PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 63–71 DOI: 10.2298/PIM1409063B

IMMERSIONS AND EMBEDDINGS OF QUASITORIC MANIFOLDS OVER THE CUBE

Ðorđe Baralić

Communicated by Rade Živaljević

Abstract. A quasitoric manifold M^{2n} over the cube I^{n} is studied. The Stiefel–Whitney classes are calculated and used as the obstructions for immersions, embeddings and totally skew embeddings. The manifold M^{2n} , when n is a power of 2, has interesting properties: imm(M^{2n} ) = 4n − 2, em(M^{2n} ) = 4n − 1 and N(M^{2n} ) ⩾ 8n − 3.

1. Introduction

Immersions and embeddings of manifolds are a classical topic in algebraic topology. Almost every monograph in topology has sections devoted to manifolds and obstructions to immersions and embeddings [2].

A nice introduction to problems and theory of characteristic classes is given in [11]. The connection among Stiefel–Whitney classes, immersions and embeddings is given by the following theorem

Theorem 1.1. If k := max\{i | \overline{w}_{i}(M^{n}) ≠ 0\}, then

imm(M^{n}) ⩾ n + k and em(M^{n}) ⩾ n + k + 1, where
imm(M^{n}) = min{d | M immerses into ℝ^{d}} ,
em(M^{n}) = min{d | M embedds intoℝ^{d}} .

The study of skew embeddings was started by Ghomi and Tabachnikov in [8]. They defined a number N(M^{n}) = min{d | M totally skew embedds into R^{d}}, for which they obtained the bounds 2n + 2 ⩽ N(M^{n}) ⩽ 4n + 1.

2010 Mathematics Subject Classification: Primary 57N35, 57R20; Secondary 52B20.

Key words and phrases: quasitoric manifolds, cube, Stiefel–Whitney classes, immersions, embeddings.

Partially supported by the Grant 174020 of the Ministry for Education and Science of the Republic of Serbia.

In [1] the lower bound is improved for various classes of manifolds, such as projective spaces (both real and complex), products of projective spaces, Grassmannians, etc. Stiefel–Whitney classes are obstructions to totally skew embeddings as shown in [1, Proposition 1.] and [1, Corollary 4.]

Theorem 1.2. If k := max\{i | \overline{w}_{i}(M) ≠ 0\}, then N(M) ⩾ 2n + 2k + 1.

In the same paper a conjecture [1, Conjecture 20] was formulated predicting that N(M^{n}) ⩽ 4n − 2α(n) + 1, for compact smooth manifold M^{n} (n > 1), where α(n) is the number of non-zero digits in the binary representation of n. Cohen [5] in 1985 resolved positively the famous Immersion Conjecture, by showing that each compact smooth n-manifold for n > 1 can be immersed in ℝ^{2n−α(n)} .

Various types of immersions and embeddings are an interesting research topic. In [12, 13, 14] some more general conditions with multiple regularity are studied.

In the last decades a lot has been written about toric actions and quasitoric manifolds due to their wide applications in combinatorics, physics, topology, geometry, etc.

Quasitoric manifolds are a class of manifolds with a well understood cohomology ring which is determined by the Davis–Januszkiewicz formula [7, Theorem 4.14, Corollary 6.8]. Other topological invariants can be computed from the formula, and we are particulary interested in the Stiefel–Whitney classes. A nice exposition of the theory of quasitoric manifolds, including a review of their topological and combinatorial properties, can be found in the monograph [3] of Buchstaber and Panov.

The construction of a quasitoric manifold from the characteristic pair (P^{n}, l) is described in [3, Construction 5.12]. Recall that P^{n} is a simple polytope with m facets and Λ = (λ_{1}, …, λ_{m}) an integer n × m matrix, where λ_{j} ∈ ℤ^{n} j = 1, …, m corresponds to the generator of the Lie algebra isotropy subgroup of the characteristic submanifold M_{j} over the facet F_{j}. For every vertex v = F_{i_{1}} ∩ ⋯∩ F_{i_{n}} ∈ P^{n} the matrix has the property detΛ_{I_{(v)}} = ±1 where Λ_{I_{(v)}} is a square submatrix formed by the column vectors λ_{i_{1}} , …, λ_{i_{n}} corresponding to the facets F_{i_{1}} , …, F_{i_{n}} . The matrix Λ is called the characteristic matrix of M.

Let λ_{j} = (λ_{1j} , …, λ_{nj} )^{t} ∈ ℤ^{n}. Then we have θ_{i} := ∑ _{j=1}^{m} λ_{ij} v_{j} and let J be the ideal in ℤ[v_{1}, …, v_{m}] generated by θ_{i} for all i = 1, …, n. Let I denote the Stanley-Reisner ideal of P. The ordinary cohomology of quasitoric manifolds has the following ring structure:

H^{∗}(M) ≃ ℤ[v_{1}, …, v_{m}]/(I + J).

The total Stiefel–Whitney class can be described by the following Davis–Januszkiewicz formula:

w(M) = ∏_{i=1}^{m} (1 + v_{i}) ∈ H^{∗}(M; ℤ_{2}),

where v_{i} is the ℤ_{2}-reduction of the corresponding class over ℤ coefficients. The Stiefel–Whitney classes are a powerful tool for studying problems of toric topology such as cohomological rigidity [6].

In Section 2 one special quasitoric manifold M_{I} over the cube I^{n} is constructed by matrix Λ_{M_{I}} . The cohomology ring and the total Stiefel–Whitney class of this manifold are described.

Section 3 is devoted to the calculation of the total Stiefel–Whitney class of the stable normal bundle using careful manipulations of binomial coefficients in the cohomology ring (with ℤ_{2} coefficients). The obstruction to immersion, embedding and totally skew embedding of the manifold M_{I} is calculated and the main result of the paper is obtained.

2. Quasitoric manifold over the cube

2.1. Matrix Λ_{M_{I}} and the cube. A quasitoric manifold M is described by two key objects: its orbit polytope P and characteristic matrix Λ. Two quasitoric manifolds over the same polytope, but with distinct characteristic matrices may be different, in general, due to nonisomorphic cohomology rings. Although, the polytope P with its combinatorics yields a lot of information about the manifold itself, the characteristic matrix Λ is essential to understand important topological invariants of the quasitoric manifold.

Let I^{n} be a cube and M_{I^{n}} a quasitoric manifold over I^{n}. The cube has 2n facets F_{1}, …, F_{n}, F′_{1}, …, F′_{n} such that F_{i} ∩ F′_{i} = ∅ for every i = 1, …, n. Let v_{1}, …, v_{n}, u_{1}, …, u_{n} be Poincaré duals to the characteristic submanifolds over the facets F_{1}, …, F_{n}, F′_{1}, …, F′_{n} respectively. The Stanley–Reisner ideal is generated by I = {v_{1}u_{1}, v_{2}u_{2}, …, v_{n}u_{n}}.

A special quasitoric manifold M_{I^{n}} over the cube is studied, such that the vector λ_{i} assigned to the facet F_{i} (or the generators of the Lie algebra isotropy subgroup of the characteristic submanifold M_{i}) is λ_{i} = (\underbrace{0, …, 0,}_{i−1}1, \underbrace{0, …, 0}_{n−i})^{t} for every i =

1, …, n and vector λ_{i+n} assigned to the facet F′_{i} is λ_{n+i} = (\underbrace{0, …, 0}_{i−1}, \underbrace{1, …, 1}_{n−i+1})^{t} for

every i = 1, …, n. Then we have:

Λ_{M_{I^{n}}} = [ \begin{matrix}1 & 0 & … & 0 & 1 & 0 & … & 0 \\ 0 & 1 & … & 0 & 1 & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 & 1 & 1 & … & 1 \end{matrix} ] .

The matrix Λ_{M_{I^{n}}} has the property that det(Λ_{M_{I^{n}}} )_{(v)} = 1 for every vertex v of I^{n} (all entries above the main diagonal are 0 while all entries lying on the main diagonal are 1).

The ideal J in ℤ[v_{1}, …, v_{n}, u_{1}, …, u_{n}] is generated by linear forms

v_{1} + u_{1},
v_{2} + u_{1} + u_{2},
…
v_{n} + u_{1} + u_{2} + ⋯+ u_{n}.

2.2. Cohomology ring H^{∗}(M_{I^{n}} ) and the total Stiefel–Whitney class w(M_{I^{n}} ). The cohomology ring H^{∗}(M_{I^{n}} ) is determined using the Davis–Januszkiewicz theorem:

Proposition 2.1. The cohomology ring H^{∗}(M_{I^{n}} ; ℤ) is isomorphic to

H^{∗}(M_{I^{n}} ; ℤ) ≃ ℤ[u_{1}, …, u_{n}]/F_{n}

where F_{n} is an ideal in the polynomial ring ℤ[u_{1}, …, u_{n}] (such that deg(u_{1}) = ⋯= deg(u_{n}) = 2) generated by quadratic forms

u_{1}^{2}, u_{2}^{2} + u_{1}u_{2}, …, u_{n}^{2} + u_{1}u_{n} + u_{2}u_{n} + ⋯+ u_{n−1} u_{n}.

Reducing modulo 2 we obtain that H^{∗}(M_{I^{n}} ; ℤ_{2}) ≃ ℤ_{2}[u_{1}, …, u_{n}]/F_{n} where F_{n} is an ideal in the polynomial ring ℤ_{2}[u_{1}, …, u_{n}] (such that deg(u_{1}) = ⋯= deg(u_{n}) = 2) generated by quadratic forms

F_{n} = {u_{1}^{2}, u_{2}^{2} + u_{1}u_{2}, …, u_{n}^{2} + u_{1}u_{n} + u_{2}u_{n} + ⋯+ u_{n−1} u_{n}}.

It is easy to show the following relations in H^{∗}(M_{I^{n}} ; ℤ_{2}):

Proposition 2.2. For every i = 2, …, n the following equality holds

(1 + u_{i})(1 + v_{i}) = 1 + u_{1} + ⋯+ u_{i−1} = 1 + v_{i−1} .

The total Stiefel–Whitney class is the characteristic class in cohomology with ℤ_{2} coefficients. By Davis–Januszkiewicz’s formula, the total Stiefel–Whitney class of M_{I^{n}} is given by w(M_{I^{n}} ) = (1 + u_{1}) ⋯ (1 + u_{n})(1 + v_{1}) ⋯ (1 + v_{n}), but according to Propositions 2.1 and 2.2 it easily reduces to

w(M_{I}) = (1 + u_{1})(1 + u_{1} + u_{2}) ⋯ (1 + u_{1} + ⋯+ u_{n−1} ).

For the purposes of the main theorem, we are going to use another form of the cohomology ring H^{∗}(M_{I^{n}} ; ℤ_{2}), with generators v_{1}, …, v_{n}. Recall that

v_{1} = u_{1},
v_{2} = u_{1} + u_{2},
…
v_{n} = u_{1} + u_{2} + ⋯+ u_{n},

so, we get that H^{∗}(M_{I^{n}} ; ℤ_{2}) ≃ ℤ_{2}[v_{1}, …, v_{n}]/G_{n} where G_{n} is an ideal in the polynomial ring ℤ_{2}[v_{1}, …, v_{n}] (such that deg(v_{1}) = ⋯= deg(v_{n}) = 2) generated by quadratic forms v_{1}^{2}, v_{2}^{2} + v_{1}v_{2}, …, v_{n}^{2} + v_{n−1} v_{n}. Consequently, the total Stiefel– Whitney class is given by w(M_{I}) = (1 + v_{1}) ⋯ (1 + v_{n−1} ).

In the following proposition we begin the analysis of the cohomology ring H^{∗}(M_{I^{n}} ; ℤ_{2}).

Proposition 2.3. For every i = 1, …, n the following equalities hold

v_{i}^{i} = v_{1}v_{2} ⋯ v_{i} ≠ 0 and v_{i}^{i+1} = 0.

Proof. We easily deduce

v_{i}^{i+1} = v_{i}^{i}v_{i−1} = ⋯= v_{i}v_{i−1}^{i} = v_{i}v_{i−1}^{i−1} v_{i−2} = ⋯= v_{i} ⋯ v_{2}v_{1}^{2} = 0.

Similarly, v_{i}^{i} = v_{1}v_{2} ⋯ v_{i} for all i = 1, …, n.

To show the nontriviality of classes v_{i}^{i}, it is enough to show that v_{n}^{n} = v_{1} ⋯ v_{n} is nonzero. First, we prove the following lemma.

Lemma 2.1. Let i < j and a and b be nonnegative integers such that a ⩽ i and b ⩽ j. Then the class v_{i}^{a}v_{j}^{b} is trivial or equal to the product of some a + b distinct generators v_{k}.

Proof. It is easy to get that v_{i}^{a}v_{j}^{b} = v_{i−a+1} ⋯ v_{i}v_{j−b+1} ⋯ v_{b}. If i ⩽ j − b the proof is completed. Otherwise, i = j − k for some positive integer k ⩽ b − 1 resulting in

v_{i}^{a}v_{j}^{b} = v_{j−(a+k)+1} · · · ·· v_{j−k} · v_{j−b+1} · · · ·· v_{b}
= v_{j−(a+k−1)} ⋯ v_{j−b} v_{j−b+1}^{2} ⋯ v_{j−k}^{2} v_{j−k+1} ⋯ v_{j} .

Now we continue to remove squares and powers from the expression above using the equalities v_{m}^{2} = v_{m}v_{m−1} . Since v_{1}^{2} = 0, if j ⩾ a + b then

v_{i}^{a}v_{j}^{b} = v_{j−(a+b)+1} ⋯ v_{j} ,

while in the other case v_{i}^{a}v_{j}^{b} = 0. □

An immediate consequence of the previous lemma is:

Corollary 2.1. Every class of type v_{i_{1}}^{r_{1}} ⋯ v_{i_{k}}^{r_{k}} is either trivial or equal to the product of some r_{1} + ⋯+ r_{k} distinct generators. Moreover, this class is non-zero if and only if for each p = 1, 2, …, n r_{1} + ⋯+ r_{p} ⩽ p.

From the general manifold theory it is known that H^{2n} (M_{I^{n}} ; ℤ_{2}) = ℤ_{2}. Thus, according to the previous observations, the generator of the highest cohomology group must be the class v_{1}v_{2} ⋯ v_{n} and the proposition is therefore proved. □

The following proposition, referred to as the ‘cancellation lemma’, summarizes most of the properties of the cohomology ring H^{∗}(M_{I} , ℤ_{2}) that will be needed in Section 3. Here and later on we use the multi-index power v^{α} to denote the monomial v^{α} = v_{i_{1}} v_{i_{2}} ⋯ v_{i_{k}} of degree |α| = k where α = {i_{1}, i_{2}, …, i_{k}} is a subset of [n] = {1, 2, …, n}. By convention v^{α} = 1 if α = ∅ and we always assume that i_{1} < i_{2} < ⋯< i_{k}.

Proposition 2.4 (Cancellation Lemma). The collection of monomials v^{α} = v_{i_{1}} v_{i_{2}} ⋯ v_{i_{k}} , where α ⊂ [n], is a graded ℤ_{2}-vector space basis of the graded vector space H^{∗}(M_{I} , ℤ_{2}). Moreover, if β ⊂ [n] then v^{β}v_{n}^{p} ≠ 0 if and only if |β| + p ⩽ n.

Proof. We already know from the proof of Proposition2.3, that the collection B = {v^{α} }_{α⊂[n]} is a spanning set for the ℤ_{2}-vector space H^{∗}(M_{I} , ℤ_{2}). As a consequence dim(H^{∗}(M_{I} , ℤ_{2})) ⩽ 2^{n}. By [3, Proposition 5.16],

dim(H^{∗}(M_{I} , ℤ_{2})) = h_{0} + h_{1} + ⋯+ h_{n}

where h = (h_{0}, …, h_{n}) is the h-vector of the associated polytope. Recall that the sum of all h_{j} is always equal to the number of vertices of the associated simple polytope P. In particular if P = I^{n} is the n-dimensional cube, we obtain that dim(H^{∗}(M_{I} , ℤ_{2})) = 2^{n} which completes the proof of the first half of the proposition. The second half is an easy consequence which can be proved by induction on p. □

3. Topological obstructions to immersions and embeddings of the manifold M_{I}

3.1. Stiefel–Whitney class \overline{w}(M_{I}) of the stable normal bundle. For the proof of the main theorem of this paper, we are interested in characteristic classes \overline{w}(M_{I}) of the stable normal bundle of M_{I}. The Stiefel–Whitney classes w(M_{I}) and \overline{w}(M_{I}) are related to each other by the following equality

w(M_{I}) · \overline{w}(M_{I}) = 1.

In the previous section the total Stiefel–Whitney class w(M_{I^{n}} ) is determined. So, by Proposition2.3, the following holds:

Lemma 3.1. The total Stiefel–Whitney class \overline{w}(M_{I^{n}} ) of the stable normal bundle is given by

\overline{w}(M_{I^{n}} ) = (1 + v_{1})(1 + v_{2} + v_{2}^{2}) ⋯ (1 + v_{n−1} + ⋯+ v_{n−1}^{n−1} ).

Since \overline{w}_{2i}(M_{I^{n}} ) = 0 when i ⩾ n, it is not evident what \overline{w}(M_{I^{n}} ) is in the cohomology ring H^{∗}(M_{I^{n}} ; ℤ_{2}). For small n, we could calculate \overline{w}(M_{I^{n}} ) by hand:

Exercise 3.1. (1) \overline{w}(M_{I^{2}} ) = 1 + v_{1},

(2) \overline{w}(M_{I^{3}} ) = 1 + (v_{1} + v_{2}),

(3) \overline{w}(M_{I^{4}} ) = 1 + (v_{1} + v_{2} + v_{3}) + v_{1}v_{3} + v_{1}v_{2}v_{3},

(4) \overline{w}(M_{I^{5}} ) = 1+(v_{1} +v_{2} +v_{3} +v_{4})+(v_{1}v_{3} +v_{1}v_{4} +v_{2}v_{4})+(v_{1}v_{2}v_{3} +v_{2}v_{3}v_{4}).

The top class \overline{w}_{2n} is always zero by the theorem of Massey [9].

By Lemma 3.1 for the total Stiefel–Whitney classes of \overline{w}(M_{I^{n}} ) and \overline{w}(M_{I^{n+1}} ) the following recurrence relation holds (in H^{∗}(M_{I^{n+1}} ; ℤ_{2})):

\overline{w}(M_{I^{n+1}} ) = \overline{w}(M_{I^{n}} )(1 + v_{n} + ⋯+ v_{n}^{n}),

or more explicitly

(3.1) \overline{w}_{2k}(M_{I^{n+1}} ) = \overline{w}_{2k}(M_{I^{n}} )+v_{n} \overline{w}_{2k−2} (M_{I^{n}} )+⋯ +v_{n}^{k} for all k = 0, …, n−1

and

(3.2)  \overline{w}_{2n} (M_{I^{n+1}} ) = v_{n} \overline{w}_{2n−2} (M_{I^{n}} ) + ⋯+ v_{n}^{n}.

Here we use the fact that there is a natural homomorphism i : H^{∗}(M_{I^{n}} ; ℤ_{2}) → H^{∗}(M_{I^{n+1}} ; ℤ_{2}) which allow us to move all classes to the latter group.

By the cancellation lemma (Proposition 2.4) and working modulo 2, \overline{w}_{2k} is the sum of a certain number of linearly independent square-free monomials. We consider the polynomial \overline{W}_{2k}(v_{1}, …, v_{n}) in the ring ℤ_{2}[v_{1}, …, v_{n}] of degree 2k, obtained after applying all possible cancellations in Lemma 3.1. Define the numbers σ_{n}^{k} for all positive integers n and 0 ⩽ k ⩽ n − 1 as follows

σ_{n}^{k} = \overline{W}_{2k}(1, …, 1) (mod 2)

So by (3.1) and (3.2), we have σ_{n+1}^{k} = ∑ _{i=0}^{k} σ_{n}^{i} for every k = 1, …, n − 1 and σ_{n+1}^{n} = σ_{n+1}^{n−1} . Here we tacitly used the second half of Proposition 2.4.

Let us write the first n rows of numbers σ_{n}^{k} for k = 0, …, n:

1

1 1
1 0 0
1 1 1 1
1 0 1 0 0
1 1 0 0 0 0
1 0 0 0 0 0 0
1 1 1 1 1 1 1 1
...........................

The previous sequence is closely related to the following sequence of binomial coefficients \binom{n+k}{k}:

1

1 3
1 4  10
1 5  15  35
1 6  21  56  70
1 7 \binom{8}{2}  \binom{9}{3}  \binom{10}{4}  \binom{11}{5}
1 8 \binom{9}{2}  \binom{10}{3}  \binom{11}{4}  \binom{12}{5}  \binom{13}{6}
1 9 \binom{10}{2}  \binom{11}{3}  \binom{12}{4}  \binom{13}{5}  \binom{14}{6}  \binom{15}{7}
.....................................................

An easy mathematical induction shows that:

Lemma 3.2. We have σ_{n}^{k} ≡ \binom{n+k}{k} (mod 2).

By the previous Lemma, in the case when n = 2^{r} we have

σ_{n}^{n−1} ≡ \binom{2^{r} +(2^{r} −1)}{2^{r} −1} ≡ \binom{2^{r+1} −1}{2^{r} −1} ≡ 1 (mod 2).

Obviously, from the definition of σ_{n}^{k}, if σ_{n}^{k} = 1, then \overline{w}_{2k} is the sum of an odd number of linearly independent square-free monomials and \overline{w}_{2k}(M_{I^{n}} ) ≠ 0. Thus, we obtain:

Theorem 3.1. If n = 2^{r} is a power of two then

\overline{w}_{2n−2} (M_{I^{n}} ) = v_{1}v_{2} ⋯ v_{n−1} ≠ 0.

Hence, Theorem 1.1 yields:

Corollary 3.1. If n is a power of two then

imm(M_{I^{n}} ) ⩾ 4n − 2 and em(M_{I^{n}} ) ⩾ 4n − 1.

Since M_{I^{n}}^{2n} is orientable, it can be embedded into ℝ^{4n−1} . Thus,

Theorem 3.2. If n is a power of two, then em(M_{I^{n}} ) = 4n − 1.

Lemma 3.1 implies that \overline{w}_{2}(M_{I^{n}} ) = v_{1}+v_{2}+⋯ +v_{n−1} . Due to the cancellation lemma, when n is a power of two, the characteristic class \overline{w}_{2}(M_{I^{n}} )\overline{w}_{2n−2} (M_{I^{n}} ) is trivial. By the result of Massey [10, Theorem V], it follows:

Theorem 3.3. If n ⩾ 4 is a power of two, then imm(M_{I^{n}} ) = 4n − 2

For totally skew embeddings, from Theorem 3.1 the lower bound obtained is:

Corollary 3.2. If n is a power of two, then N(M_{I^{n}} ) ⩾ 8n − 3.

3.2. Topological obstructions when n is not a power of 2. Theorem 3.1 is the sharpest possible result that one can obtain using Stiefel–Whitney classes for quasitoric manifolds. However, when n is not a power of 2 the previously constructed quasitoric manifold M_{I^{n}} , in general, does not achieve the maximal possible value k for which the Stiefel–Whitney class \overline{w}_{2k}(M_{I^{n}} ) ≠ 0.

This problem could be overcome using the results from the previous part. Let n = 2^{r_{1}} +2^{r_{2}} +⋯ +2^{r_{t}} , r_{1} > r_{2} > ⋯> r_{t} ⩾ 0 be the binary representation

of n and let m_{i} = 2^{r_{i}} for i = 1, …, t and m_{0} = 0. In the previous section we described the quasitoric manifold M_{I_{j}} over the cube I^{m_{j}} . From the result of Buchstaber and Ray [4, Proposition 4.7], it follows that M_{I} = M_{I_{1}} × ⋯× M_{I_{t}} is a quasitoric manifold over the cube I^{n} = I_{1} × ⋯× I_{t}.

The total Stiefel–Whitney class of the tangent bundle of M_{I} can be easily determined using the following formula (see [11, pp. 27, 54]):

w(M_{I}) = w(M_{I_{1}} ) ⋯ w(M_{I_{t}} ) ∈ H^{∗}(M_{I}) ≅ H^{∗}(M_{I_{1}} ) ⊗ ⋯⊗ H^{∗}(M_{I_{t}} ).

Let v_{i}^{(j)} , for i = 1, …, m_{j}, be the generators of the cohomology ring H^{∗}(M_{I_{j}} , ℤ_{2}). The total Stiefel–Whitney class is given by

w(M_{I}) = ∏_{j=1}^{t} (1 + v_{1}^{(j)} ) ⋯ (1 + v_{m_{j} −1}^{(j)} ).

Thus, the corresponding dual Stiefel–Whitney class is given by

\overline{w}(M_{I}) = ∏_{j=1}^{t} (1 + v_{1}^{(j)} )( 1 + v_{2}^{(j)} + (v_{2}^{(j)} )^{2}) ⋯ ( 1 + v_{m_{j} −1}^{(j)} + ⋯+ (v_{m_{j} −1}^{(j)} )^{m_{j} −1} ) .

But, according to Theorem 3.1 we have:

\overline{w}(M_{I}) = ∏_{j=1}^{t} ( 1 + (v_{1}^{(j)} + ⋯+ v_{m_{j} −1}^{(j)} ) + ⋯+ v_{1}^{(j)} v_{2}^{(j)} ⋯ v_{m_{j} −1}^{(j)} ) .

So, the highest nontrivial dual Stiefel–Whitney class is

\overline{w}_{2n−2α(n)} (M_{I}) = v_{1}^{(1)} ⋯ v_{m_{1} −1}^{(1)} v_{1}^{(2)} ⋯ v_{m_{t}−1}^{(t)} ,

where α(n) is the number of non-zero digits in the binary representation of n. As corollary we obtain:

Theorem 3.4 (Main theorem). For every positive integer n there is a quasitoric manifold M_{I} over the cube such that

imm(M_{I}) ⩾ 4n − 2α(n),
em(M_{I}) ⩾ 4n − 2α(n) + 1,
N(M_{I}) ⩾ 8n − 4α(n) + 1.

Remark 3.1. No similar result can be obtained in the class of toric varieties from a cube because the total Stiefel–Whitney class is trivial in that case.

Acknowledgements. The author is grateful to the referees for recommending various suggestions for improvement in the article, careful reading and many useful comments.

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14. , Embeddings with certain non-degeneracy conditions, Ph.D. degree thesis, Pennsylvania State University, August 2007.

Matematički institut SANU (Received 19 04 2013) Beograd (Revised 12 08 2013) Serbia

   djbaralic@mi.sanu.ac.rs