The topology of T -duality for T^{n}-bundles

Ulrich Bunke, Phillip Rumpf, and Thomas Schick ^{∗} January 27, 2005

Contents

1 Introduction 1 2 The space R_{n} 6 3 Universal T -duality 12 4 R_{n} as a classifying space 15 5 T -duality transformations in twisted cohomology 23

1 Introduction

1.1 The notion of T -duality has its origin in string theory. It asserts that a certain type of string theory on a target F partially compactified on a torus is equivalent to another type of string theory on a dual target F̂. In the presence of an H-flux the topological aspects of T -duality are of particular interest. In the present paper we concentrate on the topological aspects of T -duality, and we neglect geometric questions and physical interpretations.

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, bunke@uni-math.gwdg.de, prumpf@uni-math.gwdg.de schick@uni-math.gwdg.de

1.2 T -duality as studied in the present paper has a two sides, a constructive one and a descriptive one. The constructive side is the main focus of the present paper and described below. The descriptive picture relies on the notion of a T -duality triple ^{\underline{tr13}}^{\underline{tr13}} 5.5.

Let T^{n} := U(1)^{n} denote the n-dimensional torus. Let B be some base space. We consider a pair (F, h) over B consisting of a T^{n}-principal bundle F → B and a class h ∈ H^{3}(F, ℤ). In Definition ^{\underline{drt}}^{\underline{drt}} 4.7 we introduce the notion of a T -dual pair using the following construction. In \underline{rt3}^{\underline{rt3}} 2.1 we construct a space R_{n} and a homotopy involution T : R_{n} → R_{n}. This space classifies a set-valued functor B ↦ P_{n}(B) := [B, R_{n}] with an automorphism T_{B} : P_{n}(B) → P_{n}(B) of order two induced by T ([B, R_{n}] denotes the homotopy classes of maps from B to R_{n}). Let B → P̃_{(0)}(B) denote the functor which associates to a space B the set of isomorphism classes of pairs over B. Then we obtain a (forgetful) transformation v_{B} : P_{n}(B) → P̃_{(0)}(B). In order to define a T -dual of a pair (F, h) we must choose an element x ∈ v_{B}^{−1}([F, h]). Then we define the isomorphism class of the T -dual pair as the underlying class of pairs of T_{B}(x), i.e. [F̂, ĥ] := v_{B}(T_{B}(x)).

filtex

1.3 In this approach, by definition a T -dual pair exists precisely for each pair in the image of v_{B}. The cohomology of the total space of the fibre bundle π : F → B has a decreasing filtration

0 ⊂ F^{n}H^{n}(F, ℤ) ⊂ F^{n−1}H^{n}(F, ℤ) ⊂ ⋯ ⊂ F^{1}H^{n}(F, ℤ) ⊂ F^{0}H^{n}(F, ℤ) = H^{n}(F, ℤ)

such that F^{n}H^{n}(F, ℤ) = π^{∗}(H^{n}(B, ℤ)). Choose a CW-decomposition B^{(0)} ⊂ B^{(1)} ⊂ ⋯ of B. By definition, x ∈ F^{k}H^{n}(F, ℤ) if its restriction to π^{−1}(B^{(k−1)}) is zero. The associated graded group is calculated by the Leray-Serre spectral sequence. For this reason we call this filtration the Leray-Serre filtration.

th1 Theorem 1.1 (Corollary ^{\underline{exex}}^{\underline{exex}} 4.8) The pair (F, h) admits a T -dual pair if and only if h ∈ F^{2}H^{3}(F, ℤ).

1.4 The approach to T -duality in the present paper is similar to (and inspired by) the point of view of ^{\underline{bhm}} [1]. In that paper the authors work on smooth T^{n}-bundles equipped with real valued cohomology classes h_{ℝ}. The T -dual pair is constructed geometrically using differential forms.

1.5 An important aspect of (topological) T -duality is an (expected) isomorphism between suitable twisted cohomology groups of the bundle and its dual. In order to formulate precise statements about T -duality isomorphisms between twisted cohomology groups one must fix the isomorphism classes of the twists precisely and not only up to torsion. We refer to ^{\underline{bunkeschickt}} [3], Sec. 4.3, for an explicit example which shows that the torsion part plays a significant role. The present paper represents an improvement over ^{\underline{bhm}} [1] since it introduces and studies T -duality of pairs (F, h) with an integral cohomology class h.

Moreover, we make precise the role of the identification of twists u introduced in ^{\underline{kli}}^{\underline{kli}} 5.2. This point is neglected in ^{\underline{bhm}} [1].

1.6 Note that a T -dual pair of (F, h) is not unique. This has already been observed in \underline{bunkeschickt} [3], Sec. 4.4. One of the results of the present paper is the detailed description of set of possible choices.

In order to define a T -dual of a pair (F, h) we must choose an element x ∈ v_{B}^{−1}([F, h]). In general the isomorphism class v_{B}(T_{B}(x)) of the T -dual pair will depend non-trivially on the choice of x ∈ v_{B}^{−1}([F, h]).

1.7 In order to study the fibers of v_{B} we construct a non-canonical isomorphism (see Corollary ^{\underline{hhll}}^{\underline{hhll}} 4.6) with another functor B ↦ P̃_{n}(B). The elements of P̃_{n}(B) have a geometric meaning, and the fiber of the forgetful map ṽ_{B} : P̃_{n}(B) → P̃_{(0)}(B) can be studied by obstruction theory (see ^{\underline{clh}}^{\underline{clh}} 4.9 and ^{\underline{clh1}}^{\underline{clh1}} 4.10). It seems to be difficult to formulate a general result. But let us mention the following opposite extremes.

(1) If B = S^{3}, then the transformation ṽ_{S^{3}} : P̃_{n}(S^{3}) → P_{0}(S^{3}) is an inclusion of a subset.

(2) If B = R_{n} is the space introduced in ^{\underline{rt3}}^{\underline{rt3}} 2.1, then the fibers of ṽ_{R_{n}} : P_{n}(R_{n}) → P̃_{(0)}(R_{n}) are torsors over the group H^{2}(T^{n}, ℤ).

1.8 In ^{\underline{mr}} [4] and ^{\underline{mr1}} [5] the authors follow a completely different point of view. The class h ∈ H^{3}(F, ℤ) is the Dixmier-Douady class of a unique isomorphism class of a stable continuous trace algebra A := A(F, h) with spectrum F. The authors study the question of lifting the T^{n}-action on F to an ℝ^{n}-action on A such that the Mackey invariant is trivial. In this case the crossed product Â := A ⋊ ℝ is again a continuous trace algebra with a spectrum F̂ which is a T^{n}-principal bundle. Let ĥ ∈ H^{3}(F̂, ℤ) denote the Dixmier-Douady class of Â. By the point of view of ^{\underline{mr}} [4], ^{\underline{mr1}} [5] the pair (F̂, ĥ) is the T -dual of (F, h).

The two problems in this approach are to decide under which conditions the required ℝ^{n}-action on A exists, and to study the set of choices for such an action. A satisfactory picture can be obtained in the cases n = 1 and n = 2. The case n = 1 is easy and has been reviewed in ^{\underline{bunkeschickt}} [3]. The main results of ^{\underline{mr}} [4] deal with the case n = 2. The necessary and sufficient condition for the existence of the ℝ^{n} action is again that h ∈ F^{2}H^{3}(F, ℤ). It is then claimed in ^{\underline{mr}} [4], that the action is unique. This is not always true. In fact, it follows from the diagram given in ^{\underline{mr}} [4], Theorem 4.3.3, and the observation that d′′_{2} factors over p_{!} : H^{2}(F, ℤ) → H^{0}(B, ℤ) that the group H^{0}(B, ℤ)/im(p_{!}) acts freely on the set of ℝ-actions with trivial Mackey invariant lifting the T^{n}-action on F. One can check that the same group acts freely on the fibers of v_{B}.

For general n ≥ 3 the relation between the the classification of the ℝ^{n}-actions on A(F, h) with trivial Mackey invariant and the classification of the elements in v_{B}^{−1}([F, h]) is not clear.

1.9 In ^{\underline{bhm2}} [2] the point of view of ^{\underline{mr,mr1}} [4, 5] is generalized even further by considering torus bundles with completely arbitrary H-flux differential 3-form with integral periods. It is then argued that the resulting dual should be a bundle of non-associative non-commutative tori, and a number of (expected) properties of this duality are detailed. Questions about choices involved in the construction of the dual object, or about a lift of the invariants to integral cohomology were not in the focus of these papers.

1.10 The present paper also gives the correct description of the topological invariants of a^{1} T -dual, i.e. relations between the Chern classes and the H^{3}-classes. This way, we improves the results of ^{\underline{mr1}} [5]. The calculation in ^{\underline{mr1}} [5] is wrong since the inclusion X_{j} → X in (4) of ^{\underline{mr1}} [5] does not exist in general.

1It is not clear at the moment that the notions of a T -dual used in the present paper coincides with that of \underline{mr1} [5]

1.11 Let us consider two pairs (F, h) and (F̂, ĥ) over B. The interest of T -duality in topology stems to a great deal from an (expected) isomorphism of certain twisted cohomology groups of the two spaces. Let h(…, …) denote a twisted cohomology theory in the sense of ^{\underline{bunkeschickt}} [3, Section 3]. Let H be a twist over F in the isomorphism class determined by h, and similarly, let Ĥ be a twist over F̂ in isomorphism class ĥ. Let p : F ×_{B} F̂ → F and p̂: F ×_{B} F̂ → F̂ be the projections. We assume that there exists an isomorphism of twists u : p̂^{∗}Ĥ → p^{∗}H. The triple ((F, H), (F̂, Ĥ), u) is a T -duality triple (see Definition \underline{tr13}^{\underline{tr13}} 5.5), if u satisfies a condition called P(u) (see ^{\underline{condp}}^{\underline{condp}} 5.4). If we assume that (F̂, ĥ) is T -dual to (F, h) (in the constructive way), then by Corollary ^{\underline{rr43}}^{\underline{rr43}} 5.6 there exists an u such that ((F, H), (F̂, Ĥ), u) is a T -duality triple.

1.12 Let us assume that ((F, H), (F̂, Ĥ), u) is a T -duality triple. In Definition ^{\underline{trd}}^{\underline{trd}} 5.7 we define a T -duality transformation T : h(F, H) → h(F̂, Ĥ) of degree −n which is linear over h(B). It is given by T := p̂_{!} ∘ u^{∗} ∘ p^{∗}.

rrr Theorem 1.2 (Theorem ^{\underline{isot}}^{\underline{isot}} 5.8) If the twisted cohomology theory h is T -admissible in the sense of ^{\underline{bunkeschickt}} [3, Definition 3.7], then the T -duality transformation is an isomorphism.

Examples for T -admissible cohomology theories are twisted extensions of K-theory or cohomology with coefficients in the graded ring ℝ[x, x^{−1}], where x has degree 2.

A corresponding result is also stated in ^{\underline{bhm}} [1]. In the approach of ^{\underline{mr}} [4] the T -duality isomorphism for twisted K-theory is equivalent to an isomorphism K(A) ≅ K(Â). In fact it is a by now classical result that these groups are isomorphic by Connes’ Thom isomorphism for crossed products with ℝ^{n}.

1.13 Assume that F = B × T^{n} is the trivial T^{n}-bundle and that we consider the trivial twist h = 0. Then the T -dual bundle is again the trivial bundle, F̂ = B×T^{n}, and the dual twist vanishes: ĥ = 0. In this situation the T -duality transformation T : K(B × T^{n}) → K(B × T^{n}) is a K-theory version of the Fourier-Mukai transformation (see e.g. ^{\underline{muk}} [6]).

2 THE SPACE R_{N} 6

Note that the algebraic geometric analog is more precise. In this case F and F̂ are bundles of dual abelian varieties. On F ×_{B} F̂ one has the so-called Poincaré sheaf P. Its first Chern class c_{1} (considered as an automorphism of the trivial twist) satisfies the condition P(c_{1}). The Fourier-Mukai transformation is a functor between bounded derived categories of coherent sheaves T : D^{b}(F) → D^{b}(F̂) given on objects by T (X) = Rp̂_{∗}(P ⊗^{L} p^{∗}X). Thus the T -duality transformation considered in the present paper is a coarsification since it takes in a certain sense only the isomorphism classes of objects into account. But note that the tensor product with the Poincaré sheaf plays the role of an automorphism of the trivial twist.

A bundle of abelian varieties has a section. Therefore this case corresponds to the case of trivial T^{n}-bundles in the present paper. Non-trivial bundles can be interpreted as bundles of torsors. In this case in general a good analog of the Poincaré bundle such that P(c_{1}) (see ^{\underline{condp}}^{\underline{condp}} 5.4) is satisfied may not exist. We argue below that in the topological situation we must replace the Poincaré bundle by an isomorphisms u of non-trivial twists in order to satisfy P(u), and to have a T -duality isomorphism. In algebraic geometry a similar observation is known, where twists are represented by Azumaya algebras.

1.14 As a specific example in the topological case, let F → B be some possibly nontrivial T^{n}-bundle, and let the twist on F be trivial (i.e. h = 0). Then the pair (F, 0) admits a T -dual pair (F̂, ĥ). This is not unique in general (see the example in ^{\underline{bunkeschickt}} [3], Sec. 4.4), but one choice is the trivial bundle F̂ := B × T^{n}. In order to describe the class ĥ ∈ H^{3}(F̂, ℤ), we consider F̂ as the fiber product of n-copies of a trivial T^{1}-bundle F_{1} := B × T^{1}. Let π_{i} : F̂ → F_{1} denote the corresponding projections. Then one possible choice of ĥ is ĥ = ∑_{i=1}^{n} pr_{i}^{∗}h_{1}, where h_{i} = c_{i} × or_{T^{1}} ∈ H^{3}(B × T^{1}, ℤ), c_{i} ∈ H^{2}(B, ℤ) denotes the Chern classes of F, and or_{T^{1}} ∈ H^{1}(T^{1}, ℤ) is the orientation class. This example shows that it is necessary to introduce twists in order to generalize the Fourier-Mukai transformation to non-trivial torus bundle.

2 The space R_{n}

fd

2.1 Throughout this chapter (and the whole paper) we fix n ≥ 1.

expla

2.2 If G is an abelian group and k ∈ ℕ, then we consider the homotopy type K(G, n) of the Eilenberg MacLane space. It is characterized by π_{i}(K(G, k)) ≅ 0 for i ≠ k, and π_{k}(K(G, k)) = G. We will denote a CW -complex representing this homotopy type by the same symbol.

The Eilenberg-MacLane space K(G, k) classifies the cohomology functor H^{k}(…, G). In fact, there is a universal class z ∈ H^{k}(K(G, k), G) such that f ↦ f^{∗}(z) induces a natural isomorphism [B, K(G, k)] → H^{k}(B, G), where [B, K(G, k)] denotes homotopy classes of maps.

Occasionally, we will interpret K(ℤ, 2) also as the classifying space of T^{1} := U(1). An explicit model is U/T^{1}, where U is the unitary group of a separable infinite dimensional Hilbert space. The bundle U → U/T^{1} is the universal T^{1}-principal bundle. Note further that K(ℤ^{n}, 2) ≅ K(ℤ, 2)^{n} has the homotopy type of BT^{n}, and this space carries a universal T^{n}-bundle U^{n} → K(ℤ, 2)^{n}.

2.3 We consider the Eilenberg-MacLane space K(ℤ^{2n}, 2) ≅ K(ℤ, 2)^{2n} ≅ K(ℤ^{n}, 2) × K(ℤ^{n}, 2). Let x_{i}, x̂_{i}, i = 1, …, n be the canonical generators of H^{2}(K(ℤ, 2)^{2n}, ℤ). Let q : K(ℤ, 2)^{2n} → K(ℤ, 4) be the map classifying the class

x_{1} ∪ x̂_{1} + ⋯ + x_{n} ∪ x̂_{n} ∈ H^{4}(K(ℤ, 2)^{2n}, ℤ) .

rt3 Definition 2.1 We define the homotopy type R_{n} by the homotopy pull-back diagram

\begin{matrix}R_{n} & →^{(c,ĉ)} & K(ℤ^{n}, 2) × K(ℤ^{n}, 2) \\ ↓ & & q ↓ \\ ∗ & → & K(ℤ, 4) \end{matrix} .

2.4 For later use we determine the homotopy groups of R_{n}.

hh Lemma 2.2 The homotopy groups of R_{n} are given by

\begin{matrix}i & 0 & 1 & 2 & 3 & ≥ 4 \\ π_{i}(R_{n}) & ∗ & 0 & ℤ^{2n} & ℤ & 0 \end{matrix} .

Proof. The homotopy fiber of (c, ĉ) is homotopy equivalent to the one of ∗ → K(ℤ, 4), i.e to K(ℤ, 3). The assertion follows immediately from the long exact sequence of homotopy groups. □

tz1

2.5 Write c = (c_{1}, …, c_{n}) and ĉ = (ĉ_{1}, …, ĉ_{n}). We denote c_{i} and ĉ_{i} toe be the components of c or ĉ, respectively.

gh Lemma 2.3 We have

\begin{matrix}i & 0 & 1 & 2 & 3 & 4 \\ H^{i}(R_{n}, ℤ) & ℤ & 0 & ℤ^{2n} & 0 & ℤ^{n(2n+1)−1} \end{matrix} .

Here H^{2}(R^{n}, ℤ) is freely generated by the components of c and ĉ, and H^{4}(R_{n}, ℤ) is generated by all possible products of the components of c and ĉ subject to one relation

0 = c_{1} ∪ ĉ_{1} + ⋯ + c_{n} ∪ ĉ_{n} .

Proof. Recall from the proof of Lemma ^{\underline{hh}}^{\underline{hh}} 2.2 that the homotopy fiber of R_{n} → K(ℤ, 2)^{2n} is K(ℤ, 3). The relevant part of the second page of the Leray-Serre spectral sequence E_{2}^{p,q} ≅ H^{p}(K(ℤ^{2n}, 2), H^{q}(K(ℤ, 3))) therefore becomes

\begin{matrix}3 & ℤ & 0 & ∗ & 0 & ∗ \\ 2 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & ℤ & 0 & ℤ^{2n} & 0 & ℤ^{\frac{2n(2n+1)}{2}} \\ q/p & 0 & 1 & 2 & 3 & 4 \end{matrix} .

We read off that H^{2}(R^{n}, ℤ) ≅ ℤ^{2n} is generated by the components of c and ĉ. The group E_{2}^{4,0} ≅ H^{4}(K(ℤ, 2)^{2n}, ℤ) is freely generated by all possible products of the components of c and ĉ.

Let z_{3} ∈ H^{3}(K(ℤ, 3), ℤ) ≅ E_{2}^{0,3} be the canonical generator. It also generates the group Ē_{2}^{0,3} of the Leray-Serre spectral sequence of the homotopy fibration ∗ → K(ℤ, 4). A part of its second page is

\begin{matrix}3 & ℤ & 0 & 0 & 0 & ∗ \\ 2 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & ℤ & 0 & 0 & 0 & ℤ \\ q/p & 0 & 1 & 2 & 3 & 4 \end{matrix} .

Since H^{3}(∗, ℤ) ≅ 0 ≅ H^{4}(∗, ℤ) we conclude that d̄_{2}^{0,3}(z_{3}) = z_{4} ∈ H^{4}(K(ℤ, 4), ℤ) ≅ Ē_{2}^{4,0} is the generator. Now by construction q^{∗}z_{4} = x_{1} ∪ x̂_{1} + ⋯ + x_{n} ∪ x̂_{n}, and by naturality of the spectral sequences d_{2}^{0,3}(z) = q^{∗}z_{4}. This implies the assertion about H^{4}(R_{n}, ℤ). □

2.6 Recall that we consider K(ℤ^{n}, 2) ≅ BT^{n} (see ^{\underline{expla}}^{\underline{expla}} 2.2).

Definition 2.4 We define π_{n} : F_{n} → R_{n} to be the T^{n}-bundle which is classified by c : R → K(ℤ^{n}, 2), i.e. the pullback of the universal bundle U^{n}n.

Let U → K(ℤ, 2) be the universal T^{1}-bundle. The n-fold product U^{n} → K(ℤ, 2)^{n} is the universal T^{n}-bundle. By definition we get a pull-back diagram

\begin{matrix}F_{n} & → & U^{n} \\ π_{n} ↓ & & ↓ \\ R_{n} & →^{c} & K(ℤ, 2)^{n} \end{matrix} .

cd1

2.7 Let us fix canonical generators y_{1}, …, y_{n} ∈ H^{1}(T^{n}, ℤ). Let F^{k}H^{∗}(F, ℤ), k = 0, 1, … be the decreasing Leray-Serre filtration and E_{2}^{p,q} be the second page of the Leray-Serre spectral sequence for the bundle F_{n} → R_{n}.

re1 Lemma 2.5 We have

\begin{matrix}i & 0 & 1 & 2 & 3 \\ H^{i}(F_{n}, ℤ) & ℤ & 0 & ℤ^{n} & ℤ \end{matrix}

Here the group H^{2}(F_{n}, ℤ) is freely generated by the components of π_{n}^{∗}ĉ. In particular, restriction to the fiber of F_{n} → R_{n} induces the zero homomorphism on H^{2}(F_{n}, ℤ). Furthermore, H^{3}(F_{n}, ℤ) is generated by a class h_{n} ∈ F^{2}H^{3}(F_{n}, ℤ) which is represented by the cocycle y_{1} ⊗ ĉ_{1} + ⋯ + y_{n} ⊗ ĉ_{n} ∈ E_{2}^{2,1}.

Proof. We write out the second page E_{2}^{p,q} (isomorphic to H^{q}(T^{n}, ℤ) ⊗ H^{p}(R_{n}, ℤ)).

\begin{matrix}3 & ℤ^{\frac{n(n−1)(n−2)}{6}} \\ 2 & ℤ^{\frac{n(n−1)}{2}} & 0 & ℤ^{n^{2}(n−1)} & 0 \\ 1 & ℤ^{n} & 0 & ℤ^{2n^{2}} & 0 \\ 0 & ℤ & 0 & ℤ^{2n} & 0 & ℤ^{n(2n+1)−1} \\ q/p & 0 & 1 & 2 & 3 & 4 \end{matrix} .

We first calculate d_{2}^{0,1}(y_{i}) ∈ E_{2}^{2,0} ≅ H^{2}(R_{n}, ℤ). To this end we consider the universal T^{1}-bundle U → K(ℤ, 2). Let y ∈ H^{1}(T^{1}, ℤ) be the canonical generator. It also generates the group Ē_{2}^{0,1} of the associated Leray-Serre spectral sequence

\begin{matrix}1 & ℤ & 0 & ∗ \\ 0 & ℤ & 0 & ℤ \\ q/p & 0 & 1 & 2 \end{matrix} .

Since U is contractible d̄_{2}^{0,1}(y) ∈ Ē_{2}^{2,0} ≅ H^{2}(K(ℤ, 2), ℤ) is a generator. The bundle π_{n} : F_{n} → R_{n} is obtained as a pull-back of an n-fold product of the bundle U → K(ℤ, 2) via the components of c. We conclude by the naturality of the spectral sequences that d_{2}^{0,1}(y_{i}) ∈ {c_{i}, −c_{i}}.

We see that d_{2}^{0,1} is an isomorphism onto the subgroup of E_{2}^{0,2} generated by c so that E_{3}^{0,1} ≅ 0 and E_{3}^{2,0} is freely generated by the components of ĉ. We see already that H^{1}(F_{n}, ℤ) ≅ 0.

The group E_{2}^{0,2} ≅ H^{2}(T^{n}, ℤ) is freely generated by all products y_{i} ∪ y_{j}, i < j. We now use the multiplicativity of the Leray-Serre spectral sequence in order see that d_{2}^{0,2}(y_{i} ∪ y_{j}) = ±(y_{j} ⊗ c_{i} − y_{i} ⊗ c_{j}). We conclude that d_{2}^{0,2} is injective. This implies that E_{3}^{0,2} ≅ 0, and it follows that H^{2}(F_{n}, ℤ) is freely generated by the components of π_{n}^{∗}ĉ.

We have E_{2}^{4,0} ≅ H^{4}(R_{n}, ℤ) and d_{2}^{1,2}(y_{i} ⊗ c_{j}) = c_{i} ∪ c_{j} and d_{2}^{1,2}(y_{i} ⊗ ĉ_{j}) = c_{i} ∪ ĉ_{j}.

In order to calculate ker(d_{2}^{1,2}) recall the relation c_{1} ∪ ĉ_{1} + ⋯ + c_{n} ∪ ĉ_{n} = 0. Let h := y_{1} ⊗ ĉ_{1} + ⋯ + y_{n} ∪ ĉ_{n}. Then we have d_{2}^{1,2}(h) = 0. We claim that ker(d_{2}^{1,2}) ≅ ℤh ⊕ im(d_{2}^{0,2}). Let t := ∑_{i,j=1}^{n} a_{i,j}y_{i} ⊗ c_{j} + b_{i,j}y_{i} ⊗ ĉ_{j} for a_{i,j}, b_{i,j} ∈ ℤ and assume that d_{2}^{1,2}(t) = 0. Then ∑_{i,j=1}^{n} a_{i,j}c_{i} ∪ c_{j} + b_{i,j}c_{i} ∪ ĉ_{j} = 0. This implies that a_{i,j} + a_{j,i} = 0, b_{i,j} = 0 for i ≠ j, and that there exists b ∈ ℤ such that b_{i,i} = b for all i = 1, …, n. But then we can write t = ∑_{i<j} a_{i,j}d_{2}^{0,2}(y_{j} ∪ y_{i}) + bh. It follows that E_{3}^{2,1} ≅ ℤ is generated by the class of h.

The group E_{2}^{3,0} ≅ H^{3}(T^{n}, ℤ) is freely generated by the products y_{i} ∪ y_{j} ∪ y_{k}, i < j < k. Furthermore E_{2}^{2,2} ≅ H^{2}(T^{n}, ℤ) ⊗ H^{2}(R_{n}, ℤ) and d_{2}^{0,3}(y_{i} ∪ y_{j} ∪ y_{k}) = y_{j} ∪ y_{k} ⊗ c_{i} − y_{i} ∪ y_{k} ⊗ c_{j} + y_{i} ∪ y_{j} ⊗ c_{k}. We thus see by a simple calculation that d_{2}^{0,3} is injective. We conclude that H^{3}(F_{n}, ℤ) ≅ F^{2}H^{3}(F_{n}, ℤ) is generated by the class h_{n} represented by h ∈ E_{2}^{1,2}. □

f16

2.8 In the proof of Lemma ^{\underline{hh}}^{\underline{hh}} 2.2 we have found a homotopy Cartesian square

\begin{matrix}K(ℤ, 3) & →^{i} & R_{n} \\ ↓ & & (c, ĉ) ↓ \\ ∗ & → & K(ℤ^{n}, 2) × K(ℤ^{n}, 2) \end{matrix} .

We form the pull-back of this square by the map ψ : K(ℤ^{n}, 2) ≅ U^{n} × K(ℤ^{n}, 2) → K(ℤ^{n}, 2) × K(ℤ^{n}, 2) and obtain a cube of homotopy Cartesian squares

\begin{aligned}i^{∗}F_{n} & & & ⟶^{I} & & & F_{n} \\ & pr ↘ & & & & π_{n} ↙ \\ & & K(ℤ, 3) & →^{i} & R_{n} \\ λ ↓ & & ↓ & & (c, ĉ) ↓ & & κ ↓ \\ & & ∗ & → & K(ℤ^{n}, 2) × K(ℤ^{n}, 2) \\ & ↗ & & & & ψ ↖ \\ K(ℤ, 1)^{n} & & & ⟶ & & & K(ℤ^{n}, 2) \end{aligned} .

g16 Lemma 2.6 We have I^{∗}h_{n} = ±pr^{∗}z_{3}, where z_{3} ∈ H^{3}(K(ℤ, 3), ℤ) is the canonical generator.

3 UNIVERSAL T -DUALITY 12

Proof. The fiber of κ is equivalent to K(ℤ, 3). The second page E_{2} of the corresponding Leray-Serre spectral sequence has the form

\begin{matrix}3 & ℤ & 0 & ∗ & ∗ & ∗ \\ 2 & 0 & 0 & 0 & 0 & ∗ \\ 1 & 0 & 0 & 0 & 0 & ∗ \\ 0 & ℤ & 0 & ℤ^{n} & 0 & ℤ^{\frac{n(n+1)}{2}} \\ q/p & 0 & 1 & 2 & 3 & 4 \end{matrix} .

We know that H^{3}(F_{n}, ℤ) ≅ ℤ is freely generated by h_{n}. We see that d_{2}^{0,3} = 0 and E_{2}^{0,3} ≅ H^{3}(R_{n}, ℤ) is generated by h_{n}. On the other hand the group E_{2}^{0,3} ≅ H^{3}(K(ℤ, 3), ℤ) is freely generated by z_{3}. Therefore h_{n} = ±z_{3}.

Note that i^{∗}F_{n} ≅ K(ℤ, 3) × K(ℤ, 1)^{n} so that the Leray-Serre spectral sequence Ē of λ degenerates. Let I^{∗} : E_{2} → Ē_{2} be the induced map of the second pages and note that Ē_{2} has the form

\begin{matrix}3 & ℤ & ℤ^{n} & ∗ & ∗ & ∗ \\ 2 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & ℤ & ℤ^{n} & ℤ^{\frac{n(n−1)}{2}} & ℤ^{\frac{n(n−1)(n−2)}{6}} & ℤ^{\frac{n(n−1)(n−2)(n−3)}{24}} \\ q/p & 0 & 1 & 2 & 3 & 4 \end{matrix} ,

where Ē_{2}^{0,3} is freely generated by z_{3} ∈ H^{3}(K(ℤ, 3), ℤ). The map I induces an equivalence of the fibers of λ and κ. In particular, it induces an isomorphism I^{∗} : E_{2}^{0,3} → Ē_{2}^{0,3} identifying the generators above. This implies that I^{∗}h_{n} = ±pr^{∗}z_{3}. □

3 Universal T -duality

tz

3.1 Let (e_{1}, …, e_{n}, ê_{1}, …, ê_{n}) be the standard basis of ℤ^{2n}. Let G_{n} ⊂ GL(2n, ℤ) be the subgroup of transformations which fix the form q : ℤ^{2n} → ℤ given by q(∑_{i=1}^{n} a_{i}e_{i}+b_{i}ê_{i}) := ∑_{i=1}^{n} a_{i}b_{i}. In the usual notation G_{n} = O(n, n, ℤ). In the present paper it will be called the group of T -duality transformations.

3.2 Each g ∈ G_{n} induces an equivalence g : K(ℤ^{n}, 2)×K(ℤ^{n}, 2) → K(ℤ^{n}, 2))×K(ℤ^{n}, 2).

re Lemma 3.1 There exist a unique lift g̃ in the diagram

\begin{matrix}R_{n} & ⟶^{g̃} & R_{n} \\ ↓^{(c,ĉ)} & & ↓^{(c,ĉ)} \\ K(ℤ^{n}, 2) × K(ℤ^{n}, 2) & ⟶^{g} & K(ℤ^{n}, 2) × K(ℤ^{n}, 2) \end{matrix} .

Proof. We apply obstruction theory to the problem of existence and classification of lifts g̃. In fact, the obstruction is the class (c, ĉ)^{∗}g^{∗}(∑_{i=1}^{n} x_{i} ∪ x̂_{i}) ∈ H^{4}(R_{n}, ℤ) which vanishes since g preserves q. Therefore lifts g̃ exist. The set of lifts is a torsor over H^{3}(R_{n}, ℤ). Since H^{3}(R_{n}, ℤ) = {0}, the lift g̃ is unique. □

3.3 We let t ∈ G_{n} be the transformation given by t(e_{i}) = ê_{i} and t(ê_{i}) = e_{i}.

tt1 Definition 3.2 The universal T -duality is the lift T := t̃: R_{n} → R_{n} of t according to

\underline{re}^{\underline{re}} 3.1.

Note that T ∘ T = id_{R_{n}} since t ∘ t = 1 ∈ G_{n}.

3.4

rtz Definition 3.3 The dual T^{n}-bundle is defined by the pull-back

\begin{matrix}F̂_{n} & →^{T̃} & F_{n} \\ ↓ & & ↓ \\ R_{n} & →^{T} & R_{n} \end{matrix} .

We further define ĥ_{n} := T̃^{∗}h_{n} ∈ H^{3}(F̂_{n}, ℤ).

adef

3.5 We consider the pull-back diagram

\begin{aligned}& & F_{n} ×_{R_{n}} F̂_{n} \\ & a_{n} ↙ & & â_{n} ↘ \\ F_{n} & & & & F̂_{n} \\ & π_{n} ↘ & & π̂_{n} ↙ \\ & & R_{n} \end{aligned} .

aeq Lemma 3.4 We have

\begin{matrix}i & 0 & 1 & 2 & 3 \\ H^{i}(F_{n} ×_{R_{n}} F̂_{n}, ℤ) & ℤ & 0 & 0 & ℤ \end{matrix} .

Moreover, a_{n}^{∗}h_{n} = â_{n}^{∗}ĥ_{n}, and this element generates H^{3}(F_{n} ×_{R_{n}} F̂_{n}, ℤ).

Proof. We use the Leray-Serre spectral sequence. The second page is E_{2}^{p,q} ≅ H^{q}(T^{n} × T^{n}, ℤ) ⊗ H^{p}(R_{n}, ℤ). Let y_{i}, ŷ_{i}, i = 1, …, n denote the canonical generators of H^{1}(T^{n} × T^{n}, ℤ) (so that the elements decorated with ˆ come from the right copy). The relevant part of the second page has the form

\begin{matrix}3 & ℤ^{\frac{2n(2n−1)(2n−2)}{6}} \\ 2 & ℤ^{\frac{2n(2n−1)}{2}} & 0 & ℤ^{2n^{2}(2n−1)} & 0 \\ 1 & ℤ^{2n} & 0 & ℤ^{4n^{2}} & 0 \\ 0 & ℤ & 0 & ℤ^{2n} & 0 & ℤ^{n(2n+1)−1} \\ q/p & 0 & 1 & 2 & 3 & 4 \end{matrix} .

Now E_{2}^{0,1} is freely generated by y_{i}, ŷ_{i}, and E_{2}^{2,0} is freely generated by c_{i}, ĉ_{i}. A similar argument as in the proof of Lemma ^{\underline{re1}}^{\underline{re1}} 2.5 shows that d_{2}^{0,1}(y_{i}) = c_{i} and d_{2}^{0,1}(ŷ_{i}) = ĉ_{i}. We conclude that d_{2}^{0,1} is injective and H^{1}(F_{n} ×_{R_{n}} F̂_{n}, ℤ) ≅ 0.

The group E_{2}^{0,2} is freely generated by the products y_{i}∪y_{j}, ŷ_{i}∪ŷ_{j}, i < j and y_{i}∪ŷ_{j}. We have d_{2}^{0,2}(y_{i}∪y_{j}) = y_{j}⊗c_{i}−y_{i}⊗c_{j}, d_{2}^{0,2}(ŷ_{i}∪ŷ_{j}) = ŷ_{j}⊗ĉ_{i}−ŷ_{i}⊗ĉ_{j}, and d_{2}^{0,2}(y_{i}∪ŷ_{j}) = ŷ_{j}⊗c_{i}−y_{i}⊗ĉ_{j}. It follows that d_{2}^{0,2} is injective.

4 R_{N} AS A CLASSIFYING SPACE 15

In a similar way we see that d_{2}^{0,3} is injective.

We now calculate E_{3}^{2,1}. We claim that this group is freely generated by one class which can be represented by ∑_{i=1}^{n} y_{i} ⊗ĉ_{i} or alternatively by ∑_{i=1}^{n} ŷ_{i} ⊗c_{i}. In view of the construction of h_{n} and ĥ_{n}, and by naturality of the Leray-Serre spectral sequence this would imply the assertion of the lemma about the third cohomology.

The group E_{2}^{4,0} is generated by the products c_{i} ∪ c_{j}, ĉ_{i} ∪ ĉ_{j}, i ≤ j, and all products c_{i} ∪ ĉ_{j} subject to one relation ∑_{i=1}^{n} c_{i} ∪ ĉ_{i} = 0. Let t = ∑_{i,j} a_{i,j}y_{i} ⊗ c_{j} + ∑_{i,j} b_{i,j}y_{i} ⊗ ĉ_{j} + ∑_{i,j} c_{i,j}ŷ_{i} ⊗ c_{j} + ∑_{i,j} d_{i,j}ŷ_{i} ⊗ ĉ_{j} and assume that d_{2}^{2,1}(t) = 0. Then we have ∑_{i,j} a_{i,j}c_{i} ⊗ c_{j} + ∑_{i,j} b_{i,j}c_{i} ⊗ ĉ_{j} + ∑_{i,j} c_{i,j}ĉ_{i} ⊗ c_{j} + ∑_{i,j} d_{i,j}ĉ_{i} ⊗ ĉ_{j} = 0. This implies that a_{i,j} + a_{j,i} = 0, d_{i,j} + d_{j,i} = 0, for all i, j, b_{i,j} + c_{i,j} = 0 for all i ≠ j, and that there exists a unique e ∈ ℤ such that b_{i,i} + c_{i,i} = e for all i = 1, …, n. We can now write t = ∑_{i<j} a_{i,j}d_{2}^{0,2}(y_{j} ∪ y_{i}) + ∑_{i<j} d_{i,j}d_{2}^{0,2}(ŷ_{j} ∪ ŷ_{i}) − ∑_{i≠j} b_{i,j}d_{2}^{0,2}(y_{i} ∪ ŷ_{j}) − ∑_{i=1}^{n} b_{i,i}d_{2}^{0,2}(y_{i} ∪ ŷ_{i}) + e ∑_{i=1}^{n} ŷ_{i} ⊗ c_{i}. This already shows that E_{3}^{2,1} is freely generated by the class of ∑_{i=1}^{n} ŷ_{i} ⊗ c_{i}. Finally note that ∑_{i=1}^{n} d_{2}^{0,2}(y_{i} ∪ ŷ_{i}) = ∑_{i=1}^{n} ŷ_{i} ⊗ c_{i} − ∑_{i=1}^{n} y_{i} ⊗ ĉ_{i}. This finishes the proof of the claim. □

4 R_{n} as a classifying space

fff

4.1 The homotopy type R_{n} classifies a contravariant set-valued functor P_{n} on the category of topological spaces defined on objects by P_{n}(X) := [X, R_{n}]. In ^{\underline{tz}}^{\underline{tz}} 3.1 we have introduced the group G_{n} of T -duality transformations which by acts by automorphisms on P_{n}. Of particular interest is the T -duality transformation T given in ^{\underline{tt1}}^{\underline{tt1}} 3.2.

In the present section we want to describe the geometric meaning of the objects classified by R_{n} and the action of T. To this end we give another construction of the homotopy type R_{n}. We closely follow Section 2 of ^{\underline{bunkeschickt}} [3], which deals with the special case n = 1.

4.2 We start with the universal T^{n}-bundle U^{n} → K(ℤ, 2)^{n} and the T^{n}-space Map(T^{n}, K(ℤ, 3)), where T^{n} acts by reparametrization. In a first step we form the associated bundle

p : R̃_{n}(0) := U^{n} ×_{T^{n}} Map(T^{n}, K(ℤ, 3)) → K(ℤ, 2)^{n} .

We define a T^{n}-bundle via pull-back

\begin{matrix}F̃_{n}(0) & → & U^{n} \\ ↓ & & ↓ \\ R̃_{n}(0) & →^{p} & K(ℤ, 2)^{n} \end{matrix} .

There is a canonical map

h̃_{n}(0): F̃_{n}(0) → K(ℤ, 3)^{n} .

It is given by h̃_{n}(0)([v, ϕ], u) := ϕ(s), where s ∈ T^{n} is the unique element such that sv = u. Here u, v ∈ U^{n}, ϕ ∈ Map(T^{n}, K(ℤ, 3)), [v, ϕ] ∈ R̃_{n}(0), and ([v, ϕ], u) ∈ F̃_{n}(0).

4.3 A pair (F, h) over a space B consists of a T^{n}-bundle F → B and a class h ∈ H^{3}(F, ℤ). An isomorphism between pairs (F, h) and (F′, h′) is given by a diagram

\begin{matrix}F & →^{Φ} & F′ \\ ↓ & & ↓ \\ B & = & B \end{matrix} ,

where Φ is a T^{n}-bundle isomorphism such that Φ^{∗}h′ = h.

Given a map f : B′ → B of spaces we can form the pull-back

\begin{matrix}F′ & →^{f̃} & F \\ ↓ & & ↓ \\ B′ & →^{f} & B \end{matrix} .

We define the pair f^{∗}(F, h) := (F′, f̃^{∗}h) over B′. Pull-back preserves isomorphism classes of pairs.

4.4 Let P̃_{(0)} be the contravariant set-valued functor which associates to each space B the set P̃_{(0)}(B) of equivalence classes of pairs. cl1 Lemma 4.1 The space R̃_{n}(0) is the classifying space for P̃_{(0)}. More precisely, the pair

[F̃_{n}(0), h̃_{n}(0)] ∈ P_{(0)}(R̃_{n}(0)) induces a natural isomorphism ṽ_{B} : [B, R̃_{n}(0)] → P̃_{(0)}(B) such that ṽ_{B}(f) = f^{∗}[F̃_{n}(0), h̃_{n}(0)].

Proof. This is completely analogous to the proof of ^{\underline{bunkeschickt}} [3, Proposition 2.6]. We therefore refrain from repeating the proof here. □

4.5 By Lemma ^{\underline{cl1}}^{\underline{cl1}} 4.1 we have an isomorphism π_{0}(R̃_{n}(0)) =^{def} P̃_{(0)}(∗). Furthermore, note that P_{(0)}(∗) ≅ H^{3}(T^{n}, ℤ) canonically. We define R̃_{n}(1) ⊂ R̃_{n}(0) to be the component which corresponds to 0 ∈ H^{3}(T^{n}, ℤ). Restricting the pair [F̃_{n}(0), H̃_{n}(0)] gives the pair [F̃_{n}(1), h̃_{n}(1)] over R̃_{n}(1). We let P̃_{(1)} be the functor classified by R̃_{n}(1). Observe that P̃_{(1)}(B) ⊂ P̃_{(0)}(B) is the set of isomorphism classes of pairs [F, h] such that the restriction of h to the fibers of F vanishes.

cl2

4.6 By Lemma ^{\underline{cl1}}^{\underline{cl1}} 4.1 we have P̃_{(0)}(S^{1}) ≅ H^{3}(S^{1} × T^{n}). By the Künneth formula H^{3}(S^{1} ×

T^{n}, ℤ) ≅ H^{3}(T^{n}, ℤ) ⊕ H^{2}(T^{n}, ℤ), and π_{1}(R̃_{n}(1)) =^{def} P̃_{(1)}(S^{1} × T^{n}) ≅ H^{2}(T^{n}, ℤ) corre-

sponds to the second summand.

We consider the isomorphism ϕ : π_{1}(R̃_{n}(1)) → H^{2}(T^{n}, ℤ) as a cohomology class ϕ ∈ H^{1}(R̃_{n}(1), H^{2}(T^{n}, ℤ))), i.e. as a homotopy class of maps ϕ : R̃_{n}(1) → K(H^{2}(T^{n}, ℤ), 1). We define R̃_{n} as the homotopy pull-back

\begin{aligned}R̃_{n} & → & R̃_{n}(1) \\ ↓ & & ϕ ↓ \\ ∗ & → & K(H^{2}(T^{n}), 1) \end{aligned} .

Furthermore, we consider the pull-back

\begin{aligned}F̃_{n} & → & F̃_{n}(1) \\ ↓ & & ↓ \\ R̃_{n} & → & R̃_{n}(1) \end{aligned} ,

and let h̃_{n} ∈ H^{3}(F̃_{n}, ℤ) be the pullback of h̃_{n}(1). Note that by construction and naturality h̃_{n} pulls back to zero on the fiber of F̃_{n} → R̃_{n}. Since R̃_{n} is simply connected, it then even belongs to the second step F^{2}H^{3}(F̃_{n}, ℤ) of the Leray-Serre filtration associated with F̃_{n} → R̃_{n}.

4.7

ght Lemma 4.2 The homotopy groups of R̃_{n} are given by

\begin{matrix}i & 0 & 1 & 2 & 3 & ≥ 4 \\ π_{i}(R̃_{n}) & ∗ & 0 & ℤ^{2n} & ℤ & 0 \end{matrix} .

Proof. By construction, R̃_{n} is connected and simply connected. The homotopy fiber of R̃_{n} → R̃_{n}(1) is equivalent to the homotopy fiber of ∗ → K(H^{2}(T^{n}, ℤ), 1), i.e. to K(H^{2}(T^{n}, ℤ), 0). Hence this map induces an isomorphism π_{i}(R̃_{n}) ≅ π_{i}(R̃_{n}(1)) for i ≥ 2.

Since K(ℤ, k) is an h-space for each k we have an equivalence

Map(T^{1}, K(ℤ, 3)) ≃ K(ℤ, 3) × ΩK(ℤ, 3) ≃ K(ℤ, 3) × K(ℤ, 2) .

We use the exponential law to write Map(T^{n}, K(ℤ, 3)) as an iterated mapping space, and obtain in the same way an equivalence

Map(T^{n}, K(ℤ, 3)) ≃ K(ℤ, 3) × K(ℤ, 2)^{n} × K(ℤ, 1)^{\frac{n(n−1)}{2}} × K(ℤ, 0)^{\frac{n(n−1)(n−2)}{6}} .

The long exact sequence of homotopy groups for

Map(T^{n}, K(ℤ, 3)) → R̃_{n}(0) → K(ℤ, 2)^{n}

and the fact that π_{3}(K(ℤ, 2)^{n}) ≅ 0 ≅ π_{1}(K(ℤ, 2)^{n}) and π_{2}(K(ℤ, 2)^{n}) ≅ ℤ^{n} yields the exact sequence

0 → ℤ^{n} → π_{2}(R̃_{n}(1)) → ℤ^{n} →^{δ} ℤ^{\frac{n(n−1)}{2}} →^{α} π_{1}(R̃_{n}(1)) → 0 .

Furthermore, we observe that π_{3}(R̃(1)) ≅ ℤ and π_{i}(R̃_{n}(1)) = 0 for i ≥ 4.

We have seen in Subsection ^{\underline{cl2}}^{\underline{cl2}} 4.6 that π_{1}(R̃_{n}(1)) ≅ H^{2}(T^{2}, ℤ) ≅ ℤ^{\frac{n(n−1)}{2}} . We conclude that α must be surjective. Consequently it is injective and δ = 0.

We therefore have an exact sequence

0 → ℤ^{n} → π_{2}(R̃_{n}(1)) → ℤ^{n} → 0 ,

and this implies that π_{2}(R̃_{n}(1)) ≅ π_{2}(R̃_{n}) ≅ ℤ^{2n}. □

bc1

4.8 Since the fiber of p: R̃_{n} → R̃_{n}(1) is equivalent to K(H^{2}(T^{n}, ℤ), 0) ≅ π_{1}(R̃_{n}(1)) we can consider this map as the universal covering of R̃_{n}(1). We now consider the problem of existence and classification of lifts in the diagram

\begin{aligned}& & R̃_{n} & → & ∗ \\ & f̃ ↗ & p ↓ & & ↓ \\ B & →^{f} & R̃_{n}(1) & ⟶^{ϕ} & K(H^{2}(T^{n}, ℤ), 1) \end{aligned} .

It follows from the construction of R̃_{n} that a lift f̃ exists if and only if ϕ ∘ f is homotopic to a constant map. The lift itself depends on the choice of an explicit homotopy. If a lift exists, the set of homotopy classes of lifts is a torsor over H^{0}(B, H^{2}(T^{n}, ℤ)).

clh

4.9 The classification of homotopy classes f̃ (considered just as maps, not as lifts) lifting a homotopy class f is more subtle. In order to study this problem we assume that B is path connected and equipped with a base point b ∈ B. Let f̃_{0} be a lift of f and consider x ∈ π_{1}(R̃_{n}(1)) ≅ H^{0}(B, H^{2}(T^{n}, ℤ)). Then we consider the lift f̃_{1} = xf̃_{0}, i.e. the composition of f̃_{0} with the deck transformation associated to x.

Assume that f̃_{0} and f̃_{1} are homotopic. Let H : I × B → R̃_{n} be a homotopy. Then p ∘ H : S^{1} × B → R̃_{n}(1) can be considered as a map h : B → Map(S^{1}, R̃_{n}(1)). We have the following diagram

\begin{matrix}\{b\} & →^{x} & Map(S^{1}, R̃_{n}(1)) \\ ↓ & h ↗ & ev_{1} ↓ \\ B & →^{f} & R̃_{n}(1) \end{matrix} ,  (4.3) de9

where ev_{1} : Map(S^{1}, R̃_{n}(1)) → R̃_{n}(1) is the evaluation at 1 ∈ S^{1}.

Vice versa, if x ∈ π_{1}(R̃_{n}(1)) is such that the diagram above admits a lift h, then f̃_{0} and xf̃_{0} are homotopic.

clh1

4.10 The existence problem for a lift h can be studied using obstruction theory. The fiber of the map ev_{1} is ΩR̃_{n}(1)). In the proof of Lemma ^{\underline{ght}}^{\underline{ght}} 4.2 we have seen that the homotopy groups of R̃_{n}(1) are given by

\begin{matrix}i & 0 & 1 & 2 & 3 & ≥ 4 \\ π_{i}(R̃_{n}(1)) & ∗ & ℤ^{\frac{n(n−1)}{2}} & ℤ^{2n} & ℤ & 0 \end{matrix} .

It follows that the homotopy groups of ΩR̃_{n}(1) are given by

\begin{matrix}i & 0 & 1 & 2 & ≥ 3 \\ π_{i}(ΩR̃_{n}) & Z^{\frac{n(n−1)}{2}} & ℤ^{2n} & ℤ & 0 \end{matrix} .

We therefore have obstructions in H^{1}(B, ℤ^{\frac{n(n−1)}{2}} ), H^{2}(B, ℤ^{2n}) and H^{3}(B, ℤ), and a general discussion seems to be complicated.

p2

4.11 Let P̃_{n} be the set-valued functor classified by R̃_{n}. For each space B we have a natural transformation u_{B} : P̃_{n}(B) → P̃_{(1)}(B) induced by composition with the map R̃_{n} → R̃_{n}(1).

We conclude from Subsection ^{\underline{bc1}}^{\underline{bc1}} 4.8 that the fibers of u_{B} are homogeneous spaces over H^{0}(B, H^{2}(T^{n}, ℤ)).

Consider a class [F, h] ∈ P̃_{(1)}(B). By Subsection ^{\underline{bc1}}^{\underline{bc1}} 4.8, it belongs to the image of u_{B} if and only the restriction [F, h]_{|B^{(1)}} to a 1-skeleton B^{(1)} ⊂ B is trivial. Since every principal torus bundle on a 1-dimensional complex is trivial, this is equivalent to the condition that the restriction of h to F_{|B^{(1)}} is trivial, or equivalently h ∈ F^{2}H^{3}(F, ℤ). Let us fix a map f : B → R̃_{n}(1) representing a pair [F, h] with this property. If we choose a homotopy from ϕ ∘ f to the constant map, then we distinguish an element in the fiber u_{B}^{−1}([F, h]).

4.12 In Section^{\underline{fd}}^{\underline{fd}} 2 we have introduced a pair (F_{n}, h_{n}) over R_{n} such that h_{n} ∈ F^{2}H^{3}(F_{n}, ℤ). This isomorphism class of pairs gives rise to a classifying map f(1) : R_{n} → R̃_{n}(1).

var Lemma 4.4 The set of homotopy classes of maps f which are lifts of f(1) in the diagram

R̃_{n}
f ↗ p ↓
R_{n} →^{f(1)} R̃_{n}(1)

is a torsor over H^{2}(T^{n}, ℤ).

Proof.

Since R_{n} is simply connected we know the existence of lifts. Let now x ∈ π_{1}(R̃_{n}(1)). We choose a base point b ∈ R_{n} and identify π_{1}(R̃_{n}(1)) ≅ H^{2}(F_{n,b}, ℤ), where F_{n,b} denotes the fiber of F_{n} over b. In particular we view x ∈ H^{2}(F_{n,b}, ℤ).

We must show that the existence of a lift h in the diagram ^{\underline{de9}}^{\underline{de9}} 4.3 (with B replaced by R_{n} and f replaced by f(1)) implies that x = 0. Assume that a lift h exists, adjoint to a map H: S^{1} × R_{n} → R̃_{n}(1). This corresponds to a T^{n}-bundle F → S^{1} × R_{n} and a class h ∈ H^{3}(F, ℤ). Let pr : S^{1} × R_{n} → R_{n} be the projection. Since R_{n} is simply connected pr induces in isomorphism in second cohomology and the bundle F is the pull-back via pr of a T^{n}-bundle from R_{n}. Since H restricts to f on {1} × R_{n} and the corresponding T^{n}-bundle is F_{n}, necessarily F ≅ pr^{∗}F_{n} = S^{1} × F_{n}. By the Künneth formula H^{3}(F, ℤ) ≅ H^{3}(F_{n}, ℤ) ⊕ H^{2}(F_{n}, ℤ), with corresponding decomposition h = h_{n} ⊕ u. By the definition of h and the calculation of π_{1}(R̃_{n}(1)) in Subsection ^{\underline{cl2}}^{\underline{cl2}} 4.6, the restriction of u to F_{n,b} is x. Since the restriction H^{2}(F_{n}, ℤ) → H^{2}(F_{n,b}, ℤ) is trivial by the description of H^{2}(F_{n}, ℤ) given in Lemma ^{\underline{re1}}^{\underline{re1}} 2.5 it follows that x = 0. □

4.13 We now fix one choice of f in Lemma ^{\underline{var}}^{\underline{var}} 4.4.

var1 Proposition 4.5 The map f : R_{n} → R̃_{n} is a weak homotopy equivalence.

Proof. By Lemma ^{\underline{hh}}^{\underline{hh}} 2.2 and Lemma ^{\underline{ght}}^{\underline{ght}} 4.2 it suffices to show that f induces isomorphisms on π_{2} and π_{3}.

Note that H^{2}(R̃_{n}, ℤ) ≅ Hom_{ℤ}(π_{2}(R̃_{n}), ℤ) ≅ ℤ^{2n}. We have a natural map x : R̃_{n} → K(ℤ, 2)^{n} which classifies the T^{n}-principal bundle F̃_{n} → R̃_{n}. We consider the components x_{i} ∈ H^{2}(R̃_{n}, ℤ).

Let y_{1}, …, y_{n} be the generators of H^{1}(T^{n}, ℤ) as before. We consider the second page of the Leray-Serre spectral sequence Ẽ_{2}^{p,q} of the fibration F̃_{n} → R̃_{n}. Since h_{n} ∈ F^{2}H^{3}(F̃_{n}, ℤ), there are elements x̂_{1}, …, x̂_{n} ∈ H^{2}(R̃_{n}, ℤ) and a_{i,j} ∈ ℤ such that h̃_{n} is represented by h̃ = ∑_{i,j} a_{i,j}y_{i} ⊗ x_{j} + ∑_{i} y_{i} ⊗ x̂_{i} ∈ Ẽ_{2}^{2,1} ≅ H^{1}(T^{n}, ℤ) ⊗ H^{2}(R̃_{n}, ℤ).

Under the pull-back induced by f the spectral sequence Ẽ is mapped to the spectral sequence E considered in ^{\underline{cd1}}^{\underline{cd1}} 2.7. In particular h̃ is mapped to h = ∑_{i=1}^{n} y_{i} ⊗ ĉ_{i}. Since f^{∗}x_{i} = c_{i} we see that a_{i,j} = 0. Furthermore we see that f^{∗}x̂_{i} = ĉ_{i}. It follows that f^{∗} : H^{2}(R̃_{n}, ℤ) → H^{2}(R_{n}, ℤ) maps (x_{1}, …, x_{n}, x̂_{1}, …, x̂_{n}) to a basis and is therefore surjective and injective. This implies that f_{∗} : π_{2}(R_{n}) → π_{2}(R̃_{n}) is an isomorphism.

It now suffices to show that f : π_{3}(R_{n}) → π_{3}(R̃_{n}) is surjective. A generator g ∈ π_{3}(R̃_{n}) is represented by a map g : S^{3} → R̃_{n}. The corresponding pair is the trivial torus bundle pr_{1} : S^{3}×T^{n} → S^{3} with the cohomology class of the form h = pr_{1}^{∗}z for some z ∈ H^{3}(S^{3}, ℤ) which is a generator.

It suffices to show that the isomorphism class of pairs [S^{3} × T^{n}, pr_{1}^{∗}z] is the pull back of [F_{n}, h_{n}] on R_{n}. We consider the composition g : S^{3} → K(ℤ, 3) →^{i} R_{n}, where the map i was defined in ^{\underline{f16}}^{\underline{f16}} 2.8 and the first map realizes a generator of π_{3}(K(ℤ, 3)). It then follows immediately from Lemma ^{\underline{g16}}^{\underline{g16}} 2.6 that g^{∗}[F_{n}, h_{n}] = [S^{3} × T^{n}, ±pr_{1}^{∗}z]. Choosing the opposite generator of π_{3}(K(ℤ, 3)), if necessary, the assertion follows. □

epsd

4.14

hhll Corollary 4.6 The functors P_{n} classified by R_{n} and P̃_{n} classified by R̃_{n} are isomorphic. The group H^{2}(T^{n}, ℤ) acts freely on the set of such isomorphisms, and it acts transitively if we fix the composition with p_{∗}: P̃_{n} → P̃_{(1)}.

Let B be a space. Note that the group G_{n} of T -duality transformations acts by automorphisms on P (B). In particular each element x ∈ P_{n}(B) has a canonical T -dual x̂ := T (x) (see ^{\underline{tt1}}^{\underline{tt1}} 3.2).

Assume that we have fixed an isomorphism ϵ : P_{n} ≅ P̃_{n}. Then we can transfer the T -duality transformations to P̃_{n}. Assume that (F, h) is a pair over B.

drt Definition 4.7 We say that (F, h) has a T -dual pair (F̂, ĥ) if there exists a lift x ∈ P̃_{n}(B) of [F, h] ∈ P̃_{(0)}(B) such that (F̂, ĥ) is the underlying pair of ϵ ∘ T ∘ ϵ^{−1}(x).

5 T -DUALITY TRANSFORMATIONS IN TWISTED COHOMOLOGY 23

Recall that F^{2}H^{3}(F, ℤ) ⊂ H^{3}(F, ℤ) denotes the second step of the Leray-Serre filtration of the cohomology associated to the bundle F → B (see ^{\underline{filtex}}^{\underline{filtex}} 1.3).

exex Corollary 4.8 A pair (F, h) admits a T -dual pair if and only if h ∈ F^{2}H^{3}(F, ℤ).

In ^{\underline{bunkeschickt}} [3, Section 4.4], we have given an example where for n ≥ 2 even the topology of the underlying T^{n}-bundle of a T -dual pair (F̂, ĥ) depends on the choice of the lift of (F, h) to an element of P̃_{n}(B).

5 T -duality transformations in twisted cohomology

w2w2w

5.1 In ^{\underline{bunkeschickt}} [3, Section 3.1] we have introduced a set of axioms which describe the basic properties of twists and twisted cohomology theories which are used in connection with T -duality considerations. Since in the present paper the main focus lies on the category of twists and since simple realizations can be given, we shall recall here two explicite models.

For the first model we fix a space K(ℤ, 3) representing the homotopy type of the Eilenberg-MacLane space. It determines a model of twists as follows. To each space B we associate the category of twists T (B). Its objects are the maps H : B → K(ℤ, 3). The morphisms Hom_{T (B)}(H, H′) are homotopy classes of homotopies from f to f′. Note that T (B) is a groupoid. Its isomorphism classes are classified by H^{3}(B, ℤ), and the sets Hom_{T (B)}(H, H′) are torsors over H^{2}(B, ℤ) in a natural way. Note that B ↦ T (B) extends to a groupoidvalued functor. In order to define the monoidal structure on T (B) we use the h-space structure of K(ℤ, 3).

Our second model is as follows. It associates to each space locally trivial bundles H → B of C^{∗}-algebras with fibre K, the compact operators of a separable infinite-dimensional Hilbert space. The structure group of such a bundle is P U, the projective unitary group of the Hilbert space with the strong topology. The morphisms Hom_{T (B)}(H, H′) are homotopy classes of bundle isomorphisms. We again obtain a groupoid-valued functor B ↦ T (B). Since we have an isomorphism of homotopy types BP U ≃ K(ℤ, 3) we see again that the isomorphism classes of objects in T (B) are classified by H^{3}(B, ℤ) (this class is usually called the Dixmir-Douady invariant). And furthermore, Hom_{T (B)}(H, H′) is again a torsor over H^{2}(B, ℤ). The monoidal structure on T (B) is induced by the fiberwise tensor product.

The second picture is very suitable to give a short definition of twisted K-theory at least for locally compact B. If H ∈ T (B) is a twist, i.e. a bundle of C^{∗}-algebras H → B, then we can form the C^{∗}-algebra C_{0}(B, H) of continuous sections of H vanishing at infinity. Then we define

K(B, H) := K(C_{0}(B, H)) .

Below we will only use the axioms fixed in ^{\underline{bunkeschickt}} [3, Section 3.1] and not rely on any concrete realization.

kli

5.2 For a space B we consider a category T (B) of twists as in ^{\underline{bunkeschickt}} [3, Section 3.1]. Let H ∈ T (F_{n}) be a twist with isomorphism class [H] = h_{n} ∈ H^{3}(F_{n}, ℤ). Set further Ĥ := T̃^{∗}H ∈ T (F̂_{n}), where T̃ was defined in ^{\underline{rtz}}^{\underline{rtz}} 3.3. Then [Ĥ] = ĥ_{n} (see again ^{\underline{rtz}}^{\underline{rtz}} 3.3).

Recall the definition of a_{n} and â_{n} in ^{\underline{adef}}^{\underline{adef}} 3.5. Since a_{n}^{∗}h_{n} = â_{n}^{∗}ĥ_{n} by Lemma ^{\underline{aeq}}^{\underline{aeq}} 3.4, we conclude that there exists an isomorphism u : â_{n}^{∗}Ĥ → a_{n}^{∗}H in T (F_{n} ×_{R_{n}} F̂_{n}). Since by Lemma ^{\underline{aeq}}^{\underline{aeq}} 3.4 H^{2}(F_{n} ×_{R_{n}} F̂_{n}, ℤ) = 0, this isomorphism is in fact unique.

5.3 We consider the pull-back diagrams

\begin{matrix}T^{n} × T̂^{n} & →^{v} & F_{n} ×_{R_{n}} F̂_{n} \\ ↓ & & ↓ \\ ∗ & → & R_{n} \end{matrix} ,  \begin{matrix}T̂^{n} & →^{ŵ} & F̂_{n} \\ ↓ & & ↓ \\ ∗ & → & R_{n} \end{matrix} ,  \begin{matrix}T^{n} & →^{w} & F_{n} \\ ↓ & & ↓ \\ ∗ & → & R_{n} \end{matrix} .

Let y_{1}, …, y_{n}, ŷ_{1}, … ŷ_{n} be the canonical generators of H^{1}(T^{n} × T^{n}, ℤ) as in the proof of Lemma ^{\underline{aeq}}^{\underline{aeq}} 3.4.

We choose trivializations of twists t̂: 0 → ŵ^{∗}Ĥ and t : w^{∗}H → 0. Then we consider the automorphism

ϕ : 0 →^{pr_{2}^{∗}t̂} pr_{2}^{∗}ŵ^{∗}Ĥ ≅ v^{∗}â_{n}^{∗}Ĥ →^{v^{∗}u} v^{∗}a_{n}^{∗}H ≅ pr_{1}^{∗}w^{∗}H →^{pr_{1}^{∗}t} 0 .

We can identify ϕ with a class ϕ ∈ H^{2}(T^{n} × T̂^{n}, ℤ). If we decompose

ϕ = ϕ_{l} ⊕ ϕ_{m} ⊕ ϕ_{r} ∈ H^{2}(T^{n}, ℤ) × 1 ⊕ H^{1}(T^{n}, ℤ) ⊗ H^{1}(T̂^{n}, ℤ) ⊕ 1 × H^{2}(T̂^{n}, ℤ)

according to the Künneth formula, then the component ϕ_{m} ∈ H^{1}(T^{n}, ℤ) ⊗ H^{1}(T̂^{n}, ℤ) is well-defined, not depending on the choices of t and t̂.

5.4

cond3 Proposition 5.1 We have

ϕ_{m} = ∑_{i=1}^{n} y_{i} ⊗ ŷ_{i} .

Proof. Note that by Lemma ^{\underline{var}}^{\underline{var}} 4.4 and Proposition ^{\underline{var1}}^{\underline{var1}} 4.5 we have a canonical equivalence R_{1} ≅ R̃_{1}. In ^{\underline{bunkeschickt}} [3] we studied in detail the topology of R_{1} and the associated T -duality. The idea of the proof is to reduce the present task to the case n = 1.

We consider the n-fold product F → B of the T^{1}-bundle F_{1} → F_{1} (i.e. F = (F_{1})^{n}, B = (R_{1})^{n}). Let p_{i} : B → R_{1} denote the projections. Let (z, ẑ) : B → K(ℤ^{n}, 2) × K(ℤ^{n}, 2) be the map with components z_{i} := p_{i}^{∗}c and ẑ_{i} := p_{i}^{∗}ĉ. We now apply obstruction theory to the lifting problem

\begin{aligned}& & R_{n} \\ & f ↗ & (c, ĉ) ↓ \\ B & →^{(z,ẑ)} & K(ℤ^{n}, 2) × K(ℤ^{n}, 2) \end{aligned} .

It follows from z_{i} ∪ ẑ_{i} = 0 that ∑_{i=1}^{n} z_{i} ∪ ẑ_{i} = 0 so that this diagram admits a lift. Since H^{3}(B, ℤ) ≅ 0 such a lift is in fact unique.

We therefore have a pull-back diagram of principal T^{n}-bundles

\begin{matrix}F & →^{f̃} & F_{n} \\ ↓ & & ↓ \\ B & →^{f} & R_{n} \end{matrix} .

We define h := f̃^{∗}h_{n}. In this way we obtain a pair (F, h) over B. We further have natural projections pr_{i} : F → F_{1}, i = 1, …, n. Using the characterizations of h_{n} and h_{1} in Lemma ^{\underline{re1}}^{\underline{re1}} 2.5, the naturality of Leray-Serre spectral sequences, and H^{3}(B, ℤ) ≅ 0, we see that

h = ∑_{i=1}^{n} pr_{n}h_{1} .  (5.2) defc

Let T_{B} : B → B be the product of the T -duality transformations T : R_{1} → R_{1} on each factor. Since T is a homotopy equivalence, there is a unique lift α in

\begin{aligned}& & R_{n} \\ & α ↗ & T ↓ \\ B & →^{f∘T_{B}} & R_{n} \end{aligned} .

Since the composition B →^{α} R_{n} → K(ℤ^{n}, 2) × K(ℤ^{n}, 2) coincides with (z, ẑ) we have α = f, and the following diagram commutes upto homotopy

\begin{matrix}B & →^{f} & R_{n} \\ T_{B} ↓ & & T ↓ \\ B & →^{f} & R_{n} \end{matrix} .

This shows that we have a pull-back diagram

\begin{matrix}F̂ & →^{f̂} & F̂_{n} \\ ↓ & & ↓ \\ B & →^{f} & R_{n} \end{matrix} ,

where F̂ → B is the n-fold product of F̂_{1} → R_{1}. We get the diagram

\begin{aligned}T^{n} × T^{n} & → & F ×_{B} F̂ & →^{f̃×_{B} f̂} & F_{n} ×_{R_{n}} F̂_{n} \\ ↓ & & ↓ & & ↓ \\ ∗ & → & B & →^{f} & R_{n} \end{aligned} .

It follows from Lemma ^{\underline{aeq}}^{\underline{aeq}} 3.4 (in the case n = 1) and the Künneth formula that H^{2}(F ×_{B} F̂, ℤ) ≅ 0. Therefore (f̃×_{B} f̂)^{∗}u : (f̃×_{B} f̂)^{∗}â_{n}^{∗}Ĥ → (f̃×_{B} f̂)^{∗}a_{n}^{∗}H is the unique isomorphism.

Let V ∈ T (F_{1}) and V̂ ∈ T (F̂_{1}) be twists in the classes h_{1} and ĥ_{1}. Then by (^{\underline{defc}}^{\underline{defc}} 5.2) there exists isomorphisms κ : f̃^{∗}H ≅ ∑_{i=1}^{n} pr_{i}^{∗}V and κ̂ : f̂^{∗}Ĥ ≅ ∑_{i=1}^{n} pr_{i}^{∗}V̂. The choice of these isomorphisms is not unique. However, their restrictions to the fiber over ∗ → B is unique. This follows from the structure of H^{2}(F, ℤ) (and of H^{2}(F̂, ℤ)) implied by Lemma ^{\underline{re1}}^{\underline{re1}} 2.5 in the case n = 1 and the Künneth-formula.

At this moment we fix some choices of κ and κ̂.

Let q_{i} : F ×_{B} F̂ → F_{1} ×_{R_{1}} F̂_{1} be the projection onto the ith component, and let r : F ×_{B} F̂ → F, r̂ : F ×_{B} F̂ → F̂ denote the projections onto the factors. Note that f̃∘ r = a_{n} ∘ (f̃×_{B} f̂), f̂∘ r̂ = â_{n} ∘ (f̃×_{B} f̂), a_{1} ∘ q_{i} = pr_{i} ∘ r and â_{1} ∘ q_{i} = pr_{i} ∘ r̂. We now have fixed isomorphisms

r^{∗}κ: (f̃×_{B} f̂)^{∗}a_{n}^{∗}H ≅ ∑_{i=1}^{n} q_{i}^{∗}a_{1}^{∗}V  ,  r̂^{∗}κ̂ : (f̃×_{B} f̂)^{∗}â_{n}^{∗}Ĥ ≅ ∑_{i=1}^{n} q_{i}^{∗}â_{1}^{∗}V̂.

Note that there is a unique isomorphism ψ : â_{1}^{∗}V̂ → a_{1}^{∗}V. This induces another isomorphism

Φ : (f̃×_{B} f̂)^{∗}â_{n}^{∗}Ĥ ≅^{r̂^{∗}κ̂} ∑_{i=1}^{n} q_{i}^{∗}â_{1}^{∗}V̂ ∑_{i=1}^{n} q_{i}^{∗}ψ ; ≅ ∑_{i=1}^{n} q_{i}^{∗}a_{1}^{∗}V ≅^{(r^{∗}κ)^{−1}} (f̃×_{B} f̂)^{∗}a_{n}^{∗}H .

It follows that (f̃×_{B} f̂)^{∗}u = Φ. We can now restrict Φ to the fiber T^{n} × T̂^{n}.

It was shown in ^{\underline{bunkeschickt}} [3, 3.2.4] that the restriction of ψ to a fiber T^{1} × T̂^{1} is classified by a generator of H^{2}(T^{1} × T̂^{1}, ℤ), namely by y ∪ ŷ in the canonical basis of H^{1}(T^{1} × T̂^{1}, ℤ). If we restrict the whole composition defining Φ to the fiber and use the definition of ϕ_{m}, together with the uniqueness of the restrictions κ and κ̂ and our freedom in the choice of t and t̂, we obtain the result. □

5.5 Let us consider a diagram

\begin{aligned}& & F ×_{B} F̂ \\ & p ↙ & & p̂ ↘ \\ F & & & & F̂ \\ & ↘ & & ↙ \\ & & B \end{aligned} ,

where F → B and F̂ → B are some T^{n}-principle bundles. Let H ∈ T (F) and Ĥ ∈ T (F̂) be twists. We assume that (F, H) and (F̂, Ĥ) are intertwined in the following sense: if c_{1}, …, c_{n} ∈ H^{2}(B, ℤ) are the Chern classes of F and ĉ_{1}, …, ĉ_{n} ∈ H^{2}(B, ℤ) are the Chern classes of F̂, then the isomorphism class [H] ∈ H^{3}(F, ℤ) shall be represented by ∑ y_{i} ⊗ ĉ_{i} ∈ E_{2}^{1,2}, the E_{2}-page of the Leray-Serre spectral sequence for F, where we use the canonical generators y_{i} ∈ H^{1}(F_{b}, ℤ) of the (principal) fiber of F. We also require the corresponding statement for [Ĥ]. Note that this implies in particular that [H] ∈ F^{2}H^{3}(F, ℤ) for the Leray-Serre filtration.

Let us further assume that there exists an isomorphism u : p̂^{∗}Ĥ → p^{∗}H.

Fix b ∈ B. Let F_{b} and F̂_{b} denote the fibers over b. The existence of u implies the existence of isomorphisms v : H_{|F_{b}} → 0 and v̂ : 0 → Ĥ_{|F̂_{b}}. We now consider the composition

u(b) : (0 →^{p̂_{|F̂_{b}}^{∗} v̂} p̂_{|F̂_{b}}^{∗}Ĥ_{|F̂_{b}} →^{u_{|F_{b}×F̂_{b}}} p_{|F_{b}}^{∗}H_{|F_{b}} →^{p_{|F_{b}}^{∗}v} 0) ∈ H^{2}(F_{b} × F̂_{b}, ℤ) .

Definition 5.3 We say that u satisfies the condition P(u), if

u(b) ∈ ∑_{i=1}^{n} y_{i} ∪ ŷ_{i} + span(y_{i} ∪ y_{j}, ŷ_{i} ∪ ŷ_{j})  ∀b ∈ B .  (5.4) condp

Of course, it suffices to check this for one point in each component of B.

tr13 Definition 5.5 An n-dimensional T -duality triple over B is an intertwined triple ((F, H), (F̂, Ĥ), u)

as above such that u satisfies P(u).

zt5

5.6 Here is our main construction of T -duality triples. Let f : B → R_{n} classify an element x ∈ P_{n}(B). Let (F, h) represent the underlying isomorphism class of pairs. Then we have a pull-back

\begin{matrix}F & →^{f̃} & F_{n} \\ ↓ & & ↓ \\ B & →^{f} & R_{n} \end{matrix}

such that h = f̃^{∗}h_{n}. Let x̂ := T (x) ∈ P_{n}(B) be the T -dual element and let (F̂, ĥ) represent the underlying isomorphism class of the dual pair. We have a pull-back diagram

\begin{matrix}F̂ & →^{f̂} & F̂_{n} \\ ↓ & & ↓ \\ B & →^{f} & R_{n} \end{matrix} .

such that f̂^{∗}h_{n} = ĥ. We consider the diagram

\begin{aligned}& & F ×_{B} F̂ \\ & p ↙ & & p̂ ↘ \\ F & & & & F̂ \\ & ↘ & & ↙ \\ & & B \end{aligned} .

Let f̃ ×_{B} f̂: F ×_{B} F̂ → F_{n} ×_{R_{n}} F̂_{n} be the canonical map. We then get a T -duality triple ((F, f̃^{∗}H), (F̂, f̂^{∗}Ĥ), (f̃×_{B} f̂)^{∗}u). In fact, P((f̃×_{B} f̂)^{∗}u)) is satisfied by Propostion \underline{cond3}^{\underline{cond3}} 5.1, and the two triples are intertwined by Lemma ^{\underline{re1}}^{\underline{re1}} 2.5 and naturality of the Leray-Serre spectral sequence.

5.7 Consider a T^{n}-bundle F → B and a twist H ∈ T (B). The following follows from Corollary ^{\underline{exex}}^{\underline{exex}} 4.8 and Proposition ^{\underline{cond3}}^{\underline{cond3}} 5.1.

rr43 Corollary 5.6 The pair (F, H) belongs to some n-dimensional T -duality triple ((F, H), (F̂, Ĥ), u) if and only if [H] ∈ F^{2}H^{3}(F, ℤ).

In general, the corresponding T -duality triple is not unique. It is obtained by lifting [F, [H]] ∈ P̃_{(0)}(B) to some element [B →^{f} R_{n}] ∈ P_{n}(B) and pulling back the universal T -duality triple ((F_{n}, H), (F̂_{n}, Ĥ), u) over R_{n} using f.

5.8 There are natural notions of an isomorphism between T -duality triples and of pullback along a map of base spaces. We thus can define the set-valued functor which associates to a space the set of isomorphisms classes Triple_{n}(B) of n-dimensional T -duality triples. After some calculations we expect that R_{n} is close but not equal to a classifying space for Triple_{n}.

5.9 In ^{\underline{bunkeschickt}} [3, Definition 3.7] we have introduced the notion of a T -admissible twisted cohomology theory h(…, …). Examples are twisted K-theory (see ^{\underline{w2w2w}}^{\underline{w2w2w}} 5.1) and twisted twoperiodized cohomology with real coefficients. Let ((F, H), (F̂, Ĥ), u) be a T -duality triple.

trd Definition 5.7 We define the T -duality transformation

T := p̂_{!} ∘ u^{∗} ∘ p^{∗} : h(F, H) → h(F̂, Ĥ) .

This transformation has degree −n.

5.10

isot Theorem 5.8 If ((F, H), (F̂, Ĥ)) is a T -duality triple, and if h is a T -admissible twisted cohomology theory, then the T -duality transformation ^{\underline{trd}}^{\underline{trd}} 5.7 is an isomorphism.

Proof. Exactly as in ^{\underline{bunkeschickt}} [3, Proof of Theorem 3.13] one uses induction, the Mayer-Vietoris exact sequence and the 5-lemma to reduce this to the case B = ∗. For B = ∗ one observes that the n-dimensional T -duality transformation is an iterated 1-dimensional T -duality transformation which is an isomorphism by the definition of T -admissibility and ^{\underline{bunkeschickt}} [3, Theorem 3.13]. □

References

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