Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes

For any (n-1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar question are also discussed for 2n-dimensional real vector bundles with particular prescribed characteristic Pontrjagin and Euler classes. We also analyze which of these bundles admit a complex structure. It turns out that all these bundles are closely related to the tangent bundles of quasitoric manifolds and moment angle complexes.


Introduction
For any finite simplicial complex K, Davis and Januszkiewicz constructed a family of realizations of the Stanley-Reisner algebra Z [K], that is the integral cohomology of these spaces is isomorphic to Z [K]. They also showed that all these spaces are homotopy equivalent [DJ,Section 4]. We denote a generic model for this homotopy type by DJ (K). In Section 6 of the mentioned paper, Davis and Januszkiewicz also studied particular vector bundles over DJ (K). They constructed an m-dimensional complex bundle λ ↓ DJ (K) whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra Z [K] and compared it with vector bundles obtained from the tangent bundles of the moment angle complex or a quasitoric manifold by an application of the Borel construction. In particular, they showed that these bundles are stably isomorphic and have the same Pontrjagin classes. We were wondering to which extend the characteristic classes determine the isomorphism type of these bundles.
We will split off a large trivial vector bundle from λ and will show that the isomorphism type of the remaining vector bundle ξ ↓ DJ (K) as well as of its realification ξ R is completely determined by its characteristic classes. We will also study real vector bundles ρ with the same Pontrjagin classes as ξ and prove for them similar existence and uniqueness results in terms Pontrjagin and Euler classes. We are able to offer several applications of the uniqueness results: Colorings of K are reflected by (stable) splittings of ξ and ξ R into a direct sum of complex line bundles or of 2-dimensional real bundles. We will also improve on the stable isomorphisms results mentioned above and classify complex structures on the real vector bundles ρ.
To make our statements more precise we need to fix notation and to recall some basic constructions. An abstract simplicial complex on the finite vertex set V of order |V | = m is a set K = {α 1 , ...., α s } of subsets α i ⊂ V which is closed under the formation of subsets and which contains the empty set ∅. We will always identify V with the set [m] : = {1, ..., m} of the first m natural numbers. The dimension dim K of K is expressed in terms of the cardinality of its faces α ∈ K. We set dim α: =|α|−1 and dim K is the maximum of the dimensions of its faces. Some of our statements involve sums taken over all maximal faces. We will denote this set by M K . Examples are given by full simplices and their boundaries. For a set α we denote by ∆[α] the simplicial complex given by all subsets of α and by ∂∆[α] the complex of all proper subsets of α. Then dim ∆[α] = |α| − 1 and dim ∂∆[α] = |α| − 2. The set M ∆ [α] consist only of the set α and M ∂∆[α] consists of all subsets of α of order |α| − 1. The complex K is a subcomplex of the full simplex ∆[m] on m-vertices and for each α ∈ K, the complex ∆[α] ⊂ K is a subcomplex of K.
For a commutative ring R with unit we denote by R[m]: =R[v 1 , ..., v m ] the graded polynomial algebra generated by algebraically independent elements v 1 , ..., v m of degree 2, one for each vertex of K. For each subset α ⊆ [m] we denote by v α : = j∈α v j the square free monomial whose factors are in one to one correspondence with vertices contained in α. The graded Stanley-Reisner algebra R [K] associated with K is defined as the quotient R [K] : = R[m]/I K , where I K ⊂ R[m] is the ideal generated by all elements v β such that β ⊆ [m] is not a face of K.
For a complex vector bundle η over a space X we denote by c i (η) ∈ H 2i (X; Z) the i-th Chern class and by c(η) : = 1 + Σ i≥1 c i (η) the total Chern class of η. For a real vector bundle ρ over X we use p i (ρ) ∈ H 4i (X; Z) and p(ρ) : = 1 + Σ i≥1 p i (ρ) to denote the i-th and the total Pontrjagin class of ρ. If ρ is k-dimensional and oriented, the Euler class is denoted by e(ρ) ∈ H k (X; Z).
We fix an isomorphism H * (DJ (K); Z) ∼ = Z [K]. Since BT m is an Eilenberg-MacLane space realizing the algebra Z[m], there exist a map q K : DJ (K) −→ BT m which induces in cohomology the projection Z[m] −→ Z [K]. We can think of T m as the maximal torus T m ⊂ U(m). The pull back of the m-dimensional universal complex vector bundle over BU(m) along the composition of q K with the maximal torus inclusion BT m −→ BU(m) produces a vector bundle λ ↓ DJ (K) whose total Chern class is c(λ) = m i=1 (1 + v i ). And the total Pontrjagin class of the realification λ R is p( [DJ,Section 6]. These characteristic classes play a particular role in our statements and we define c(K) : = m i=1 (1 + v i ) and p(K) : = m i=1 (1 − v 2 i ). Confusing notation we also denote by C and R 1-dimensional trivial complex or real vector bundles over a space X.
The vector bundle ξ ↓ DJ(K) also satisfies uniqueness properties. Its isomorphism type is completely determined by its characteristic classes. The realification ξ R of a complex bundle ξ carries a canonical orientation. And ξ R denotes the underlying oriented real vector bundle with this canonical orientation. The Euler class e(ξ R ) is then given by the n-the Chern class c n (ξ). Trivial real vector bundles get the standard orientation.
For clarification, the isomorphisms in Part (ii) and (iii) are isomorphisms of oriented bundles.
The realification ξ R satisfies the assumption of the last theorem, realizes the function ω ≡ 1, i.e. ω(µ) = 1 for all µ ∈ M K , and admits obviously a complex structure. We say that a real vector bundle ρ admits a complex structure if there exists a complex bundle η such that ρ and η R are isomorphic as non-oriented real vector bundles. For a function f : Theorem 1.4. Let K be an n−1-dimensional finite simplicial complex. Let ρ be an 2n-dimensional oriented real vector bundle over DJ (K) such that p(ρ) = p (K). Then, the bundle ρ admits a complex structure if and only if there exists a function f : Remark 1.5. Since Davis-Januszkiewicz spaces are simply connected, every real vector bundle ρ over DJ (K) is orientable. The Euler classes for different orientations may only differ by a sign. In Theorem 1.2, Theorem 1.3 and Theorem 1.4 we made the assumption that ρ is oriented. This is not a serious restriction. For example, if η is a complex and ρ a real vector bundle over DJ (K) such that ρ and η R are isomorphic as non-oriented vector bundles, then the canonical orientation on η R must agree with one of the possible orientation for ρ. That is that ξ R and ρ with the appropriate orientation are isomorphic as oriented bundles.
Several of the vector bundles will be constructed as homotopy orbit spaces. For a compact Lie group G and a G-space X, the Borel construction or homotopy orbit space EG × G X will be denoted by X hG . If η is an n-dimensional G-vector bundle over X with total space E(η), the Borel construction establishes a fibre bundle E(η) hG −→ X hG . In fact, this is an n-dimensional vector bundle over X hG [S], denoted by η hG .
There are several connections with the work of Davis and Januszkiewicz, in particular with Section 6 of [DJ]. For every simplicial complex K there exists an associated moment angle complex Z K which carries a T m -action. If K is the triangulation of an (n − 1)-dimensional sphere, the moment angle complex Z K is an (m + n)-dimensional manifold. The tangent bundle τ Z is a T m -equivariant real vector bundle. Applying the Borel construction establishes a vector bundle (τ Z ) hT m over (Z K ) hT m ≃ DJ (K) which is stably isomorphic to λ R . In particular, (τ Z ) hT m and ξ R have equal Pontrjagin classes. Some details of the construction are recalled in the next section.
An oriented quasi toric manifold M 2n is an oriented 2n-dimensional manifold with an T n -action such that the action is locally standard and the orbit space M 2n /T n = P is a simple polytope. We apply the Borel construction to the tangent bundle τ M of M 2n and get a vector bundle (τ m ) hT n over (M 2n ) hT n ≃ DJ (K P ), where K P is the simplicial complex dual to the boundary of P . Again, (τ M ) hT n and λ R are stably isomorphic and (τ M ) hT n and ξ R have the same Pontrjagin classes. All this can be found in [DJ,Sections 4 and 6].
Theorem 1.2 allows to draw the following corollary, which improves the stable isomorphism result about λ and (τ M ) hT n [DJ, Theorem 6.6] Corollary 1.6. (i) If K is the triangulation of an (n − 1)-dimensional sphere, then (τ Z ) hT m and ξ R ⊕ R m−n are isomorphic. (ii) If M 2n −→ P is an oriented quasi toric manifold over the simple polytope P , then (τ M ) hT n ⊕ R and ξ R ⊕ R are isomorphic.
The connected sum CP (2)♯CP (2) of CP (2) with itself is a quasitoric manifold over a square, but does not admit an almost complex structure [DJ, Section 0 (C') and Example 1.19] [BP,page 68]. In particular, for M = CP (2)♯CP (2), the real vector bundle (τ M ) hT n has no complex structure. This shows that in this case ξ R and (τ M ) hT n cannot be isomorphic and that we cannot omit the trivial summand in the second part of the above Corollary as well as in Theorem 1.2(ii).
Equivariant and non-equivariant almost complex structures for quasi toric manifolds are discussed in [BP,Section 5]. Every equivariant complex structure for a quasi toric manifold M establishes a complex structure for the Borel construction (τ M ) hT n . Since the total Pontrjagin class of (τ M ) hT n equals m i=1 (1−v 2 i ), Theorem 1.4 tells us that the Euler class e((τ M ) hT n ) decides whether (τ M ) hT n admits a complex structure. The calculation of e((τ M ) hT n ) can be worked out, but will be discussed in a forthcoming paper [DN]. It turns out that (τ M ) hT n has a complex structure if and only τ M has one. The combinatorial conditions for both cases are the same. For a few more details see Section 7.
In a recent paper [K], Kustarev gave a combinatorial condition, described in terms of omniorientations of the quasi toric manifold M, which is sufficient and necessary for the existence of an equivariant almost complex structure for M. One can show that his condition and our condition established in Theorem 1.4 are equivalent [DN]. In contrast to his result we are able to give a complete classification of all complex structures for (τ M ) hT n . For example, we have the following classification result. We say that two complex structure η 1 and η 2 for an real vector vector bundle ρ are isomorphic if η 1 ∼ = η 2 as complex vector bundles.
Corollary 1.7. Let K be an (n − 1)-dimensional abstract finite simplicial complex and let s ≥ n. Let ρ be an 2s-dimensional oriented real vector bundle over DJ (K) such that p(ρ) = p (K).  In Section 7 we will give an explicit description of the complex structures of the bundle ρ. In fact, for each function f : [m] −→ {±1}, we will construct an n-dimensional complex vector bundle ξ f over DJ (K). The complex structures are represented by the bundles ξ f with e f (K) = ±e(ρ) respectively ξ f ⊕ C s−n .
Theorem 1.2 also allows to relate colorings of simplicial complexes to stable splittings of the vector bundle ξ. A regular r-paint coloring, an r-coloring for short, of a simplicial complex K is a non degenerate simplicial map K −→ ∆[r], i.e. each face of K is mapped isomorphically onto a face of ∆[r]. If dim K = n − 1, then K only allows r-colorings for r ≥ n. The inclusion K ⊂ ∆([m]) always provides an m-coloring.
Corollary 1.8. For an (n − 1)-dimensional simplicial complex K the following conditions are equivalent Proof. The proof is based on a similar result for colorings and splittings of λ and λ R . We can assume that n ≤ r ≤ m. An r-coloring K −→ ∆([r]) establishes a splitting λ ∼ = r i=1 ν i ⊕ C m−r [N2]. The bundles r i=1 ν i and ξ ⊕ C r−n have the same Chern classes. Theorem 1.2 shows that both vector bundles are isomorphic and produces the desired splitting for ξ ⊕ C r−n . The realification of both sides establishes a splitting for ξ R ⊕ R 2(m−r) . A splitting of ξ R ⊕ R 2(r−n) produces a splitting for λ R ∼ = ξ R ⊕ R 2(m−n) . And such a splitting establishes a coloring K −→ ∆([r]) [N2].
The paper is organized as follows. In the next section we describe different models for DJ (K) needed for our purposes. A geometric construction of the vector bundle ξ with the properties stated in Theorem 1.1 is contained in Section 3. Using Sullivan's arithmetic square we reduce the global uniqueness problem for our bundles to the analogous p-adic question in Section 4. This section also contains a proof of Theorem 1.3. The p-adic case involves the calculation of some higher derived inverse limits. In Section 5 we provide some general remarks about higher limits and filtrations of functors, needed in Section 6 for the proof of the p-adic existence and uniqueness statements. In the last section we discuss complex structures for real vector bundles and prove Theorem 1.4 and Corollary 1.7.
If not specified otherwise, K will always denote an abstract finite simplicial complex of dimension n − 1 with m vertices and M K the set of maximal faces, We also fix the classes c (K) . We would like to express our gratitude to Natalia Dobrinskaya, Taras Panov and Nigel Ray for many helpful discussions.
2. Models for DJ (K) For the proof of our theorems we will need different models for the space DJ (K), which we will describe in this section.
Let cat(K) be the category whose objects are the faces of K and whose arrows are given by the subset relations between the faces. cat (K) has an initial object given by the empty face. Given a pair (X, Y ) of pointed topological space we can define covariant functors The functor X K assigns to each face α the cartesian product X α and to each morphism i α,β the inclusion X α ⊂ X β where missing coordinates are set to the base point * .
We are interested in two particular cases, namely the functor X K for the classifying space BT = CP (∞) of the 1-dimensional circle T (considered as a topological group) and the functor (X, Y ) K for the pair (D 2 , S 1 ). The colimit is called the moment angle complex associated to K. We have inclusions Z K ⊂ (D 2 ) m ⊂ C m and the standard T m -action on C m restricts to Z K . The Borel construction produces a fibration [DJ,Theorem 4.8]. We will use this model for the geometric construction of our vector bundle ξ.
Buchstaber and Panov gave a different construction for DJ (K). They showed that the colimit c(K) : = colim cat(K) BT K is homotopy equivalent to B T K and that the inclusion [BP,Theorem 6.29].
We wish to study homotopy theoretic properties of c (K). Colimits behave poorly from a homotopy theoretic point of view, but the left derived functor, known as homotopy colimit, provides the appropriate tool for such questions. Following [V], the homotopy colimit hc(K) : = hocolim cat(K) BT K may be described as the two sided bar construction B( * , cat(K), BT K ) in Top. For the functor BT K , the map hocolim cat(K) BT K −→ colim cat(K) BT K is a homotopy equivalence [NR1]. In particular, hc(K) ≃ c (K). We will use the model hc (K) to prove the uniqueness properties of the vector bundle ξ and to provide a homotopy theoretic construction of ξ.
For later purpose we include the following remark.
. And this map is induced by the inclusion BT α −→ DJ(K) which comes from the description of DJ(K) as colimit or homotopy colimit. The direct sum of the maps h α induces a monomorphism of algebras

Geometric construction of vector bundles
The m-dimensional torus T m acts on C m via coordinate wise multiplication. Let λ ′ ↓ Z K denote the m-dimensional T m -equivariant complex vector bundle over Z K given by the diagonal action of T m on C m × Z K . The Borel construction produces an m-dimensional complex bundle λ: =λ ′ hT m over DJ (K) which is isomorphic to the pull back of the universal bundle over BU(m) along the composition of q K : DJ(K) −→ BT m and the maximal torus inclusion BT m −→ BU(m). As already mentioned, we have c(λ) = c(K) (for details see [DJ,Section 6]). For x, z ∈ C m we denote by z the coordinate wise complex conjugate of z and by xz ∈ C m the coordinate wise product of x and z; that is that the i-th coordinate (xz) i of xz is given by the product We define a map Here, we use that Z K ⊂ C m . If T m acts on the target only via the second coordinate, then the space hT m as vector bundles over DJ (K).
In particular, z i = 0 for i ∈ α. Since the matrix A α is invertible, this shows that the restriction of f A to the fiber over z is an epimorphism and that f A is a bundle epimorphism. This proves the first part.
The kernel ξ ′ : = ker f A is again a T m -equivariant vector bundle over Z K . Since for a compact Lie group G every short exact sequence of G-equivariant vector bundles over compact spaces splits [S], we get λ ′ ∼ = ξ ′ ⊕ C m−n . This proves the second part. The other three parts are direct consequences of Part (ii).
Remark 3.2. This gives a geometric construction of the vector bundle ξ : = ξ ′ hT m , whose existence was claimed in Theorem 1.1. We will give an alternative proof of Theorem 1.1 in the later sections.
The construction of ξ depends on the chosen matrix A. But, as Theorem 1.2 says, the isomorphism type is independent of A.

Uniqueness properties of ξ
In this section we describe vector bundles by their classifying maps. That is an r-dimensional complex vector bundle η over a space X is a map η : X −→ BU(r). Then, Theorem 1.1 and Theorem 1.2 become statements about existence of maps and the number of homotopy classes of maps realizing prescribed cohomological data given by characteristic classes. As usual, such problems are best approached by considering separately their rational and p-adic versions. An application of Sullivan's arithmetic square will then provide the desired global information. As a side effect we will also provide a homotopy theoretic construction of the vector bundle ξ : DJ(K) −→ BU(n), already constructed in Section 3.
Let U(r) −→ SO(2r) −→ SO(2r + 1) denote the standard inclusions of the compact Lie groups U(r), SO(2r) and SO(2r + 1). Composition with BU(r) −→ BSO(2r) induces the realification of a complex vector bundle η : X −→ BU(r) and, as in the introduction, will be denoted by η R . Composition with BSO(2r) −→ BSO(2r + 1) means that we add an 1-dimensional trivial bundle. If we think of T r as the set of diagonal matrices in U(r), the compositions describe at each stage the standard maximal torus of the compact Lie group. The integral cohomology is a polynomial algebra with r generators in degree 2. Passing to classifying spaces and cohomology establishes maps where W U (r) , W SO(2r) and W SO(2r+1) denote the Weyl groups, c i the universal Chern classes for complex, p i the universal Pontrjagin classes for real and e r the universal Euler class for oriented 2r-dimensional real vector bundles. The class c i is given by the i-th elementary symmetric polynomial σ i (v 1 , ...v r ), the class e r by σ r (v 1 , ...v r ) and p i by σ i (−v 2 1 , ..., −v 2 r ). The Euler class satisfies the equation e 2 r = (−1) r p r . In all cases the arrow is an epimorphism , for U(r) even an isomorphism. If we pass to rational coefficients all arrows become isomorphisms. If  This diagram can partly be realized by spaces and maps, namely by We want to construct a map DJ(K) −→ BU(n) making the above diagram commutative up to homotopy. Such a map is a realizations of f and establishes an n-dimensional complex vector bundle over DJ (K) which has the desired Chern classes. In fact, the bundle ξ ↓ DJ (K) constructed in Section 3 is such a map. But we will give here another construction. We also want to show that such a map is unique up to homotopy. We can play a similar game with BSO(2m) and BSO(2n). In this case, the composition Z[p 1 , ...., p m−1 , e m ] −→ Z[m] −→ Z [K] factors through Z[p 1 , ..., p n−1 , e n ] −→ Z [K]. We have some freedom for the choice of the image of the Euler class e n . Since p n is mapped to the non trivial class p n (K) : = (−1) n σ n (v 2 1 , ...v 2 m ) i, each square root of (−1) n p n (K) establishes a different factorization Z[p 1 , ..., p n−1 , e n ] −→ Z [K]. Again, all these factorizations can be realized and are unique up to homotopy. Before we make these statements precise, we need a description of all square roots. The Euler classes e ω associated to functions ω : M K −→ {±1} and defined in the introduction provide a complete list of such square roots.
Lemma 4.1. Let K be an (n − 1)-dimensional simplicial complex. Then, every square root of (−1) n p n (K) is of the form e ω .
Proof. Every square root e of (−1) n p n (K) can be write as a linear combination e = r i=1 a i q i of monomials q i which are of the form m j=1 v l j j with l j ≥ 0 and j l j = n. We can compare the square e 2 = i a 2 i q 2 i + i<k 2a i a j q i q k with µ∈M K v 2 µ . By Remark 2.1, both expressions are equal in Z [K] precisely, when all non trivial monomials have equal coefficients and q 2 i = 0 if and only if q i = 0. Hence, if µ ∈ K is a maximal face, the coefficient of v µ in e is ±1 and all other coefficients vanish. This shows that e = e ω for an appropriate function ω : M K −→ {±1}.
We say that a map ρ ω : The uniqueness question can be handled simultaneously for the different cases. For n ≤ r we denote by G r one of the groups U(r), SO(2r) or SO(2r + 1) and by γ r : DJ(K) −→ BG r one of the maps ξ ⊕ C r−n , ρ ω ⊕ R 2(n−r) or ξ R ⊕ R 2(r−n)+1 . The above two results are versions of our first two main theorems. Proof of Theorem 1.1 and Theorem 1.2: Theorem 1.1 is contained in Theorem 4.2 and Theorem 1.2(i) and (iii) in Theorem 4.3. The non vanishing of c n (ξ) and p n (ξ R ) follows from the fact that K contains a face of dimension n − 1. The inequalities r ≥ n and s ≥ 2n follow from the non vanishing of these Chern and Pontrjagin classes. Since DJ (K) is simply connected, every real vector bundle ρ ↓ DJ(K) can be given an orientation. If dim ρ > 2n, then e(ρ) = 0 = e(ξ R ⊕ R s−2n ) and if dim ρ = 2n then the same formula holds for ρ ⊕ R and ξ R ⊕ R. In both cases, Theorem 4.3 establishes the desired isomorphism between the vector bundles.
As already mentioned the proofs of the above theorems are based on an arithmetic square argument and rational and p-adic versions of the above statements. In the rest of this section we will reduce the proof of Theorem 4.2 and Theorem 4.3 to their p-adic versions.
For a topological space X we denote by X 0 the rationalization, by X ∧ p the p-adic completion in the sense of Bousfield and Kan, by X ∧ : = p X ∧ p the product of all p-adic completions, and by X A f the finite adele type of X, which is the formal completion of X 0 or the rationalization of X ∧ . All spaces under consideration are simply connected.In this case Sullivan's arithmetic square is a homotopy pull back diagram. Proof of both theorems: All claims follow from the fact that (BG r ) 0 is a product of rational Eilenberg-Maclane spaces. Theorem 4.7. Let p be a prime and K an (n − 1)-dimensional finite simplicial complex. A map (δ r ) ∧ p :

For abbreviation, we define H
. The proof of the last two theorems will be postponed to Section 6.
The following statement is necessary for the proof of Theorem 4.3.
Theorem 4.8. The map Proof. Since the arithmetic square for BG r is a pull back, the homotopy fiber F of BG r −→ BG ∧ r is equivalent to the homotopy fiber of the rationalization (BG r ) 0 −→ (BG r ) A f . Since (BG r ) 0 is a product of rational Eilenberg-MacLane spaces of even degrees, π i (F ) = 0 for i even. The obstruction groups for lifting homotopies between maps DJ(K) −→ BG ∧ r to BG r are given by H * (DJ (K); π * (F )). All these obstruction groups vanish, since H * (DJ (K); Z) is concentrated in even degrees.
Proof of Theorem 4.2. Since the rationalization BU(n) 0 is a product of rational Eilenberg-MacLane spaces of even degrees, the same holds for the finite adele type BU(n) A f . Up to homotopy, maps into BU(n) A f are determined by cohomological information. Since ξ 0 and ξ ∧ p realize f ⊗ Q respectively f ⊗ Q ∧ p , the two compositions are homotopic. Using the homotopy pull back property of the arithmetic square for BU(n) we can construct a map ξ : DJ(K) −→ BU(n) with the desired cohomological property. This proves the claim for BU(n).. For BSO(2n) we can argue analogously.

higher limits
The proofs of Theorem 4.6 and Theorem 4.7 involve the calculation of the some higher derived limits for covariant functors defined on the opposite category cat(K) op of cat (K). For a definition and properties of higher derived limits see [BK] or [O]. We will drop cat(K) op from the notation of limits.
Let ab denote the category of abelian groups. We would like to construct a filtration for a functor Φ : cat(K) op −→ ab. For 0 ≤ s ≤ n we define functors Φ ≤s , Φ s : cat (K) Since for |α| < |β| there is no arrow α → β in cat(K) op , both functors are well defined. Moreover, we have Φ ≤n = Φ. There exist short exact sequences of functors of higher derived limits.
Let M : = Φ(∅) and let cst M : cat(K) op −→ ab denote the constant functor with value M; i.e. every face is mapped to M and every morphism to the identity. Then, lim i cst M = 0 for i ≥ 1 since cat(K) op has a terminal object and is contractible [BK]. By part (i) and (ii), we get for i ≥ 1 which proves the third part.

The p-adic case
We will work with the model hc(K) = hocolim cat(K) BT K . Again all colimits, limits and higher derived limits are taken over cat (K) or the opposite category cat (K) op . For simplification we will continue to drop these categories from the notation of limits. We will also skip the notation ∧ p from completion of maps. Again, G r will denote either U(r), SO(2r) or SO(2r + 1), and T r ⊂ G(r) the standard maximal torus.
Theorem 4.6 and Theorem 4.7 state the existence and uniqueness of particular maps hc(K) −→ BG r ∧ p . We need some tools to calculate π 0 (map(hc (K), BG r ∧ p ). We have map(hc (K) where [−, −] denotes the set of homotopy classes of maps. An element φ ∈ lim[BT K , BG r ∧ p ] is given by a set of homotopy classes of maps φ α : BT α −→ BG r ∧ p . We want to know whether R −1 (φ) is non empty and, if this is the case, how many elements are contained in R −1 (φ). The Bousfield-Kan spectral sequence for homotopy inverse limits [BK], together with work by Wojtkowiak [W] clarifying the situation in small degrees, provides an obstruction theory which may answer both questions. The obstruction groups are given by the higher derived limits of the functors Π Gr i : cat(K) op −→ ab which are defined by Π Gr i (α): =π i (map(BT α , BG r ∧ p ) φα ). If lim j Π Gr i = 0 for j = i, i+1 and all i ≥ 1, then R −1 (φ) consist of exactly one element. That is there there exists a map φ : hc(K) −→ BG r ∧ p , uniquely determined up to homotopy, such that the restriction φ| BT α is homotopic to φ α .
For the proof of Theorem 4.6 we need to construct particular maps ξ : hc(K) −→ BU(n) ∧ p and ρ ω : hc(K) −→ BSO(2n) ∧ p . We will discuss the construction in detail only for the latter case. It can be easily adapted to the first case.

−→ Z[α]
of h α , defined in Remark 2.1, and g ω maps p i onto p i (α) and e n onto 0 if |α| < n and onto ω(α)e(α) if |α| = n. This algebraic map can be realized by a composition where the second map is induced by the maximal torus inclusion T n ⊂ SO(2n) and the first map by a coordinate wise inclusion T α −→ T n combined with complex conjugations on some of the coordinates. For |α| < n any number of conjugations is allowed, but for |α| = n, we need an odd number of complex conjugation if ω(α) = −1 and an even number otherwise.
The Weyl group W SO(2n) of SO(2n) is isomorphic to (Z/2) n−1 ⋊ Σ n , where Σ n acts on T n via permutations of the coordinates and (Z/2) n−1 via even numbers of complex conjugation on coordinates. In fact, for each pair of coordinates, there exists an element in (Z/2) n−1 which induces complex conjugation exactly on these two coordinates. For i = 1, 2 let φ i : T α −→ T n be a coordinatewise inclusion combined with l i complex conjugations. If |α| = n then there exists w ∈ W SO(2n) such that φ 1 = wφ 2 precisely when l 1 and l 2 are congruent modulo 2. And if |α| < n, then there exists a missing coordinate which we can use to pass from an even number to an odd number of complex conjugations. In this case, there always exists w ∈ W SO(2n) such that φ 1 = wφ 2 . This shows that the underlying homomorphisms for different choices of ρ ωα are conjugate in SO(2n) and that different choices produce homotopic maps between the classifying spaces.
This argument also shows that, if α ⊂ β, the triangle y y r r r r r r r r r r BSO(2n) ∧ p commutes up to homotopy. In particular the set of maps ρ ωα defines an and all i ≥ 1 then there exists a map ρ ω : hc(K) −→ BSO(2n) ∧ p , uniquely determined up to homotopy such that ρ ω | BT α ≃ ρ ω α .
For n ≤ r and a face α ∈ K, we denote by γ rα : BT α −→ BG r one of the maps ξ α ⊕ C r−n , ρ ω α ⊕ R 2(r−n) or (ξ α ) R ⊕ R 2(r−n)+1 . By the above construction these maps fit together to define an element in lim [BT K , BG r ] and define functors Π Gr i : cat(K) op −→ ab. Proposition 6.2. For all j ≥ i ≥ 1, we have lim j Π Gr i = 0 The proof needs some preparation. The involved mapping spaces can be described in terms of centralizers of subtori. A map γ rα is induced by a homomorphism ι α : T α −→ G r which is given by the composition of a coordinate wise inclusion T α −→ T G(r) , possibly some conjugations on coordinates, and the inclusion of the maximal torus into G(r). Conjugation on coordinates of the maximal torus has no effect on the centralizer of ι α . The centralizer C Gr (T α ) : = C Gr (α) of the image of ι α can be identified with T α × G r−|α| ⊂ G r , where the inclusions of the factors are induced by the subset relations α ⊂ [r] and [r] \ α ⊂ [r].
By [N1], there exists a mod-p equivalence Moreover, up to homotopy the above mod-p equivalence is natural with respect to morphisms β ⊃ α in cat (K) op . Such an inclusion induces the composition between the classifying spaces of the centralizers. After p-adic completion this map is homotopic to the induced map between the associated mapping spaces. Passing to homotopy groups, this describes the map Π Gr i (α) −→ Π Gr i (β). It will be convenient to define functors ∧ p ). In the following we partly need to distinguish between U(r) on the one hand and SO(2r) and SO(2r + 1) on the other hand. We will denote by SO r one of special orthogonal groups. Part (ii) and Part (iii) follow from the fact that π 2s+1 (BU(t)) = 0 for 0 ≤ s ≤ t − 1 and π 2s (BU(t)) ∼ = π 2s (BU(t + 1)) for 1 ≤ s ≤ t.
For every connected compact Lie group H, the classifying space BH is simply connected. This proves Part (iv).
The above arguments and Proposition 6.2 allow to draw the following corollary.
Corollary 6.4. Let φ, ψ : DJ(K) −→ BG r ∧ p be two map such that for each face α of K the restrictions φ| BT α and ψ| BT α are homotopic to γ rα . Then, φ and ψ are homotopic.

Complex structures for vector bundles
The stable uniqueness result for vector bundles over DJ (K) realizing the total Pontrjagin class p(K) = i (1 − v 2 i ) shows that all the bundles ρ ω : DJ(K) −→ BSO(2n) of Theorem 4.2 admit stably a complex structure. In fact, ρ ω ⊕ R 2 ∼ = ξ R ⊕ R 2 already admits a complex structure. In this section we will discuss the unstable version, namely which of the bundles ρ ω actually are isomorphic to the realification of a complex vector bundle. We also will classify the complex structures of ρ ω up to isomorphism, in the unstable case as well as in the stable case. For this purpose we need to classify all complex bundles η over DJ (K) with p(η R ) = p (K).
For every function f : [m] −→ {±1} we define an element We will construct a complex vector bundle , then ξ f = ξ. For each α ∈ K we define a homomorphism ι f,α : T α −→ T α which is given by complex conjugation on the i-th coordinate if f (i) = −1 and the identity otherwise. The collection of the maps θ f,α : BT α −→ BT α induced by ι f,α provides a natural transformation Θ f : BT K −→ BT K and therefore a map By construction, the square θ 2 f is homotopic to the identity on DJ (K). In cohomology, the induced map θ * f : Z[K] −→ Z [K] maps v i to f (i)v i . The total Chern class of the composition ξ f : = ξθ f : (K) and the total Pontrjagin class by p((ξ f ) R ) = p(ξ f ) = p (K). As already explained in the introduction, the function ω f : Proof. If ξ f ∼ = ξ g , then both have the same Chern classes given by i (1 + f (i)v i ) and i (1 + g(i)v i ). A comparison of the first Chern class of both bundles already shows that f (i) = g(i) for all i.
Since the Euler classes of ξ f and ξ g are given by e f (K) and e g (K).
The bundles ξ f and ξ g are isomorphic as non-oriented real vector bundles if and only if the Euler classes of both bundle differ at most by a sign (Theorem 1.2 and Remark 1.5). And this is true if and only if ω f = ±ω g .
The next theorem classifies all complex bundles whose realification realizes the Pontrjagin class p (K)  Proof. The first part follows from the fact that θ 2 f = id DJ (K) . If η and ξ f induce the same map in rational cohomology, then the same holds for ηθ f and ξ f θ f ≃ ξθ 2 f ≃ ξ. Hence, by Theorem 1.2, both maps are homotopic as well as η and ξ f . Now let η : The Euler class e(η R ) is a square root of (−1) n p n (K) and determines a function ω : M K −→ {±1} such that e(η R ) = e ω (K). By Theorem 4.3, the maps η R and ρ ω are homotopic.
For each α ∈ K, the composition is homotopic to a map induced by a homomorphism ι α : T α −→ T n composed with the maximal torus inclusion T n −→ U(n) (Theorem 6.5). And the composition is homotopic to a map induced by ι α composed with T n −→ U(n) −→ SO(2n). By construction, the restriction ρ ω | BT α : BT α −→ BSO(2n) is homotopic to map induced by a coordinate wise inclusion T α −→ T n followed by complex conjugation on some coordinates and composed with T n −→ SO(2n). We can apply again Theorem 6.5. This shows that ι α : T α −→ T n is of the same form, namely coordinate wise inclusion followed by complex conjugation on some coordinates. If β ⊂ α, the composition ι α | T β : T β −→ T α ια −→ T n and the homomorphism ι β : T β −→ T n differ only by an element of the Weyl group of U(n), that is by a permutation of the coordinates. Such an operation has no effect on complex conjugations. This allows to define a map f : [m] −→ {±1} by f (i) = −1 if the homomorphism ι {i} : T {i} −→ T n involves complex conjugation and f (i) = 1 otherwise.
By construction of the function f , we have ξ f | BT α ≃ η| BT α for all α ∈ K (Theorem 6.5). In particular, the maps η and ξ f induce the same map in rational cohomology and are therefore homotopic (Part (i)).
Remark 7.3. Proposition 7.1 and Theorem 7.2 also have stable versions. The same arguments as above work. Suppose that t ≥ 1. (i) For two function f, g : [m] −→ {±1}, the bundles ξ f ⊕C t and ξ g ⊕C t are isomorphic if and only if f = g, but the realifications (ξ f ) R ⊕ R 2t and (ξ g ) R ⊕ R 2t are always isomorphic. (ii) Let η : DJ(K) −→ BU(n + t) be an (n + t)-dimensional complex vector bundle. Then, η and ξ f ⊕ C t are homotopic if and only if both induce the same map in rational cohomology. And if p(η R ) = p(K), then there exists a function f : [m] −→ {±1} such that η ∼ = ξ f ⊕ C t . Now we are in the position to prove Theorem 1.4 and Corollary 1.7, which we recall both for the convenience of the reader. Corollary 1.7. Let K be an n − 1-dimensional abstract finite simplicial complex and let s ≥ n. Let ρ be an 2s-dimensional oriented real vector bundle over DJ (K) such that p(ρ) = p (K). Proof of Theorem 1.4 An even dimensional orientable real vector bundle ρ over DJ (K) admits two orientations. And switching the orientation means to multiply the Euler class by −1. We denote the bundle with the opposite orientation by ρ. In particular, ρ and ρ are isomorphic as non-oriented real vector bundles.
If the Euler class e(ρ) of ρ has the required form then either ρ or ρ and (ξ f ) R have the same Euler class and the same Pontrjagin classes. By Theorem 4.3, this shows that either ρ ≃ (ξ f ) R or that ρ ≃ (ξ f ) R . In particular, ρ admits a complex structure.
Proof of Corollary 1.7. First we assume that s = n. If η is a complex structure for ρ, then, by Theorem 7.2, there exists a function f : [m] −→ Remark 7.5. For a simple polytope P we denote by K P the dual of the boundary of P . Since P is simple, K P is a simplicial polytope, in particular a simplicial complex, homeomorphic to S n−1 . An orientation for P or K P inherits an orientation to every maximal face µ ∈ K P , and hence, an orientation on the set µ. For a matrix Λ ∈ Z n×m and a maximal face µ ∈ K P we denote by Λ µ the maximal minor given by the columns of Λ associated to the vertices in µ, but we order the columns according to the orientation of µ. That is the order of the columns is only fixed up to even permutations, but such a permutation will not change the determinant.
A dicharacteristic pair (P, Λ) consists of an oriented simple polytope P of dimension n with m facets and a matrix Λ ∈ Z n×m such that det A µ = ±1 for all maximal faces of K P . Up to diffeomorphism, every oriented quasi toric manifold can be constructed from a dicharacteristic pair (P, Λ) (e.g. see [BP]). The Euler class of (τ M ) hT n is then given by e((τ M )hT n ) = µ∈M K P det Λ µ v µ [DN]. In particular, the real vector bundle (τ M ) hT n admits a complex structure if and only if their exists a function f : [m] −→ {±1} and ǫ = ±1 such that det A µ = ǫf (µ) for all maximal faces µ ∈ K P . This is exactly the condition sufficient and necessary for the existence of an almost complex structure for M 2n (e.g. see [BP,Corollary 5.54] and the following remarks).
If we stabilize the bundle (τ M ) hT n , the picture will change. Since (τ M ) hT n ⊕ R 2 ∼ = ξ R ⊕ R 2 (Theorem 1.2), Corollary 1.7 gives a complete list of the isomorphism classes of complex structures for (τ M ) hT n ⊕ R 2s for s ≥ 1.