equivariant map

Prin(B_{kl})/(Prin(A_{k}) ⊗ Prin(C_{l})) → PGL_{kl}(ℂ)/(PGL_{k}(ℂ) ⊗ PGL_{l}(ℂ)),
g · (Prin(A_{k}) ⊗ Prin(C_{l})) ↦ g · (PGL_{k}(ℂ) ⊗ PGL_{l}(ℂ)), g ∈ PGL_{kl}(ℂ).

Clearly, it gives us a trivialization of the bundle Gr_{k, l}(B_{kl}).

Now consider the fibered product Gr_{k, l}(B_{kl})×_{X}B_{kl} and the corresponding tautological M_{k}(ℂ)-subbundle Ã_{k, l}(B_{kl}) ⊂ Gr_{k, l}(B_{kl})×_{X}B_{kl}. Note that for any x_{0} ∈ X we have the diagrams

Gr_{k, l} ×M_{kl}(ℂ) ↪ Gr_{k, l}(B_{kl})×_{X}B_{kl}
↓  ↓
x_{0} ↪ X

and

A_{k, l} ↪ Ã_{k, l}(B_{kl})
↓  ↓
Gr_{k, l} ↪ Gr_{k, l}(B_{kl}).

Moreover, the subbundle μ(A_{k}) ⊂ B_{kl} determines a canonical section σ: X → Gr_{k, l}(B_{kl}) of the projection Gr_{k, l}(B_{kl}) ⟶^{Gr_{k, l}} X such that σ^{∗}(Ã_{k, l}(B_{kl})) = A_{k}.

The group PGL_{l}(A_{k}) obviously acts on the canonical M_{k}(ℂ)-bundle Ã_{k, l}(B_{kl}) over the trivial Gr_{k, l}-bundle Gr_{k, l}(B_{kl}) (note that this action induces the nontrivial one on Gr_{k, l}(B_{kl})). Consider the projection π: A_{k, l} → 𝕄_{k, l} onto 𝕄_{k, l} such that the diagram

A_{k, l} →^{π} 𝕄_{k, l}
↓  ↓
Gr_{k, l} → pt

commutes. Clearly, π is a PGL_{kl}(ℂ)-equivariant map. We have a similar PGL_{l}(A_{k})- equivariant projection π̃ for Ã_{k, l}(B_{kl}). It can be shown that 𝔄_{k, l} = π̃(Ã_{k, l}(B_{kl})) and the diagram

Ã_{k, l}(B_{kl}) \underrightarrow{π̃} 𝔄_{k, l}
↓  ↓
Gr_{k, l}(B_{kl}) → X

commutes. Now it follows easily that

𝔄_{k, l} = ⋃_{g∈GL_{l}(A_{k})}g(A_{k}⊗_{ℂ}ℂE_{l})g^{−1}.  (8)

In particular, it follows from (8) that the fan bundle 𝔄_{k, l} := R_{k, l}^{X}(B_{kl}) does not depend on the choice of C_{l}. □

Corollary 17. For any M_{kl}(ℂ)-bundle B_{kl} containing A_{k} as a matrix algebra subbundle (i.e. for any point x ∈ X the fiber (A_{k})_{x} is a central simple subalgebra in (B_{kl})_{x}) we have R_{k, l}^{X}(B_{kl}) ≅ F_{k, l}^{X}(A_{k}).

Proof. Let B_{kl} be an arbitrary M_{kl}(ℂ)-bundle containing A_{k} as a matrix algebra subbundle. For example, we can take A_{k} ⊗ M̃_{l} for B_{kl}, where M̃_{l} is a trivial M_{l}(ℂ)-bundle over X. Then the 𝕄_{k, l}-bundle R_{k, l}^{X}(B_{kl}) contains A_{k} as an M_{k}(ℂ)-subbundle. It follows from Proposition 16 that R_{k, l}^{X}(B_{kl}) ≅ F_{k, l}^{X}(A_{k}). □

Proposition 18. A bundle F_{k, l}^{X}(A_{k}) ∈ 𝔉𝔞𝔫_{k, l}(X) is trivial iff A_{k} belongs to the image of T_{k, l}^{X}, i.e. A_{k} is the core (see Appendix 1) of a floating algebra bundle (A_{k}, μ, M̃_{kl}).

Remark 19. In other words, the sequence

Ψ_{k, l}(X) \underrightarrow{T̃_{k, l}^{X}} Φ_{k}(X) \underrightarrow{F̃_{k, l}^{X}} Υ_{k, l}(X)

is exact as a sequence of pointed sets. Here Ψ_{k, l}(X), Φ_{k}(X), and Υ_{k, l}(X) are the sets of isomorphism classes of objects of 𝔉𝔄𝔅_{k, l}(X), 𝔄𝔅_{k}(X), 𝔉𝔞𝔫_{k, l}(X) respectively, and T̃_{k, l}^{X}, F̃_{k, l}^{X} are the maps, induced by the corresponding functors. Proof. Suppose F_{k, l}^{X}(A_{k}) ≅ X × 𝕄_{k, l}. By μ: A_{k} ↪ X × M_{kl}(ℂ) denote the composite map A_{k} ↪ F_{k, l}^{X}(A_{k}) →^{≅} X × 𝕄_{k, l} ⊂ X × M_{kl}(ℂ). Clearly, the triple (A_{k}, μ, M̃_{kl}) is a floating algebra bundle over X.

Now suppose A_{k} is the core of a floating algebra bundle (A_{k}, μ, M̃_{kl}). Using Corollary 17, we see that F_{k, l}^{X}(A_{k}) ≅ R_{k, l}^{X}(M̃_{kl}) is a trivial bundle. □

Corollary 20. Consider the sequence

Gr_{k, l} \underrightarrow{T̂_{k, l}} BPGL_{k}(ℂ) \underrightarrow{F̂_{k, l}} B 𝔊_{k, l},  (9)

where T̂_{k, l} is the classifying map for the core A_{k, l} of the tautological floating algebra bundle over Gr_{k, l} as a PGL_{k}(ℂ)-bundle, and F̂_{k, l} is the classifying map for the 𝕄_{k, l}-bundle obtained by application of functor F_{k, l}^{BPGL_{k}(ℂ)} to the universal M_{k}(ℂ)-bundle A_{k}^{univ} over BPGL_{k}(ℂ). Suppose X is a finite CW -complex such that dim(X) ≤ 2 min{k, l}, then the sequence of sets

[X; Gr_{k, l}] \underrightarrow{T̂_{k, l∗}} [X; BPGL_{k}(ℂ)] \underrightarrow{F̂_{k, l∗}} [X; B 𝔊_{k, l}]

is exact.

Proof. Let us recall that Gr_{k, l} is a classifying space for floating algebra bundles of the form (A_{k}, μ, M̃_{kl}) over finite CW -complexes of dimension ≤ 2 min{k, l} (see Appendix 1). Similarly, BPGL_{k}(ℂ) and B 𝔊_{k, l} are classifying spaces for locally trivial bundles with fibers M_{k}(ℂ) and 𝕄_{k, l}, respectively. Now the required assertion follows from the previous proposition. □

To any 𝕄_{k, l}-bundle 𝔄_{k, l} over X we can associate a fiber bundle Gr(𝔄_{k, l}) →^{p} X over X with fiber Gr_{k, l} . Let A_{k, l} be the core of the tautological floating algebra bundle over Gr_{k, l} (see Appendix 1). We have the canonical M_{k}(ℂ)-bundle Ã_{k, l} over Gr(𝔄_{k, l}) which is the tautological subbundle Ã_{k, l} ⊂ Gr(𝔄_{k, l})×_{X}𝔄_{k, l} (in particular, for any fiber Gr_{k, l} of the fibration Gr(𝔄_{k, l}) →^{p} X the restriction Ã_{k, l} |_{Gr_{k, l}} coincides with A_{k, l} ⊂ Gr_{k, l} ×𝕄_{k, l}). Moreover, there is the M_{kl}(ℂ)-bundle B_{kl} over Gr(𝔄_{k, l}) such that R_{k, l}^{Gr(𝔄_{k, l})}(B_{kl}) = p^{∗}(𝔄_{k, l}). The bundle B_{kl} is equipped with the canonical embedding μ̃: Ã_{k, l} ↪ B_{kl} such that for any fiber Gr_{k, l} the restriction (Ã_{k, l} |_{Gr_{k, l}}, μ̃ |_{Gr_{k, l}}, B_{kl} |_{Gr_{k, l}}) is just the canonical FAB (A_{k, l}, μ, M̃_{kl}) over Gr_{k, l} . Moreover, R_{k, l}^{Gr(𝔄_{k, l})}(B_{kl}) ≅ F_{k, l}^{Gr(𝔄_{k, l})}(Ã_{k, l}) (compare with Remark 29).

Lemma 21. The pull-back p^{∗}(𝔄_{k, l}) has the form F_{k, l}^{Gr(𝔄_{k, l})}(Ã_{k, l}).

Proof. Clearly, p^{∗}(𝔄_{k, l}) contains Ã_{k, l} as an M_{k}(ℂ)-subbundle. Moreover, p^{∗}(𝔄_{k, l}) = R_{k, l}^{Gr(𝔄_{k, l})}(B_{kl}) for the M_{kl}(ℂ)-bundle B_{kl} defined above. Now it follows from Corollary 17 that p^{∗}(𝔄_{k, l}) ≅ F_{k, l}^{Gr(𝔄_{k, l})}(Ã_{k, l}). □ Remark 22. Recall that we have the map A_{k, l} →^{π} 𝕄_{k, l} induced by the map Gr_{k, l} → pt (see Subsection 1.1). There is the map π̃: Ã_{k, l} → 𝔄_{k, l} such that for any fiber Gr_{k, l} the restriction π̃ |_{Gr_{k, l}} coincides with π. Thus, the initial bundle 𝔄_{k, l} can be reconstructed from Ã_{k, l} by fibered application of π.

Given an 𝕄_{k, l}-bundle 𝔄_{k, l} over X, one can ask the following question: is there an M_{k}(ℂ)- bundle A_{k} over X such that 𝔄_{k, l} = F_{k, l}^{X}(A_{k})?

Proposition 23. An 𝕄_{k, l}-bundle 𝔄_{k, l} over X has the form 𝔄_{k, l} = F_{k, l}^{X}(A_{k}) for some M_{k}(ℂ)- bundle A_{k} iff the fiber bundle Gr(𝔄_{k, l}) →^{p} X has a section σ.

Proof. According to the above lemma, p^{∗}(𝔄_{k, l}) = F_{k, l}^{Gr(𝔄_{k, l})}(Ã_{k, l}). Hence, given a section σ, we can take A_{k} = σ^{∗}(Ã_{k, l}).

Conversely, if 𝔄_{k, l} = F_{k, l}^{X}(A_{k}), then A_{k} is an M_{k}(ℂ)-subbundle in 𝔄_{k, l}. Clearly, any such subbundle determines a section of p. □

Let us remark that if 𝔄_{k, l} = F_{k, l}^{X}(A_{k}) for A_{k} = σ^{∗}(Ã_{k, l}) then 𝔄_{k, l} = R_{k, l}^{X}(B_{kl}), where B_{kl} := σ^{∗}(B_{kl}).

Given an 𝕄_{k, l}-bundle 𝔄_{k, l} over X, let Fr(𝔄_{k, l}) \underrightarrow{p̂} X be the corresponding fiber bundle of k-frames over X.

Proposition 24. An 𝕄_{k, l}-bundle 𝔄_{k, l} is trivial iff Fr(𝔄_{k, l}) \underrightarrow{p̂} X has a section σ̂.

Proof. Note that actually we have three fiber bundles: Gr(𝔄_{k, l}) →^{p} X with fiber Gr_{k, l}, Fr(𝔄_{k, l}) \underrightarrow{p̂} X with fiber Fr_{k, l}, and Fr(𝔄_{k, l}) \underrightarrow{p̃} Gr(𝔄_{k, l}) with fiber PGL_{k}(ℂ) (see Subsection 1.4). Clearly, lifting p̃^{∗}(Ã_{k, l}) is a trivial M_{k}(ℂ)-bundle M̃_{k} over Fr(𝔄_{k, l}). Furthermore, bundle p̂^{∗}(𝔄_{k, l}) has the form F_{k, l}^{Fr(𝔄_{k, l})}(p̃^{∗}(Ã_{k, l})).

Suppose p̂ has a section σ̂, then 𝔄_{k, l} = F_{k, l}^{X}(σ̂^{∗}(M̃_{k})) is a trivial bundle. The converse implication is clear. □

2.2 Relation to the frame spaces

Now we describe the functors F_{k, l}^{X}: 𝔄𝔅_{k}(X) → 𝔉𝔞𝔫_{k, l}(X), R_{k, l}^{X}: 𝔄𝔅_{kl}(X) → 𝔉𝔞𝔫_{k, l}(X) in terms of frames.

Definition 25. An n-frame in M_{n}(ℂ) is an ordered collection of n^{2} linearly independent matrices {α_{i, j} | 1 ≤ i, j ≤ n} such that α_{i, j}α_{k, l} = δ_{jk}α_{i, l} for all 1 ≤ i, j, k, l ≤ n.

For example, the “matrix units” {E_{ij} | 1 ≤ i, j ≤ n} form the “standard” frame in M_{n}(ℂ). Clearly, the set of all n-frames in M_{n}(ℂ) is the principal homogeneous space over PGL_{n}(ℂ).

First consider the functor F_{k, l}^{X}. To any M_{k}(ℂ)-bundle A_{k} ∈ 𝔄𝔅_{k}(X) we can associate the corresponding principal bundle of k-frames. Similarly, to any 𝕄_{k, l}-bundle 𝔄_{k, l} ∈ 𝔉𝔞𝔫_{k, l}(X) we can associate the corresponding k-frames bundle with fiber Fr_{k, l} . The functor F_{k, l}^{X}: 𝔄𝔅_{k}(X) → 𝔉𝔞𝔫_{k, l}(X) in these terms can be described as follows. To any k-frame {α_{i, j} | 1 ≤ i, j ≤ k} in M_{k}(ℂ) it assigns the k-frame {α_{i, j} ⊗ E_{l} | 1 ≤ i, j ≤ k} in 𝕄_{k, l}, where … ⊗ E_{l} denotes the Kronecker product with the unit l × l-matrix.

Now consider the functor R_{k, l}^{X}. To a given a kl-frame {γ_{i, j} | 1 ≤ i, j ≤ kl} in M_{kl}(ℂ) it assigns the k-frame {α_{i, j} | 1 ≤ i, j ≤ k} in 𝕄_{k, l}, where α_{i, j} = ∑_{1≤m≤l} γ_{l(i−1)+m, l(j−1)+m}, 1 ≤ i, j ≤ k.

3 A stable theory of 𝕄_{k, l}-bundles

3.1 A weak homotopy equivalence 𝔊_{k, l^{∞}} ≃ Fr_{k, l^{∞}}

Recall that 𝔊_{k, l} is a topological group, therefore there is the universal principal 𝔊_{k, l}-fibration (E 𝔊_{k, l}, B 𝔊_{k, l}, 𝔊_{k, l}, p_{k, l}). Using the continuous action ν_{k, l}: 𝔊_{k, l} × Fr_{k, l} → Fr_{k, l}, we can associate a Fr_{k, l}-fibration with the universal 𝔊_{k, l}-fibration. We denote the obtained Fr_{k, l}- fibration by (E_{k, l}, B 𝔊_{k, l}, Fr_{k, l}, p′_{k, l}).

Let n be a positive integer.

Proposition 26. The total space E_{k, l^{n}} is (2l^{n} − 1)-connected.

Proof. Looking at the homotopy sequence of the fibration (E_{k, l^{n}}, B 𝔊_{k, l^{n}}, Fr_{k, l^{n}}, p′_{k, l^{n}}), we see that it is sufficient to prove that ∂: π_{r}(B 𝔊_{k, l^{n}}) → π_{r−1}(Fr_{k, l^{n}}) is an isomorphism for dimensions r ≤ 2l^{n}.

(i) ∂ is a monomorphism. Let α: S^{r} → B 𝔊_{k, l^{n}} be a map and let 𝔄_{k, l^{n}} be the 𝕄_{k, l^{n}}- bundle over S^{r} classified by α. Suppose ∂ ∘ α ≃ ∗, then the k-frame fibration Fr(𝔄_{k, l^{n}}) related to 𝔄_{k, l^{n}} has a section σ: S^{r} → Fr(𝔄_{k, l^{n}}). It follows from Proposition 24 that the bundle 𝔄_{k, l^{n}} is trivial, hence α ≃ ∗.

(ii) ∂ is an epimorphism. Let β ∈ π_{r−1}(Fr_{k, l^{n}}) be an arbitrary element. We must find a bundle with fiber 𝕄_{k, l^{n}} over S^{r} such that β is the obstruction for the corresponding k-frame fibration over S^{r}. It follows from the surjectivity of p_{∗} in the homotopy sequence of the fibration

\xymatrix{PGL_{l^{n}}(ℂ) \ar[r]^{E_{k}⊗...} & PGL_{kl^{n}}(ℂ) \ar[d]^{p} \\  & _{Fr_{k, l^{n}}}}

that there is β̃ ∈ π_{r−1}(PGL_{kl^{n}}(ℂ)) such that p_{∗}(β̃) = β, if r ≤ 2l^{n}. Since the boundary homomorphism

π_{r}(BPGL_{kl^{n}}(ℂ)) →^{≅} π_{r−1}(PGL_{kl^{n}}(ℂ))

of the homotopy sequence of the universal PGL_{kl^{n}}(ℂ)-fibration is an isomorphism, we see that there is a bundle A_{kl^{n}} over S^{r} with fiber M_{kl^{n}}(ℂ) such that the obstruction to the corresponding principal bundle coincides with β̃. Consider the 𝕄_{k, l^{n}}-bundle R_{k, l^{n}}^{S^{r}}(A_{kl^{n}}) corresponding to A_{kl^{n}} (recall that PGL_{kl^{n}}(ℂ) ⊂ 𝔊_{k, l^{n}}). Let α: S^{r} → B 𝔊_{k, l^{n}} be its classifying map. It can be easily checked that the obtained bundle is what is required, i.e. with the obstruction β. (More precisely, consider the Fr_{k, l^{n}}-fibration Fr_{k, l^{n}}(A_{kl^{n}}^{univ}) ⟶^{Fr_{k, l^{n}}} BPGL_{kl^{n}}(ℂ) over BPGL_{kl^{n}}(ℂ) associated with the universal principal PGL_{kl^{n}}(ℂ)-bundle. Then we have the following commutative diagram

\xymatrix{Fr_{k, l^{n}}(A_{kl^{n}}^{univ}) \ar[r] \ar[d]_{_{Fr_{k, l^{n}}}} & _{E_{k, l^{n}}} \ar[d]^{_{Fr_{k, l^{n}}}} \\ BPGL_{kl^{n}}(ℂ) \ar[r]^{R̂_{k, l^{n}}} & B 𝔊_{k, l^{n}}.}

Now the required assertion follows from the commutative diagram

\xymatrix{π_{r}(BPGL_{kl^{n}}(ℂ)) \ar[d]_{R̂_{k, l^{n}∗}} \ar[r]^{∂} & π_{r−1}(Fr_{k, l^{n}}) \ar[d]^{id} \\ π_{r}(B 𝔊_{k, l^{n}}) \ar[r]^{∂} & π_{r−1}(Fr_{k, l^{n}})}

obtained from the homotopy sequences of the above fibrations.) Hence, ∂([α]) = [β], i.e. ∂: π_{r}(B 𝔊_{k, l^{n}}) → π_{r−1}(Fr_{k, l^{n}}) is surjective if r ≤ 2l^{n}. □

By definition, put 𝔊_{k, l^{∞}} := lim_{\overrightarrow{n}} 𝔊_{k, l^{n}}, Fr_{k, l^{∞}} := lim_{\overrightarrow{n}} Fr_{k, l^{n}} .

Theorem 27. There is a weak homotopy equivalence 𝔊_{k, l^{∞}} ≃ Fr_{k, l^{∞}} .

Proof. It follows from the previous proposition that the total space of the direct limit

fibration (E_{k, l^{∞}}, B 𝔊_{k, l^{∞}}, Fr_{k, l^{∞}}, p′_{k, l^{∞}}), where E_{k, l^{∞}} := lim_{\overrightarrow{n}} E_{k, l^{n}}, p′_{k, l^{∞}} := lim_{\overrightarrow{n}} p′_{k, l^{n}}, etc,

is aspherical. Therefore there is a weak homotopy equivalence Fr_{k, l^{∞}} ≃ Ω(B 𝔊_{k, l^{∞}}) ≃ 𝔊_{k, l^{∞}}. □

Corollary 28. (compare with Corollary 20) The sequence

Gr_{k, l^{∞}} \underrightarrow{T̂_{k, l^{∞}}} BPGL_{k}(ℂ) \underrightarrow{F̂_{k, l^{∞}}} B 𝔊_{k, l^{∞}}

is a fibration.

Proof. Since Fr_{k, l^{∞}} ≃ Ω(B 𝔊_{k, l^{∞}}), we see that the fibered sequence

PGL_{k}(ℂ) →^{α} Fr_{k, l^{∞}} → Gr_{k, l^{∞}} → BPGL_{k}(ℂ)

can be extended to the right. More precisely, let A_{k}^{univ} be the universal M_{k}(ℂ)-bundle over BPGL_{k}(ℂ). Then the map BPGL_{k}(ℂ) \underrightarrow{F̂_{k, l^{∞}}} B 𝔊_{k, l^{∞}} is a classifying map for the 𝕄_{k, l^{∞}}- bundle F_{k, l^{∞}}^{BPGL_{k}(ℂ)}(A_{k}^{univ}) over BPGL_{k}(ℂ) (see Remark 15), i.e. α = Ω(F̂_{k, l^{∞}}). More precisely, for any n we have the classifying map BPGL_{k}(ℂ) → B 𝔊_{k, l^{n}} of F_{k, l^{n}}^{BPGL_{k}(ℂ)}(A_{k}^{univ}) as an 𝕄_{k, l^{n}}-bundle. We have the corresponding map of k-frames PGL_{k}(ℂ) ⟶^{...⊗E_{l^{n}}} Fr_{k, l^{n}} . Let α: PGL_{k}(ℂ) → Fr_{k, l^{∞}} be the direct limit of such maps as n → ∞. Then the following diagram

\xymatrix{ & _{Fr_{k, l^{∞}}} \ar[r] & ∗ \ar[d] \\ PGL_{k}(ℂ) \ar@{-->}[ur]^{α} \ar[r] & ∗ \ar[ur] \ar[d] & B 𝔊_{k, l^{∞}} \\  & BPGL_{k}(ℂ) \ar[ur]^{F̂_{k, l^{∞}}}}

commutes. □ Remark 29. Note that we have the following commutative diagram

\xymatrix{BPGL_{kl}(ℂ) \ar[r]^{R̂_{k, l}} & B 𝔊_{k, l} \\ BPGL_{k}(ℂ). \ar[u]^{...⊗M̃_{l}} \ar@{-->}[ur]_{F̂_{k, l}}}

Remark 30. Now we want to establish some relation of the fibration of Corollary 28 with the results of Section 2. Let 𝔄_{k, l^{n}}^{univ} be the universal 𝕄_{k, l^{n}}-bundle over B 𝔊_{k, l^{n}}. It was shown prior to Lemma 21 that there is Gr_{k, l^{n}}-fibration Gr(𝔄_{k, l^{n}}^{univ}) →^{p_{n}} B 𝔊_{k, l^{n}} associated with the 𝕄_{k, l^{n}}-bundle 𝔄_{k, l^{n}}^{univ}. Moreover, we can define the canonical M_{k}(ℂ)-bundle Ã_{k, l^{n}}^{univ} over Gr(𝔄_{k, l^{n}}^{univ}) which is the tautological subbundle Ã_{k, l^{n}}^{univ} ⊂ Gr(𝔄_{k, l^{n}}^{univ}) ×_{B 𝔊_{k, l^{n}}}𝔄_{k, l^{n}}^{univ} (in particular,

its restriction to a fiber Gr_{k, l^{n}} of the fibration Gr(𝔄_{k, l^{n}}^{univ}) →^{p_{n}} B 𝔊_{k, l^{n}} coincides with the core A_{k, l^{n}} of the tautological floating algebra bundle over Gr_{k, l^{n}}).

Consider the fibration

\xymatrix{_{Gr_{k, l^{n}}} \ar[r] & Gr(𝔄_{k, l^{n}}^{univ}) \ar[d]^{p_{n}} \\  & B 𝔊_{k, l^{n}}.}

It follows from the obtained results that this fibration turns into the fibration of Corollary 28 as we pass to the limit with n → ∞. In particular, Gr(𝔄_{k, l^{∞}}^{univ}) := lim_{\overrightarrow{n}} Gr(𝔄_{k, l^{n}}^{univ}) ≃ BPGL_{k}(ℂ)

and, moreover, under this identification the M_{k}(ℂ)-bundle Ã_{k, l^{∞}}^{univ} over Gr(𝔄_{k, l^{∞}}^{univ}) coincides with the universal M_{k}(ℂ)-bundle A_{k}^{univ} over BPGL_{k}(ℂ) and the projection Gr(𝔄_{k, l^{∞}}^{univ}) ⟶^{p_{∞}} B 𝔊_{k, l^{∞}} is just F̂_{k, l^{∞}}.

Corollary 31. π_{r}(B 𝔊_{k, l^{∞}}) = ℤ/kℤ for even r ≥ 2 and π_{r}(B 𝔊_{k, l^{∞}}) = 0 for odd r.

Proof follows from Theorem 27 and from the homotopy sequence of the fibration

\xymatrix{PGL_{l^{∞}}(ℂ) \ar[r]^{E_{k}⊗...} & PGL_{kl^{∞}}(ℂ) \ar[d] \\  & Fr_{k, l^{∞}} . □}

3.2 The tensor product of fans

Let k, l, m, n be positive integers such that (km, ln) = 1. Denote

𝕄_{k, l}⊗_{ℂ}𝕄_{m, n} := 𝕄_{km, ln}.

We have the map

t_{k,l; m,n}: 𝕄_{k, l} × 𝕄_{m, n} → 𝕄_{k, l}⊗_{ℂ}𝕄_{m, n}

such that its restriction to each pair of subalgebras M_{k}(ℂ)×M_{m}(ℂ) ⊂ 𝕄_{k, l} ×𝕄_{m, n} coincides with the usual bilinear map M_{k}(ℂ) × M_{m}(ℂ) →^{⊗} M_{k}(ℂ)⊗_{ℂ}M_{m}(ℂ) = M_{km}(ℂ) ⊂ 𝕄_{km, ln}. (Indeed, consider the commutative diagram

\xymatrix{A_{k, l} × A_{m, n} \ar[d] \ar[r]^{⊠} & A_{km, ln} \ar[d] \\ Gr_{k, l} × Gr_{m, n} \ar[r]^{μ_{k,l; m,n}} & Gr_{km, ln},}

where μ_{k,l; m,n} is the map induced by the tensor product of algebras M_{kl}(ℂ) × M_{mn}(ℂ) →^{⊗} M_{klmn}(ℂ). Now we define t_{k,l; m,n} as the map such that the corresponding diagram

\xymatrix{A_{k, l} × A_{m, n} \ar[r]^{⊠} \ar[d]_{π_{k, l}×π_{m, n}} & A_{km, ln} \ar[d]^{π_{km, ln}} \\ 𝕄_{k, l} × 𝕄_{m, n} \ar[r]^{t_{k,l; m,n}} & 𝕄_{km, ln}}

commutes, where π_{k, l} is the composite map A_{k, l} ↪ Gr_{k, l} ×𝕄_{k, l} →^{π_{2}} 𝕄_{k, l}, etc.) Note that this definition makes sense due to Proposition 3. Clearly, the tensor product of fans is associative.

Now we must prove that the tensor product of fans is functorial. Lemma 32. Suppose ϑ: 𝕄_{k, l} → 𝕄_{k, l} is a fan automorphism. Then there is a unique automorphism ϑ̃: A_{k, l} → A_{k, l} of the core of the tautological bundle over Gr_{k, l} such that ϑ̃ |_{(A_{k, l})_{x}}= ϑ |_{M_{k, x}} for any x ∈ Gr_{k, l}, where M_{k, x} is the k-subalgebra in 𝕄_{k, l} parameterized by x ∈ Gr_{k, l}. Moreover, the assignment ϑ ↦ ϑ̃ is a group homomorphism Aut(𝕄_{k, l}) → Aut(A_{k, l}). Proof. To a given ϑ: 𝕄_{k, l} → 𝕄_{k, l} we assign the continuous map \overline{ϑ}: Gr_{k, l} → Gr_{k, l} such

that ϑ |_{M_{k, x}}: M_{k, x} →^{≅} M_{k, \overline{ϑ}(x)} for any x ∈ Gr_{k, l} . Then ϑ̃: A_{k, l} → A_{k, l} can be defined as a

unique map such that the diagram

\xymatrix{Gr_{k, l} ×𝕄_{k, l} \ar[r]^{\overline{ϑ}×ϑ} & Gr_{k, l} ×𝕄_{k, l} \\ A_{k, l} \ar[u] \ar[r]^{ϑ̃} & A_{k, l} \ar[u]}

commutes (the vertical arrows are the canonical embeddings). Obviously, ϑ̃ is a continuous map. □

Lemma 33. Conversely, suppose τ is an automorphism τ: A_{k, l} → A_{k, l} (which is not necessarily identity on the base Gr_{k, l}). Then there is an automorphism τ̂: 𝕄_{k, l} → 𝕄_{k, l} such that the diagram

\xymatrix{Gr_{k, l} ×𝕄_{k, l} \ar[r]^{\overline{τ}×τ̂} & Gr_{k, l} ×𝕄_{k, l} \\ A_{k, l} \ar[u] \ar[r]^{τ} & A_{k, l} \ar[u]}

commutes, where \overline{τ} is the map covered by τ:

\xymatrix{A_{k, l} \ar[d] \ar[r]^{τ} & A_{k, l} \ar[d] \\ _{Gr_{k, l}} \ar[r]^{\overline{τ}} & Gr_{k, l} .}

(In other words, by definition τ̂ |_{M_{k, x}}= τ |_{x}: (A_{k, l})_{x} → (A_{k, l})_{\overline{τ}(x)} for any x ∈ Gr_{k, l}, if we identify M_{k, x} with (A_{k, l})_{x}.) Moreover, the assignment τ ↦ τ̂ is a group homomorphism Aut(A_{k, l}) → Aut(𝕄_{k, l}).

Proof. By 𝕄̃_{k, l} denote the trivial 𝕄_{k, l}-bundle over Gr_{k, l} . Using the equality 𝕄̃_{k, l} = ⋃_{g∈GL_{l}(A_{k, l})}g(A_{k, l}⊗_{ℂ}ℂE_{l})g^{−1} of Proposition 16 (see (8)), we can extend the isomorphism τ

to the PGL_{l}(A_{k, l})-equivariant isomorphism of trivial bundles T: 𝕄̃_{k, l} → 𝕄̃_{k, l} which covers \overline{τ}: Gr_{k, l} → Gr_{k, l} and such that the diagram

\xymatrix{𝕄̃_{k, l} \ar[r]^{T} & 𝕄̃_{k, l} \\ A_{k, l} \ar[r]^{τ} \ar[u] & A_{k, l} \ar[u]}

commutes. It is not difficult to verify that T preserves the standard trivialization Gr_{k, l} ×𝕄_{k, l} of 𝕄̃_{k, l}, i.e. the diagram

\xymatrix{Gr_{k, l} ×𝕄_{k, l} \ar[r]^{T} \ar[d]_{π_{2}} & Gr_{k, l} ×𝕄_{k, l} \ar[d]^{π_{2}} \\ 𝕄_{k, l} \ar[r]^{τ̂} & 𝕄_{k, l}}

commutes. Then τ̂: 𝕄_{k, l} → 𝕄_{k, l} is the required automorphism, and T = (\overline{τ}, τ̂). □ Comparing the two previous lemmas, we get the following one.

Lemma 34. The composition ϑ ↦ τ̂, where τ = ϑ̃, is the identity map id_{Aut(𝕄_{k, l})} . Furthermore, the composition τ ↦ ϑ̃, where τ̂ = ϑ, is the identity map id_{Aut(A_{k, l})} . In particular, Aut(𝕄_{k, l}) = 𝔊_{k, l} is isomorphic to Aut(A_{k, l}) (recall that we consider automorphisms of the bundle A_{k, l} which are not necessarily identity on the base Gr_{k, l}).

Proof. By π: A_{k, l} → 𝕄_{k, l} denote the composite map A_{k, l} →^{μ} Gr_{k, l} ×𝕄_{k, l} →^{π_{2}} 𝕄_{k, l}. Note that it is surjective. Then it is sufficient to observe that the maps ϑ and τ̂ both make the diagrams

\xymatrix{A_{k, l} \ar[d]_{π} \ar[r]^{ϑ̃=τ} & A_{k, l} \ar[d]^{π} & A_{k, l} \ar[d]_{π} \ar[r]^{ϑ̃=τ} & A_{k, l} \ar[d]^{π} \\ 𝕄_{k, l} \ar[r]^{ϑ} & 𝕄_{k, l} & 𝕄_{k, l} \ar[r]^{τ̂} & 𝕄_{k, l}}

commutative. Furthermore, the maps τ and ϑ̃ both make the diagrams

[figure: Gr_{k, l} ×𝕄_{k, l}; \overline{ϑ}×ϑ=\overline{τ̂}×τ̂; Gr_{k, l} ×𝕄_{k, l}; Gr_{k, l} ×𝕄_{k, l}; \overline{ϑ}×ϑ=\overline{τ̂}×τ̂; Gr_{k, l} ×𝕄_{k, l}; A_{k, l}; τ; A_{k, l}; A_{k, l}; ϑ̃; A_{k, l}]

commutative. □ Remark 35. It is interesting to compare the result of the previous lemma 𝔊_{k, l} ≅ Aut(A_{k, l}) with Remark 30. In particular, consider the total space of the tautological bundle Ã_{k, l}^{univ} as a fibration over B 𝔊_{k, l} with fiber the total space of the tautological bundle A_{k, l} over Gr_{k, l}. Then the isomorphism 𝔊_{k, l} ≅ Aut(A_{k, l}) implies that the fibration Ã_{k, l}^{univ} can be glued by the same 𝔊_{k, l}-cocycle over B 𝔊_{k, l} as the universal 𝕄_{k, l}-bundle 𝔄_{k, l}^{univ}, i.e. it is the universal A_{k, l}-fibration.

For an upcoming important proposition we need one trivial lemma.

Lemma 36. Let τ_{1} : M_{k, α} → M_{k, α} be an automorphism of the standard subalgebra

M_{k, α} = M_{k}(ℂ)⊗_{ℂ}ℂE_{l} ⊂ 𝕄_{k, l} in the fan 𝕄_{k, l} and let τ_{2} : M_{m, β} → M_{m, β} be an au-

tomorphism of the standard subalgebra M_{m, β} = M_{m}(ℂ)⊗_{ℂ}ℂE_{n} ⊂ 𝕄_{m, n} in the fan 𝕄_{m, n}, respectively. Then τ_{1} can be canonically extended to the PGL_{kl}(ℂ)-equivariant automorphism τ̂_{1} : 𝕄_{k, l} → 𝕄_{k, l} and τ_{2} can be canonically extended to the PGL_{mn}(ℂ)- equivariant automorphism τ̂_{2} : 𝕄_{m, n} → 𝕄_{m, n}, respectively. Furthermore, the automorphism τ_{1} ⊗ τ_{2} : M_{k, α} ⊗ M_{m, β} → M_{k, α} ⊗ M_{m, β} can be canonically extended to the PGL_{klmn}(ℂ)- equivariant automorphism { \atop τ_{1} ⊗ τ_{2}} : 𝕄_{km, ln} → 𝕄_{km, ln} of the fan 𝕄_{km, ln} = 𝕄_{k, l} ⊗ 𝕄_{m, n} such

that { \atop τ_{1} ⊗ τ_{2}} |_{𝕄_{k, l}^{(1)}}= τ̂_{1}, { \atop τ_{1} ⊗ τ_{2}} |_{𝕄_{m, n}^{(2)}}= τ̂_{2}, where 𝕄_{k, l}^{(1)}, 𝕄_{m, n}^{(2)} are the images of the embeddings

(see Proposition 3) 𝕄_{k, l} ↪^{...⊗_{ℂ}ℂE_{mn}} 𝕄_{km, ln} and 𝕄_{km, ln} ↩^{ℂE_{kl}⊗_{ℂ}...} 𝕄_{m, n}, respectively.

Proof is trivial and is based on the equality 𝕄_{k, l} = ⋃_{g∈GL_{kl}(ℂ)}g(M_{k}(ℂ)⊗_{ℂ}ℂE_{l})g^{−1}. Let us only

remark that the proof of the next proposition can be considered as a generalization of the proof of this trivial lemma to the replacements

M_{k, α} ↔ A_{k, l}, M_{m, β} ↔ A_{m, n}, GL_{kl}(ℂ) ↔ GL_{l}(A_{k, l}),
GL_{mn}(ℂ) ↔ GL_{n}(A_{m, n}), GL_{klmn}(ℂ) ↔ GL_{ln}(A_{k, l} ⊠ A_{m, n}),

etc. □

Proposition 37. For any pair (ϑ_{1}, ϑ_{2}), ϑ_{1} ∈ Aut(𝕄_{k, l}), ϑ_{2} ∈ Aut(𝕄_{m, n}) there exists an element ϑ_{1} ⊗ ϑ_{2} ∈ Aut(𝕄_{km, ln}) canonically determined by the pair and such that ϑ_{1} ⊗ ϑ_{2} |_{𝕄_{k, l}^{(1)}}= ϑ_{1}, ϑ_{1} ⊗ ϑ_{2} |_{𝕄_{m, n}^{(2)}}= ϑ_{2}, where 𝕄_{k, l}^{(1)}, 𝕄_{m, n}^{(2)} are the images of the embeddings

𝕄_{k, l} ↪^{...⊗_{ℂ}ℂE_{mn}} 𝕄_{km, ln} and 𝕄_{km, ln} ↩^{ℂE_{kl}⊗_{ℂ}...} 𝕄_{m, n}, respectively.

Proof. Applying Lemma 32, we get automorphisms τ_{1} : A_{k, l} → A_{k, l}, τ_{2} : A_{m, n} → A_{m, n}. Thus, we have automorphism

τ_{1} ⊠ τ_{2} : A_{k, l} ⊠ A_{m, n} → A_{k, l} ⊠ A_{m, n}

of the bundle A_{k, l} ⊠ A_{m, n} over Gr_{k, l} × Gr_{m, n} with fiber M_{km}(ℂ). Note that we have the canonical embedding

A_{k, l} ⊠ A_{m, n} → Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln}

such that the diagram

\xymatrix{A_{k, l} ⊠ A_{m, n} \ar[d] \ar[r] & Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} \ar[d]^{μ_{k,l; m,n}×id_{𝕄_{km, ln}}} \\ A_{km, ln} \ar[r] & Gr_{km, ln} ×𝕄_{km, ln}}

commutes, where μ_{k,l; m,n}: Gr_{k, l} × Gr_{m, n} → Gr_{km, ln} is the map induced by the tensor product of algebras.

Applying Proposition 16 and using the same arguments as in the proof of Lemma 33, we get the automorphism T of the trivial bundle T: Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} → Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} such that the diagram

\xymatrix{Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} \ar[r]^{T} & Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} \\ A_{k, l} ⊠ A_{m, n} \ar[r]^{τ_{1}⊠τ_{2}} \ar[u] & A_{k, l} ⊠ A_{m, n} \ar[u]}

commutes. Now we can define ϑ_{1} ⊗ ϑ_{2} as the map that makes the diagram

\xymatrix{Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} \ar[r]^{T} \ar[d]_{π_{2}} & Gr_{k, l} × Gr_{m, n} ×𝕄_{km, ln} \ar[d]^{π_{2}} \\ 𝕄_{km, ln} \ar[r]^{ϑ_{1}⊗ϑ_{2}} & 𝕄_{km, ln}}

commutative. Note that having ϑ_{1} ⊗ϑ_{2} and using Lemma 32, we can construct the automorphism { \atop ϑ_{1} ⊗ ϑ}_{2} : A_{km, ln} → A_{km, ln} of the tautological M_{km}(ℂ)-bundle over Gr_{km, ln} . Clearly, its restriction { \atop ϑ_{1} ⊗ ϑ}_{2} |_{A_{k, l}⊠A_{m, n}} coincides with τ_{1} ⊠ τ_{2} (here we consider A_{k, l} ⊠ A_{m, n} as a subset in A_{km, ln} due to the canonical embedding A_{k, l} ⊠ A_{m, n} ↪ A_{km, ln} which covers the map μ_{k,l; m,n}: Gr_{k, l} × Gr_{m, n} ↪ Gr_{km, ln}).

Using the above lemmas, one can easily verify that ϑ_{1} ⊗ ϑ_{2} has the required property, i.e. ϑ_{1} ⊗ ϑ_{2} |_{𝕄_{k, l}^{(1)}}= ϑ_{1}, ϑ_{1} ⊗ ϑ_{2} |_{𝕄_{m, n}^{(2)}}= ϑ_{2}. □

It easily follows from the corresponding properties of the usual tensor product of algebras that the tensor product of fans is associative and commutative up to canonical equivalence.

Corollary 38. For any positive integers k, l, m, n such that (km, ln) = 1 we have a group monomorphism 𝔊_{k, l} ×𝔊_{m, n} ↪ 𝔊_{km, ln} and therefore a mapping of classifying spaces B 𝔊_{k, l} × B 𝔊_{m, n} ↪ B 𝔊_{km, ln}.

3.3 A stable equivalence relation

Let X be a finite CW -complex. It follows from Proposition 37 that the tensor product of fan bundles with fibers 𝕄_{k, l}, 𝕄_{m, n}, (km, ln) = 1 is a well-defined fan bundle with fiber 𝕄_{km, ln}.

By 𝕄̃_{k, l} we denote a trivial 𝕄_{k, l}-bundle over X.

Definition 39. Two fan bundles 𝔄_{k^{m}, l^{n}} and 𝔅_{k^{r}, l^{s}} over X are said to be stable equivalent if there are integers t, u, v, w such that

𝔄_{k^{m}, l^{n}} ⊗ 𝕄̃_{k^{t}, l^{u}} ≅ 𝔅_{k^{r}, l^{s}} ⊗ 𝕄̃_{k^{v}, l^{w}}

(of course, m + t = r + v, n + u = s + w).

For a fan bundle 𝔄_{k^{m}, l^{n}}, by [𝔄_{k^{m}, l^{n}}] we denote its stable equivalence class.

Let Fan^{k, l}(X) be the set of stable equivalence classes of fan bundles with fibers of the form 𝕄_{k^{m}, l^{n}} (for arbitrary m, n) over X.

By 𝔊_{k^{∞}, l^{∞}}, Fr_{k^{∞}, l^{∞}} we denote the direct limits lim_{\overrightarrow{n}} 𝔊_{k^{n}, l^{∞}}, lim_{\overrightarrow{n}} Fr_{k^{n}, l^{∞}}, respectively.

Remark 40. We can also define the space Fr_{k^{∞}, l^{∞}} by taking the direct limit lim_{\overrightarrow{n}} Fr_{k^{n}, l^{n}} with respect to the maps α_{n}:

[figure: PGL_{l^{n+1}}(ℂ); E_{k^{n+1}}⊗...; PGL_{(kl)^{n+1}}(ℂ); _{Fr_{k^{n+1}, l^{n+1}}}; PGL_{l^{n}}(ℂ); E_{k^{n}}⊗...; ...⊗E_{l}; PGL_{(kl)^{n}}(ℂ); E_{k}⊗...⊗E_{l}; Fr_{k^{n}, l^{n}} .; α_{n+1}]

One can easily verify that in the stable dimensions r π_{r}(Fr_{k^{n}, l^{n}}) = ℤ/k^{n}ℤ for odd r ≥ 1 and 0 otherwise. Furthermore, it follows from the homotopy sequence of the above diagram that α_{n+1} induces the usual embedding

π_{2s−1}(Fr_{k^{n}, l^{n}}) ↪ π_{2s−1}(Fr_{k^{n+1}, l^{n+1}}), ℤ/k^{n}ℤ ↪ ℤ/k^{n+1}ℤ, s ≥ 1.

Hence π_{r}(Fr_{k^{∞}, l^{∞}}) = lim_{\overrightarrow{n}} ℤ/k^{n}ℤ for odd r and 0 otherwise.

We can simplify our notation with the help of the following lemma.

Lemma 41. The homotopy type of the space Fr_{k^{∞}, l^{∞}} does not depend on the choice of l, (k, l) = 1.

Proof. Clearly, Fr_{k^{∞}, l^{∞}} is a fiber of the fibration

Gr_{k^{∞}, l^{∞}} \underrightarrow{T̂_{k^{∞}, l^{∞}}} BPGL_{k^{∞}}(ℂ),  (10)

where T̂_{k^{∞}, l^{∞}} is the classifying map for the core A_{k^{∞}, l^{∞}} of the universal FAB over Gr_{k^{∞}, l^{∞}} . But the homotopy type of Gr_{k^{∞}, l^{∞}} does not depend on the choice of k, l, (k, l) = 1, moreover, Gr_{k^{∞}, l^{∞}} ≃ BSU. The map T̂_{k^{∞}, l^{∞}} actually does not depend on l, because for k, l, m, (k, lm) = 1 the diagram

\xymatrix{ & _{Gr_{k^{∞}, (lm)^{∞}}} \ar[dd]^{T̂_{k^{∞}, (lm)^{∞}}} \\ _{Gr_{k^{∞}, l^{∞}}} \ar@{-->}[ur]^{≃} \ar[dr]_{T̂_{k^{∞}, l^{∞}}} &  & _{Gr_{k^{∞}, m^{∞}}} \ar@{-->}[ul]_{≃} \ar[dl]^{T̂_{k^{∞}, m^{∞}}} \\  & BPGL_{k^{∞}}(ℂ)}

commutes. Thus, we see that Gr_{k^{∞}, l^{∞}} and T̂_{k^{∞}, l^{∞}} in (10) are independent of l with (l, k) = 1, and therefore this is also true for the fiber Fr_{k^{∞}, l^{∞}} . □ Remark 42. Another proof of the previous lemma can be obtained in the following way. Suppose (k, lm) = 1, then the maps in the diagram

\xymatrix{ & _{Fr_{k, lm}} \\ _{Fr_{k, l}} \ar[ur] &  & _{Fr_{k, m}} \ar[ul]}  (11)

are homotopy equivalences in small dimensions. Thus, we have Fr_{k^{∞}, l^{∞}} ≃ Fr_{k^{∞}, (lm)^{∞}} ≃ Fr_{k^{∞}, m^{∞}} .

Corollary 43. The homotopy type of the space 𝔊_{k^{∞}, l^{∞}} does not depend on the choice of l, (k, l) = 1.

Proof follows from the previous lemma and Theorem 27. □

Thus we can omit l in the notation Fr_{k^{∞}, l^{∞}}, 𝔊_{k^{∞}, l^{∞}}, etc.

The proof of the following theorem is standard.

Theorem 44. The functor X ↦ Fan^{k, l}(X) is represented by space B 𝔊_{k^{∞}}.

Corollary 45. The functor X ↦ Fan^{k, l}(X) is independent of l.

Thus we can write just Fan^{k}(X) instead of Fan^{k, l}(X).

We define the product of two stable equivalence classes [𝔄_{k^{m}, l^{n}}], [𝔅_{k^{r}, l^{s}}] over X as

[𝔄_{k^{m}, l^{n}}] · [𝔅_{k^{r}, l^{s}}] := [𝔄_{k^{m}, l^{n}} ⊗ 𝔅_{k^{r}, l^{s}}].

Clearly, the product is well defined.

Lemma 46. The order of any stable equivalence class of fan bundles over a finite CW - complex X with respect to the above-defined operation is finite, i.e. there is w ∈ ℕ such that [𝔄_{k^{m}, l^{n}}]^{k^{w}} coincides with the stable equivalence class of a trivial bundle.

Proof follows from Theorem 44 and from the fact that π_{r}(B 𝔊_{k^{∞}}) = lim_{\overrightarrow{n}} ℤ/k^{n}ℤ for even r and 0 otherwise. □

Thus, every stable equivalence class is invertible.

Corollary 47. The product of stable equivalence classes induces a structure of Abelian group on the set Fan^{k}(X). In other words, B 𝔊_{k^{∞}} is an H-space (with inversion).

Remark 48. Note that the multiplication on the H-space B 𝔊_{k^{∞}} is determined by the tensor product map

B 𝔊_{k^{m}, l^{n}} × B 𝔊_{k^{r}, l^{s}} → B 𝔊_{k^{m+r}, l^{n+s}}

(which is a classifying map for the tensor product of the universal fan bundles over B 𝔊_{k^{m}, l^{n}} and B 𝔊_{k^{r}, l^{s}}) on finite subspaces.

Recall that in Corollary 28 we have considered the fibration

Gr_{k^{n}, l^{∞}} \underrightarrow{T̂_{k^{n}, l^{∞}}} BPGL_{k^{n}}(ℂ) \underrightarrow{F̂_{k^{n}, l^{∞}}} B 𝔊_{k^{n}, l^{∞}}.

Taking the limit as n → ∞, we obtain the following fibration

Gr \underrightarrow{T̂_{k^{∞}}} BPGL_{k^{∞}}(ℂ) \underrightarrow{F̂_{k^{∞}}} B 𝔊_{k^{∞}},  (12)

where Gr := Gr_{k^{∞}, l^{∞}}, BPGL_{k^{∞}}(ℂ) := lim_{\overrightarrow{n}} BPGL_{k^{n}}(ℂ) with respect to the maps induced by the canonical group homomorphisms PGL_{k^{n}}(ℂ) → PGL_{k^{n+1}}(ℂ), A ↦ A ⊗ E_{k}, and

T̂_{k^{∞}} := lim_{\overrightarrow{n}} T̂_{k^{n}, l^{∞}}, F̂_{k^{∞}} := lim_{\overrightarrow{n}} F̂_{k^{n}, l^{∞}}.

Recall that Gr and BPGL_{k^{∞}}(ℂ) are H-spaces. More precisely, Gr is a classifying space for stable equivalence classes of floating algebra bundles with the multiplication induced by the tensor product of FABs (see Appendix 1). The H-space BPGL_{k^{∞}}(ℂ) is a classifying space for locally trivial bundles with fibers M_{k^{m}}(ℂ), m ∈ ℕ, with respect to the following equivalence relation,

A_{k^{m}} ∼ B_{k^{n}} ⇔ ∃r, s such that A_{k^{m}} ⊗ M̃_{k^{r}} ≅ B_{k^{n}} ⊗ M̃_{k^{s}},

where M̃_{k^{m}} is a trivial bundle. The multiplication on it is induced by the tensor product of bundles. This gives us an interpretation of the groups ÃB^{1}(X) := [X; Gr] and ÃB^{k}(X) := [X; BPGL_{k^{∞}}(ℂ)].

Theorem 49. For any finite CW -complex X the sequence of Abelian groups

ÃB^{1}(X) \underrightarrow{T̂_{k^{∞}∗}} ÃB^{k}(X) \underrightarrow{F̂_{k^{∞}∗}} Fan^{k}(X)

is exact.

Proof follows from the fact that the composition (12) is a fibration. □

3.4 A GL-version

Consider spaces Ĝr_{k, l} and Ĝr which are defined in Appendix 1. It easily follows from the definition that there is a fibered sequence

GL_{k}(ℂ) ↪ Fr_{k, l} → Ĝr_{k, l} \underrightarrow{T̂′_{k, l}} BGL_{k}(ℂ),

where T̂′_{k, l} is the classifying map for the tautological GL_{k}(ℂ)-bundle over Ĝr_{k, l}. It is mentioned in Appendix 1 that the map T̂′_{k^{∞}}(:= lim_{\overrightarrow{n}} T̂′_{k^{n}, l^{n}}): Ĝr(= lim_{\overrightarrow{n}} Ĝr_{k^{n}, l^{n}}) → BGL_{k^{∞}}(ℂ) is

just the localization at k (in particular, BGL_{k^{∞}}(ℂ) is a ℤ[\frac{1}{k}]-local space).

Now consider the sequence

Ĝr \underrightarrow{T̂′_{k^{∞}}} BGL_{k^{∞}}(ℂ) \underrightarrow{F̂′_{k^{∞}}} B 𝔊_{k^{∞}},  (13)

where the map F̂′_{k^{∞}} is defined as follows. For any n we have the group homomorphism GL_{k^{n}}(ℂ) → 𝔊_{k^{n}, l^{n}} determined by the embedding M_{k^{n}}(ℂ) ↪ 𝕄_{k^{n}, l^{n}} (more precisely, it is determined by the composition

GL_{k^{n}}(ℂ) → PGL_{k^{n}}(ℂ) = Aut(M_{k^{n}}(ℂ)) → Aut(𝕄_{k^{n}, l^{n}}) = 𝔊_{k^{n}, l^{n}})

and the direct limit of such maps determines the required map F̂′_{k^{∞}}: BGL_{k^{∞}}(ℂ) → B 𝔊_{k^{∞}} of classifying spaces. In particular, the diagram

\xymatrix{BGL_{k^{∞}}(ℂ) \ar[dr]^{F̂′_{k^{∞}}} \ar[d] \\ BPGL_{k^{∞}}(ℂ) \ar[r]^{F̂_{k^{∞}}} & B 𝔊_{k^{∞}}}

commutes.

The proof of the following proposition easily follows from the fact that the space Fr_{k^{∞}} ≃ Ω(B 𝔊_{k^{∞}}) is a fiber of the localization map T̂′_{k^{∞}}: Ĝr → BGL_{k^{∞}}(ℂ).

Proposition 50. The sequence (13) is a fibration.

Let us remark that the sequence (13) is a GL-counterpart of the sequence (12):

Gr \underrightarrow{T̂_{k^{∞}}} BPGL_{k^{∞}}(ℂ) \underrightarrow{F̂_{k^{∞}}} B 𝔊_{k^{∞}}.

3.5 0 → ℤ → ℚ → ℚ/ℤ → 0

Define the direct limits

B GL(ℂ) := lim_{\overrightarrow{k}} BGL_{k}(ℂ), B PGL(ℂ) := lim_{\overrightarrow{k}} BPGL_{k}(ℂ), and B 𝔊 := lim_{\overrightarrow{k}} B 𝔊_{k},

where k runs over all positive integers, with respect to the maps corresponding to the tensor product of bundles (in particular, B GL(ℂ) is a ℚ-space).

Remark 51. In this remark we give a more detailed definition of the direct limit B 𝔊 =

lim_{\overrightarrow{(k,l)=1}} B 𝔊_{k, l}.

Let p_{1}, p_{2}, …, p_{r}, … be the sequence of primes p_{1} = 2, p_{2} = 3, p_{3} = 5, …. For any r ∈ ℕ we can choose q_{r} such that (p_{1} … p_{r}, q_{r}) = 1. Suppose (p_{1} … p_{r}p_{r+1}, q_{r}) > 1. It follows from Proposition 26 and (11) that there are homotopy equivalences

B 𝔊_{(p_{1}...p_{r})^{∞}, q_{r}^{∞}} →^{≃} B 𝔊_{(p_{1}...p_{r})^{∞}, (q_{r}q_{r+1})^{∞}} ←^{≃} B 𝔊_{(p_{1}...p_{r})^{∞}, q_{r+1}^{∞}}.

Therefore we have the composite map

B 𝔊_{(p_{1}...p_{r})^{∞}, q_{r}^{∞}} →^{≃} B 𝔊_{(p_{1}...p_{r})^{∞}, q_{r+1}^{∞}} → B 𝔊_{(p_{1}...p_{r}p_{r+1})^{∞}, q_{r+1}^{∞}},

and we can take the direct limit with respect to such maps.

The sequences (12) and (13) give us the following fibrations:

Gr \underrightarrow{T̂} B PGL(ℂ) \underrightarrow{F̂} B 𝔊  (14)

and

Ĝr \underrightarrow{T̂′} B GL(ℂ) \underrightarrow{F̂′} B 𝔊,  (15)

respectively.

The relation between (14) and (15) is very clear. More precisely, recall that

Ĝr = Gr ×ℂP^{∞}, B GL(ℂ) = ∏_{q≥1} K(2q, ℚ), B PGL(ℂ) = K(2, ℚ/ℤ) × ∏_{q≥2} K(2q, ℚ), (16)

ℂP^{∞} = K(2, ℤ), and we have the fibration ℂP^{∞} ↪ K(2, ℚ) → K(2, ℚ/ℤ) corresponding to the exact sequence of groups in the title of this subsection. Therefore (14) can be obtained from (15) with the help of the “factorization” by ℂP^{∞}.

Note that the sequence (15) relates something “integer” (Ĝr), something “rational” (B GL(ℂ)) and something “finite” (B 𝔊). Indeed, the sequence of homotopy groups π_{r}(Ĝr) \underrightarrow{T̂′_{∗}} π_{r}(B GL(ℂ)) \underrightarrow{F̂′_{∗}} π_{r}(B 𝔊) coincides with ℤ → ℚ → ℚ/ℤ for even r and with 0 → 0 → 0 for odd r.

4 A generalization of the topological Brauer group

4.1 More about H-spaces B GL(ℂ) and B PGL(ℂ)

Recall that we consider the spaces B GL(ℂ) and B PGL(ℂ) as H-spaces with respect to the multiplication induced by the tensor product of vector GL-bundles and PGL-bundles with fiber a matrix algebra, respectively.

Consider ∏_{q≥1} K(2q, ℚ) as an H-space with respect to the multiplication induced by the product of formal series of the form 1 + x_{1} + x_{2} + …, x_{i} ∈ H^{2i}(X; ℚ). We claim that Hspaces B GL(ℂ) and ∏_{q≥1} K(2q, ℚ) are isomorphic to each other. Indeed, it follows from the identity ch(ξ ⊗ η) = ch(ξ)ch(η) for any vector bundles ξ, η, where ch is the Chern character.

Note that for the special linear group SL we have similar isomorphism

B SL(ℂ) →^{ch} ∏_{q≥2} K(2q, ℚ),

where ∏_{q≥2} K(2q, ℚ) is considered as an H-space with respect to the multiplication induced by the product of formal series of the form 1 + x_{2} + x_{3} + …, x_{i} ∈ H^{2i}(X; ℚ).

Lemma 52. The H-space B PGL(ℂ) is isomorphic to the product of H-spaces K(2, ℚ/ℤ)× ∏_{q≥2} K(2q, ℚ).

Proof. Consider the commutative diagram of the H-spaces:

\xymatrix{B PGL(ℂ) \ar[r]^{ch} & ∏_{q≥2} K(2q, ℚ) \\ B SL(ℂ). \ar[u] \ar@{-->}[ur]^{≅}}

It follows from the last isomorphism in (16) that ker(ch) coincides with the H-space K(2, ℚ/ℤ). □

4.2 Reminder: the classical Brauer group

Let X be a finite CW -complex. Consider the set of isomorphism classes of locally trivial bundles over X with fiber M_{k}(ℂ) with an arbitrary integer k > 1. On the set of such bundles consider the following equivalence relation:

A_{k} ∼ B_{l} ⇔ A_{k} ⊗ M̃_{m} ≅ B_{l} ⊗ M̃_{n} for some m, n > 1,  (17)

where by M̃_{r} we denote the trivial bundle over X with fiber M_{r}(ℂ). By {C_{m}} denote the stable equivalence class of C_{m}. It can easily be checked that the product {A_{k}} ∘ {B_{l}} := {A_{k} ⊗B_{l}} is well defined and it equips the set of stable equivalence classes with the structure of Abelian group. We denote this group by ÃB(X).

The H-space B PGL(ℂ) is a represented space for the homotopy functor X ↦ ÃB(X) on the category of finite CW -complexes. Since we take the direct limit B PGL(ℂ) = lim_{\overrightarrow{k}} BPGL_{k}(ℂ) by the maps of classifying spaces BPGL_{k}(ℂ) ⟶^{i_{k, m}} BPGL_{km}(ℂ) such that

i_{k, m}^{∗}(A_{km}^{univ}) = A_{k}^{univ} ⊗ M̃_{m}, where by A_{r}^{univ} we denote the universal M_{r}(ℂ)-bundle over BPGL_{r}(ℂ), the stable equivalence relation (17) appears.

Now consider more coarse stable equivalence relation, where

A_{k} ∼ B_{l} ⇔ A_{k} ⊗ End(ξ_{m}) ≅ B_{l} ⊗ End(η_{n})  (18)

for some vector bundles ξ_{m} and η_{n} over X of rank m, n, respectively. By [C_{m}] we denote the stable equivalence class of M_{m}(ℂ)-bundle C_{m}. The tensor product of bundles induces a group structure on the set of such stable equivalence classes. It is just the classical topological Brauer group Br(X).

Let us give a homotopic description of Br(X). Consider the fibration

\begin{aligned}ℂP^{∞} & ↪ & BGL_{k}(ℂ) \\ & & ↓ \\ & & BPGL_{k}(ℂ) \end{aligned}  (19)

corresponding to the exact sequence of groups

1 → ℂ^{∗} → GL_{k}(ℂ) → PGL_{k}(ℂ) → 1.

After taking the limit with k → ∞, the above fibration gives us the fibration

\begin{aligned}ℂP^{∞} & ↪^{i} & B GL(ℂ) & i.e. & ℂP^{∞} ↪ ∏_{q≥1} K(ℚ, 2q) \\ & & ↓ p & & ↓ \\ & & B PGL(ℂ), & & K(ℚ/ℤ, 2) × ∏_{q≥2} K(ℚ, 2q). \end{aligned}  (20)

It is easy to see that Br(X) = coker{p_{∗}: [X; B GL(ℂ)] → [X; B PGL(ℂ)]}, i.e. Br(X) = coker{[X; K(ℚ, 2)] → [X; K(ℚ/ℤ, 2)]} = coker{H^{2}(X; ℚ) → H^{2}(X; ℚ/ℤ)}. Using the exact sequence of cohomology groups induced by the sequence of coefficients

0 → ℤ → ℚ → ℚ/ℤ → 0,

we see that coker{H^{2}(X; ℚ) → H^{2}(X; ℚ/ℤ)} = im{H^{2}(X; ℚ/ℤ) →^{δ} H^{3}(X; ℤ)} (here δ is the coboundary homomorphism), i.e. H_{tors}^{3}(X; ℤ).

The explicit form of the isomorphism Br(X) ≅ H_{tors}^{3}(X; ℤ) can be described as follows. Recall that the structural group of a bundle A_{k} with fiber M_{k}(ℂ) is Aut(M_{k}(ℂ)) = PGL_{k}(ℂ). The obstruction theory asserts that the first (and unique!) obstruction for the lifting of PGL_{k}(ℂ)-bundle to a GL_{k}(ℂ)-bundle belongs to the group H^{3}(X; π_{2}(ℂP^{∞})) (see fibration (19)). Therefore the assignment

A_{k} ↦ {the first obstruction for the lifting}

gives us the required description of the isomorphism Br(X) ≅ H_{tors}^{3}(X; ℤ).

Thus, any element of the group H_{tors}^{3}(X; ℤ) can be realized as an obstruction for the lifting of some PGL-bundle to a GL-bundle. Remark 53. Now suppose A_{k} has a lifting ξ_{k} (i.e. End(ξ_{k}) = A_{k}). Then the obstruction theory says that ξ_{k} is determined by A_{k} uniquely up to taking the tensor product with a line bundle ζ over X. Indeed, H^{2}(X; π_{2}(ℂP^{∞})) = H^{2}(X; ℤ) is isomorphic to the group of line bundles with respect to the tensor product; from the other hand, for any line bundle ζ we have End(ξ_{k} ⊗ ζ) = End(ξ_{k}). Remark 54. Note that for any fixed integer k > 1 one can develop the corresponding theory of bundles with fibers of the form M_{k^{n}}(ℂ) for arbitrary n ∈ ℕ. In this way one can define the kcomponent of the Brauer group Br_{k}(X) = coker{p_{k∗}: [X; BGL_{k^{∞}}(ℂ)] → [X; BPGL_{k^{∞}}(ℂ)]} (which is the k-primary subgroup in the “whole” group Br(X)).

4.3 The generalized Brauer group

Let us return to fibration (15):

Ĝr \underrightarrow{T̂′} B GL(ℂ) \underrightarrow{F̂′} B 𝔊.

By Fan we denote the homotopy functor to the category of Abelian groups represented by the H-space B 𝔊, i.e. X ↦^{Fan} [X; B 𝔊].

Thus, we have exact sequence of groups

[X; Ĝr] \underrightarrow{T̂′_{∗}} [X; B GL(ℂ)] \underrightarrow{F̂′_{∗}} Fan(X).

Definition 55. The generalized topological Brauer group GBr(X) of X is coker F̂′_{∗} = Fan(X)/ im(F̂′_{∗}).

Remark 56. Since Fan(X) is a finite (Abelian) group for a finite CW -complex X, so is GBr(X). Moreover, the k-primary component GBr_{k}(X) is just coker F̂′_{k^{∞}∗} = Fan^{k}(X)/ im(F̂′_{k^{∞}∗}).

Let us also remark that fibration (15) is closely connected with fibration (20):

ℂP^{∞} ↪^{i} B GL(ℂ) →^{p} B PGL(ℂ).

In particular, the space Ĝr ≅ BU_{⊗} represents the homotopy functor

X ↦ { the group of virtual U-bundles of virtual dimension 1 over X},

where the group operation is the tensor product of virtual bundles of virtual dimension 1, therefore it is a counterpart of the H-space ℂP^{∞}, which represents the group (of isomorphism classes) of geometric line bundles with respect to the operation induced by the tensor product.

Now consider fibration (14):

Gr \underrightarrow{T̂} B PGL(ℂ) \underrightarrow{F̂} B 𝔊.

By definition, put \overline{GB}r(X) := coker\{F̂_{∗}: [X; B PGL(ℂ)] → [X; B 𝔊]\}. Using the fibrations (15) and (14), one can easily prove the following proposition.

Proposition 57. There is an exact sequence of functors

0 → Br → GBr → \overline{GB}r → 0.

In other words, GBr is an extension of the classical Brauer group Br by \overline{GB}r.

Proof follows from the commutative diagram

\xymatrix{B GL(ℂ) \ar[d]_{ℂP^{∞}} \ar[dr]^{F̂′} \\ B PGL(ℂ) \ar[r]^{F̂} & B 𝔊. □}

Now we want to obtain a description of the generalized Brauer group in terms of equivalence classes of fan bundles with respect to some equivalence relation which is a counterpart of relation (18).

Recall that the H-space B 𝔊_{k^{∞}} is a classifying space for equivalence classes of fan bundles with fibers of the form 𝕄_{k^{m}, l^{n}} (for some l coprime with k) with respect to the following equivalence relation:

𝔄_{k^{m}, l^{n}} ∼ 𝔅_{k^{r}, l^{s}} ⇔ 𝔄_{k^{m}, l^{n}} ⊗ 𝕄̃_{k^{t}, l^{u}} ≅ 𝔅_{k^{r}, l^{s}} ⊗ 𝕄̃_{k^{v}, l^{w}},  (23)

where by 𝕄̃_{k^{m}, l^{n}} we denote the trivial 𝕄_{k^{m}, l^{n}}-bundle over X. This equivalence relation is a counterpart of relation (17).

The counterpart of relation (18) is the following one:

𝔄_{k^{m}, l^{n}} ∼ 𝔅_{k^{r}, l^{s}} ⇔ there are vector bundles ξ_{k^{t}}, η_{k^{v}}
with fibers ℂ^{k^{t}}, ℂ^{k^{v}}, respectively, and integers u, w ≥ 1 such that
𝔄_{k^{m}, l^{n}} ⊗ F_{k^{t}, l^{u}}^{X}(End(ξ_{k^{t}})) ≅ 𝔅_{k^{r}, l^{s}} ⊗ F_{k^{v}, l^{w}}^{X}(End(η_{k^{v}}))  (24)

(recall that F_{k, l}^{X} is the functor 𝔄𝔅_{k}(X) → 𝔉𝔞𝔫_{k, l}(X), i.e. F_{k, l}^{X}(End(ξ_{k})) is a fan bundle with fiber 𝕄_{k, l}). It is natural to denote the composite functor ξ_{k} ↦ End(ξ_{k}) ↦ F_{k, l}^{X}(End(ξ_{k})) by F′_{k, l}^{X}.

Note that the tensor product of fan bundles induces a well-defined operation on equivalence classes of equivalence relation (24). Note also that the equivalence classes of bundles of the form F_{k^{t}, l^{u}}^{X}(End(ξ_{k^{t}})) form the image of the homomorphism F̂′_{k^{∞}∗}: [X; BGL_{k^{∞}}(ℂ)] → [X; B 𝔊_{k^{∞}}]. Now the proof of the following proposition is clear.

Proposition 58. The k-primary component GBr_{k}(X) of the generalized Brauer group GBr(X) coincides with the group of equivalence classes of equivalence relation (24).

The similar description can be given for the group \overline{GB}r(X). More precisely, consider the following equivalence relation:

𝔄_{k^{m}, l^{n}} ∼ 𝔅_{k^{r}, l^{s}} ⇔ there are bundles A_{k^{t}}, B_{k^{v}}
with fibers M_{k^{t}}(ℂ), M_{k^{v}}(ℂ), respectively, and integers u, w ≥ 1 such that
𝔄_{k^{m}, l^{n}} ⊗ F_{k^{t}, l^{u}}^{X}(A_{k^{t}}) ≅ 𝔅_{k^{r}, l^{s}} ⊗ F_{k^{v}, l^{w}}^{X}(B_{k^{v}}).  (25)

Note that the equivalence classes of bundles of the form F_{k^{t}, l^{u}}^{X}(A_{k^{t}}) form the image of the homomorphism F̂_{k^{∞}∗}: [X; BPGL_{k^{∞}}(ℂ)] → [X; B 𝔊_{k^{∞}}].

Proposition 59. The k-primary component \overline{GB}r_{k}(X) of the group \overline{GB}r(X) coincides with the group of equivalence classes of relation (25).

Remark 60. (compare with Remark 53). Suppose a fan bundle 𝔄_{k, l} over X has a lifting A_{k} (i.e. 𝔄_{k, l} = F_{k, l}^{X}(A_{k}) for some M_{k}(ℂ)-bundle A_{k}). Then the bundle A_{k} is determined by the equivalence class (23) of 𝔄_{k, l} up to taking the tensor product with the core of some FAB. In particular, the liftings of the stable equivalence class (23) of a trivial fan bundle are precisely the cores of FABs over X; thus, we also see that FABs play the same role as (geometric) line bundles. This result follows from fibration (14). The same result is true for fibration (15).

Suppose a finite CW -complex X is obtained by applying the forgetful functor to some algebraic variety X̃ over ℂ. Then the group Br(X) is not only a homotopy invariant of X but also a birational invariant of X̃.

Conjecture 61. The group GBr(X) is a birational invariant of X̃.

5 Appendix 1: The functor represented by the Hspace Gr

In this appendix we give a brief survey of the stable theory of floating algebra bundles, for more details see [3].

5.1 Category 𝔉𝔄𝔅(X)

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

Definition 62. Let A_{k} (k > 1) be a locally trivial bundle over X with a fiber M_{k}(ℂ). Assume that 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, and the positive integers k, l are relatively prime (i.e. their greatest common divisor (k, l) = 1 ) . Then the triple (A_{k}, μ, M̃_{kl}) is called a floating algebra bundle (abbrev. FAB) over X.

Remark 63. 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 a 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}}.

Definition 64. A morphism from a FAB (A_{k}, μ, M̃_{kl}) to a FAB (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

• f, g 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 for A_{k}, C_{m}, respectively (see the remark above), 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 FABs (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 diagram

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

commutes.

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

For a continuous map φ: X → Y we have the natural transformation φ^{∗}: 𝔉𝔄𝔅(Y) → 𝔉𝔄𝔅(X).

5.2 Classifying spaces

For any pair k, l > 1 there is the “matrix Grassmannian” 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 tautological FAB (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. Remark 65. The noncompact space Gr_{k, l} can be replaced by a homotopy equivalent compact one Gr_{k, l}^{U} . More precisely, let PU(n) be the projective unitary group, i.e. the quotient group U(n)/{exp(iφ)E_{n} | φ ∈ ℝ}. Then

Gr_{k, l}^{U} = PU(kl)/PU(k) ⊗ PU(l)

(which is the subspace in Gr_{k, l} corresponding to the “unitary” subalgebras) with a tautological FAB (A_{k, l}^{U}, μ^{U}, M̃_{kl}^{U}) over it.

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

Proposition 66. If dim X ≤ 2 min{k, l}, then the assignment

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

is a bijection.

5.3 Stabilization

Define the product ∘ of two FABs over X (A_{k}, μ, M̃_{kl}), (B_{m}, ν, M̃_{mn}) such that (km, ln) = 1 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 67. A FAB of the form (M̃_{k}, τ, M̃_{kl}) is called trivial if the map τ: M̃_{k} → M̃_{kl} acts as follows,

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 68. 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 corresponding 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 69. 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

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

homotopy type we denote by Gr.

2) Gr 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 Gr.

The proof is based on Proposition 66 and on the following lemma (recall that we consider embeddings Gr_{k, l} → Gr_{m, n} induced by homomorphisms of algebras M_{kl}(ℂ) ↪ M_{mn}(ℂ); in particular, k|m, l|n).

Lemma 70. If (m, n) = 1 (⇒ (k, l) = 1), then the embedding

Gr_{k, l} → Gr_{m, n}  (31)

is a homotopy equivalence in dimensions ≤ 2 min{k, l}.

5.4 The group structure

Let (A_{k}, μ, M̃_{kl}) be a FAB 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})]
= [(A_{k} ⊗ B_{m}, μ ⊗ ν, M̃_{klmn})].

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

Lemma 71. 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}).

Clearly, the product ⋄ is associative, commutative, and has identity element [(M̃_{k}, τ, M̃_{kl})], where (M̃_{k}, τ, M̃_{kl}) is the 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 by taking centralizers for every fiber 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.

Proposition 72. The space Gr can be equipped with an H-space structure such that there is a natural equivalence of functors X ↦ [X, Gr] and X ↦ ÃB^{1}(X) taking values in the category 𝔄𝔟.

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

Definition 73. For a FAB (A_{k}, μ, M̃_{kl}), the locally trivial Aut(M_{k}(ℂ)) ≅ PGL_{k}(ℂ)-bundle A_{k} is said to be a core of (A_{k}, μ, M̃_{kl}).

Lemma 74. If A_{k} is the core of some FAB (A_{k}, μ, M̃_{kl}), then its structural group can be reduced from Aut M_{k}(ℂ) ≅ PGL_{k}(ℂ) to SL_{k}(ℂ).

Lemma 75. Let X be a finite CW -complex. Suppose dim X ≤ 2 min{k, m}; then the following conditions are equivalent:

• A_{k} is the core of some FAB over X;

• 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};

• A_{k} ⊗ M̃_{m} ≅ B_{m} ⊗ M̃_{k} for some locally trivial bundle B_{m} with fiber M_{m}(ℂ), where (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 sufficient large n, (km, n) = 1) FABs of the forms (A_{k}, μ, M̃_{kn}), (B_{m}, ν, M̃_{mn}) as representatives (for some embeddings μ, ν).

5.6 Localization

Let X be a finite CW -complex, let k ≥ 2 be 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}}  (32)

(⇒ m + r = n + s). The set of equivalence classes [A_{k^{m}}] of such bundles 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}} BPGL_{k^{n}}(ℂ) with respect to the maps induced by

\begin{matrix}PGL_{k^{n}}(ℂ) & ↪ & PGL_{k^{n+1}}(ℂ), \\ A & ↦ & A ⊗ E_{k}. \end{matrix}

Clearly, the functor X ↦ ÃB^{k}(X) is represented by lim_{\overrightarrow{n}} BPGL_{k^{n}}(ℂ).

According to Lemma 71, for any stable equivalence class of FABs over X there is a representative of the form (A_{k^{m}}, μ, M̃_{(kl)^{m}}). Therefore for any k we have the forgetful functor

T̂_{k^{∞}, l^{∞}∗}^{X}: ÃB^{1}(X) → ÃB^{k}(X), [(A_{k^{m}}, μ, M̃_{(kl)^{m}})] ↦ [A_{k^{m}}].

This functor corresponds to the following diagram

[figure: _{Gr_{k^{2}, l^{2}}}; BPGL_{k^{2}}(ℂ); _{Gr_{k, l}}; BPGL_{k}(ℂ),]

where the horizontal arrows are classifying maps for the cores A_{k, l} and A_{k^{2}, l^{2}} as PGL_{k}(ℂ) and PGL_{k^{2}}(ℂ)-bundles, respectively.

The kernel of the homomorphism ÃB^{1}(X) → ÃB^{k}(X) is just the k-torsion subgroup in ÃB^{1}(X).

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

Recall 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 76. There is an H-space isomorphism Gr ≅ BSU_{⊗} .

By { \atop KSU}(X) denote the reduced K-functor constructed by means of SU-bundles over X. Recall that { \atop KSU}(X) is a ring with the multiplication induced by the tensor product of bundles.

The previous theorem shows that the group ÃB^{1}(X) is isomorphic to the multiplicative group of the ring { \atop KSU}(X), i.e. to the group (X is a finite CW -complex) of elements of { \atop KSU}(X) with respect to the operation ξ ∗ η = ξ + η + ξη (ξ, η ∈ { \atop KSU}(X)).

This gives us a geometric description of the H-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.

5.8 A GL-version

Consider the canonical map BGL_{k}(ℂ) → BPGL_{k}(ℂ) induced by the group homomorphism GL_{k}(ℂ) → PGL_{k}(ℂ). By Ĝr_{k, l} denote the total space of the Fr_{k, l}-fibration induced by the fibration Gr_{k, l} ⟶^{Fr_{k, l}} BPGL_{k}(ℂ) and the map BGL_{k}(ℂ) → BPGL_{k}(ℂ) (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}.

Consider the following morphism of GL_{k}(ℂ)-fibrations:

\xymatrix{ & GL_{k}(ℂ) \ar[r] & ∗ \ar[d] \\ GL_{k}(ℂ) \ar[ur]^{=} \ar[r] & _{Fr_{k, l}} \ar[d] \ar[ur] & BGL_{k}(ℂ) \\  & Ĝr_{k, l}, \ar@{-->}[ur]^{T̂′_{k, l}}}

where T̂′_{k, l} is a classifying map for the canonical GL_{k}(ℂ)-bundle over Ĝr_{k, l} and by ∗ we denote a contractible space. A simple computation with homotopy sequences of the fibrations shows that T̂′_{k, l∗}: π_{2r}(Ĝr_{k, l}) → π_{2r}(BGL_{k}(ℂ)), 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 limit map T̂′_{k^{∞}}: Ĝr → BGL_{k^{∞}}(ℂ) is just the localization at k (in the sense that k is invertible; in particular, BGL_{k^{∞}}(ℂ) 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 BU_{⊗} ≅ BSU_{⊗} ×ℂP^{∞} and Gr ≅ BSU_{⊗} as H-spaces, 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: The space Fr_{k^{∞}} is a loop space

Here we consider another way to prove that Fr_{k^{∞}} is a loop space.

First we recall some properties of the localization. Let X be a CW -complex such that X = Ω^{N}Z for some space Z and positive integer N. For a CW -complex Y by λ[\frac{1}{k}]_{Y} : Y → Y [\frac{1}{k}] we denote the localization at k (i.e. homotopy and homology groups of Y [\frac{1}{k}] are ℤ[\frac{1}{k}]- modules).

Lemma 77. There is a weak homotopy equivalence α: X[\frac{1}{k}] → Ω^{N}(Z[\frac{1}{k}]) such that the diagram

[figure: X = Ω^{N}Z; λ[ \frac{1}{k}]_{X}; Ω^{N} (λ[ \frac{1}{k}]_{Z}); Ω^{N}(Z[\frac{1}{k}]); X[\frac{1}{k}]; α; ≅]

commutes. In other words, “delooping and localization commute”.

Proof. Since the space Ω^{N}(Z[\frac{1}{k}]) is ℤ[\frac{1}{k}]-local, the existence of α follows from the universal property of the localization. It is easy to see that α induces an isomorphism of homotopy groups. □

The following obvious fact is needed for the sequel.

Lemma 78. In the above notation, let Y_{k} be a fiber of the fibration Y_{k} → Z ⟶^{λ[ \frac{1}{k}]_{Z}} Z[\frac{1}{k}]. Then a fiber of the fibration X ⟶^{λ[ \frac{1}{k}]_{X}} X[\frac{1}{k}] is just Ω^{N}Y_{k}.

Now, applying the above lemmas to the localization map

Ĝr \underrightarrow{T̂′_{k^{∞}}} BGL_{k^{∞}}(ℂ),  (33)

we obtain the following result.

Proposition 79. For any positive integer N the space Fr_{k^{∞}} is an N-fold loop space.

Proof. Recall that

(i) map (33) is the localization at k;

(ii) fibration (33) has fiber Fr_{k^{∞}} .

Furthermore, it is well-known that Ĝr ≅ BU_{⊗} is an infinite loop space. Now, applying the above lemmas, we complete the proof. □ Remark 80. In particular, Fr = lim_{\overrightarrow{(k,l)=1}} Fr_{k, l} = Ω(B 𝔊), and therefore B 𝔊 is an infinite loop

space too. Thus, the group [X; B 𝔊] can be considered as 0^{th} term of some generalized cohomology theory.

References

[1] A. Grothendieck Le groupe de Brauer I. Sem. Bourbaki 1964/65, no. 290, 21p.

[2] N. Steenrod The topology of fibre bundles. Princeton University Press, Princeton, New Jersey, 1951.

[3] A.V. Ershov Homotopy theory of bundles with fiber matrix algebra. MPI-Preprint, 2003 (1).