Categories of Pointwise Convergence Topology: Theory(ies) of Bundles.

Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {U_i}_i∈I of X, the transition functions between trivializations P|_Ui can be lifted to bundle isomorphisms g_ij on double intersections U_ij = U_i ∩ U_j which are projectively coherent, i.e., over each of the triple intersections U_ijk = U_i ∩ U_j ∩ U_k the composition g_ki g_jk g_ij is given as multiplication by a U(1)-valued function f_ijk : U_ijk → U(1). The collection {(U_ij, f_ijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class B_P in the sheaf cohomology group H²(X, U(1)_X). On the other hand, the sheaf cohomology H¹(X, PU(H)_X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H¹(X,PU(H)_X).

There is an isomorphism

H¹(X, PU(H)_X) →^≈ H²(X, U(1)_X) provided by the

boundary map [P] ↦ B_P. There is also an isomorphism

H²(X, U(1)_X) →^≈ H³(X, Z_X) ≅ H³(X, Z)

The image δ(P) ∈ H³(X, Z) of B_P is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H³(X, R) by a closed three-form H on X, called the H-flux of the given B-field B_P, we will write P = P_H. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H³(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Z_n with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cⁿ). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element B_P ∈ H²(X, U(1)_X) comes from H²(X, (Z_n)_X), and therefore its order divides n.

One also has δ(P₁ ⊗ P₂) = δ(P₁) + δ(P₂) and δ(P^∨) = −δ(P). This follows from the commutative diagram

and the fact that P^∨ ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H³(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that P_x = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z₂-graded Hilbert space. We define Fred⁰(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T² − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred⁰(H) ֒→ B(H) × K(H) given by T → (T, T² − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = P_H → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred⁰(P) whose fibres are Fred⁰(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K⁰(X, H) = [X, Fred⁰(P_H)]

The group K⁰(X, H) depends functorially on the pair (X, P_H), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K⁰(X, H) → K⁰(X, H′). Addition in K⁰(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K⁰(X, H₂) with [H₂] = δ(P ⊗ P(C²)) = δ(P) = [H]. Under the isomorphism H ⊗ C² ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C²) for any projective bundle P and so K⁰(X, H₂) is canonically isomorphic to K⁰(X, H). When [H] is a non-torsion element of H³(X, Z), so that P = P_H is an infinite-dimensional bundle of projective spaces, then the index map K⁰(X, H) → Z is zero, i.e., any section of Fred⁰(P) takes values in the index zero component of Fred⁰(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle P_H we can associate a bundle of compact operators E_H → X given by

E_H = P_H ×_PU K

with δ(E_H) = [H]. The Banach algebra A_H := C₀(X, E_H) of continuous sections of E_H vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K^•(X, H) of X is canonically isomorphic to the algebraic K-theory group K_•(A_H).

We will also need a smooth version of this definition. Let A^∞_H be the smooth subalgebra of A_H given by the algebra CT^∞(X, H) = C^∞(X, L¹_PH),

where L¹_PH = P_H ×_PUL¹. Then the inclusion CT^∞(X, H) → CT(X, H) induces an isomorphism K_•CT(X, H) →^≈ K_•CT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism K_•CT^∞(X, H) ≅ K^•(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = P_H over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K⁰(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, M_m(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism

δ : Br(X) →^≈ tor(H³(X, Z))

where tor(H³(X, Z)) is the torsion subgroup of H³(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K₀(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H³(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K⁰(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K⁰(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K₀(C(X)) = K⁰(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre C^nm ≅ (C^m)^⊕n, which is naturally an M_m(C)-module.

 

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Homotopically Truncated Spaces.

The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.

A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram

such that p_n∗ : π_r(X) → π_r(P_n(X)) is an isomorphism for r ≤ n and π_r(P_n(X)) = 0 for r > n. Let F_n be the homotopy fiber of q_n. Then the exact sequence

π_r+1(P_nX) →^q_n∗ π_r+1(P_n−1X) → π_r(F_n) → π_r(P_nX) →^q_n∗ π_r(P_n−1X)

shows that F_n is an Eilenberg–MacLane space K(π_nX, n). Constructing P_n+1(X) inductively from P_n(X) requires knowing the nth k-invariant, which is a map of the form k_n : P_n(X) → Y_n. The space P_n+1(X) is then the homotopy fiber of k_n. Thus there is a homotopy fibration sequence

K(π_n+1X, n+1) → P_n+1(X) → P_n(X) → Y_n

This means that K(π_n+1X, n+1) is homotopy equivalent to the loop space ΩY_n. Consequently,

π_r(Y_n) ≅ π_r−1(ΩY_n) ≅ π_r−1(K(π_n+1X, n+1) = π_n+1X, r = n+2,

= 0, otherwise.

and we see that Y_n is a K(π_n+1X, n+2). Thus the nth k-invariant is a map k_n : P_n(X) → K(π_n+1X, n+2)

Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the P_n(X).

Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:

• • the direction of all arrows
  • π_∗ to H_∗
  • loops Ω to suspensions S
  • fibrations to cofibrations and fibers to cofibers
  • Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
  • inverse limits to direct limits

A homology decomposition (or Moore space decomposition) for a simply connected CW-complex X is a commutative diagram

such that j_n∗ : H_r(X_≤n) → H_r(X) is an isomorphism for r ≤ n and H_r(X_≤n) = 0 for

r > n. Let C_n be the homotopy cofiber of i_n. Then the exact sequence

H_r(X_≤n−1) →^i_n∗ H_r(X_≤n) → H_r(C_n) →^i_n∗ H_r−1(X_≤n−1) → H_r−1(X_≤n)

shows that C_n is a Moore space M(H_nX, n). Constructing X_≤n+1 inductively from X_≤n requires knowing the nth k-invariant, which is a map of the form k_n : Y_n → X_≤n.

The space X_≤n+1 is then the homotopy cofiber of k_n. Thus there is a homotopy cofibration sequence

Y_n →^k_n X_≤n →^i_n+1 X_≤n+1 → M(H_n+1X, n+1)

This means that M(H_n+1X, n+1) is homotopy equivalent to the suspension SY_n. Consequently,

H^˜_r(Y_n) ≅ H^∼_r+1(SY_n) ≅ H_r+1(M(H_n+1X, n+1)) = H_n+1X, r = n,

= 0, otherwise

and we see that Y_n is an M(H_n+1X, n). Thus the nth k-invariant is a map k_n : M(H_n+1X, n) → X_≤n

It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:

k_n∗ : H_n(M(H_n+1X, n)) ∼= H_n+1(X) → H_n(X) ∼= H_n(X_≤n)

The original space X is homotopy equivalent to the direct limit of the X_≤n. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let p_n(X) : X → P_n(X) be a stage-n Postnikov approximation for X, that is, p_n(X)_∗ : π_r(X) → π_r(P_n(X)) is an isomorphism for r ≤ n and π_r(P_n(X)) = 0 for r > n. If Z is a space with π_r(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through P_n(X). In particular, if f : X → Y is any map and p_n(Y) : Y → P_n(Y) is a stage-n Postnikov approximation for Y, then, taking Z = P_n(Y) and g = p_n(Y) ◦ f, there exists, uniquely up to homotopy, a map p_n(f) : P_n(X) → P_n(Y) such that

homotopy commutes. Let X = S² ∪₂ e³ be a Moore space M(Z/2,2) and let Y = X ∨ S³. If X_≤2 and Y_≤2 denote stage-2 Moore approximations for X and Y, respectively, then X_≤2 = X and Y_≤2 = X. We claim that whatever maps i : X_≤2 → X and j : Y_≤2 → Y such that i_∗ : H_r(X_≤2) → H_r(X) and j_∗ : H_r(Y_≤2) → H_r(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage-2 Moore approximations, i.e. there is no map f_≤2 : X_≤2 → Y_≤2 such that

commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence

0 → Ext(G, π_n+1Y) →^ι [M(G, n), Y] →^η Hom(G, π_nY) → 0,

where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on π_n and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:

Here we will briefly write E₂(−) = Ext(Z/2,−) so that E₂(G) = G/2G, and E^Y (−) = Ext(−, π₃Y). By the Hurewicz theorem, π₂(X) ∼= H₂(X) ∼= Z/2, π₂(Y) ∼= H₂(Y) ∼= Z/2, and π₂(i) : π₂(X_≤2) → π₂(X), as well as π₂(j) : π₂(Y_≤2) → π₂(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E₂(φ) : E₂(A) = A/2A → B/2B = E₂(B) remains onto. By the Hurewicz theorem, Hur : π₃(Y) → H₃(Y) = Z is onto. Consequently, the induced map E₂(Hur) : E₂(π₃Y) → E₂(H₃Y) = E₂(Z) = Z/2 is onto. Let ξ ∈ E₂(H₃Y) be the generator. Choose a preimage x ∈ E₂(π₃Y), E₂(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f_≤2] ∈ [X_≤2, Y_≤2] such that

j_∗[f_≤2] = i_∗[f].

Then

η_≤2[f_≤2] = π₂(j)_∗η_≤2[f_≤2] = ηj_∗[f_≤2] = ηi^∗[f] = π₂(i)^∗η[f] = π₂(i)^∗ηι(x) = 0.

Thus there is an element ε ∈ E₂(π₃Y_≤2) such that ι_≤2(ε) = [f_≤2]. From ιE₂π₃(j)(ε) = j_∗ι_≤2(ε) = j_∗[f_≤2] = i^∗[f] = i^∗ι(x) = ιE^Y π₂(i)(x)

we conclude that E₂π₃(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square

commutes and induces a commutative diagram upon application of E₂(−):

It follows that

ξ = E₂(Hur)(x) = E₂(Hur)E₂π₃(j)(ε) = E₂H₃(j)E₂(Hur)(ε) = 0,

a contradiction. Therefore, no compression [f_≤2] of [f] exists.

Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.

Truncation Functors

Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ_≤nA• be the truncated complex

··· → Aⁿ⁻² → Aⁿ⁻¹ → ker(Aⁿ → Aⁿ⁺¹)→ 0 → 0 → ··· , and dually we let τ_≥nA be the complex

··· → 0 → 0 → coker(Aⁿ⁻¹ → Aⁿ) → Aⁿ⁺¹ → Aⁿ⁺² → ···

Note that

H^m(τ_≤nA•) = H^m(A•) if m ≤ n

= 0 if m > n

and that

H^m(τ_≥nA•) = H^m(A•)  if m ≥ n

= 0 if m < n

One checks that τ_≥n (respectively τ_≤n) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D_≤n (respectively D_≥n) is the full subcategory of D whose objects are the complexes A• such that H^m(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ_≤n : D → D_≤n ⊂ D

τ_≥n : D → D_≥n ⊂ D

together with obvious functorial maps

iⁿ_A : τ_≤n A• → A•

jⁿ_A : A• → τ_≥n A•

The preceding iⁿ_A , jⁿ_A induce functorial isomorphisms

Hom_D≤n (B•,τ_≤nA•) →^~ Hom_D(B•, A•) (B• ∈ D_≤n) —– (1)

Hom_D≥n (τ_≥nA•,C•) →^~ Hom_D(A•,C• ) (C• ∈ D_≥n) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D_≤n factors uniquely via i_A := iⁿ_A

Given φ, we have a commutative diagram

and since B• ∈ D_≤n, therefore i_B is an isomorphism in D, so we can write

φ = i ◦ (τ_≤nφ ◦ i⁻¹_B),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that i_A ◦ τ_≤n φ = 0 and deduce that τ_≤n φ = 0. The assumption means that there is a commutative diagram in K(A)

where s and s′′ are quasi-isomorphisms, and f/s = τ_≤nφ

Applying the (idempotent) functor τ_≥n, we get a commutative diagram

Since τ_≤ns and τ_≤ns′′ are quasi-isomorphisms, we have

τ_≤nφ = τ_≤n f/τ_≤ns = 0/τ_≤ns′′ = 0

as desired.