Characteristic number associated to mass linear pairs

Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\mathbf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta,\,{\mathbf b})$. Under certain hypotheses, we prove the equivalence between the fact that $(\Delta,\,{\mathbf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions. I.} Int. Math. Res. Not. IMRN 8 (2010) 1506-1574.) and the vanishing of a characteristic number of $E$. Denoting by ${\rm Ham}(M_{\Delta})$ the Hamiltonian group of the symplectic manifold defined by $\Delta$, we determine loops in ${\rm Ham}(M_{\Delta})$ that define infinite cyclic subgroups in $\pi_1({\rm Ham}(M_{\Delta}))$, when $\Delta$ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii) it is a $\Delta_p$ bundle over $\Delta_1$, (iii) $\Delta$ is the truncated simplex associated with the one point blow up of ${\mathbb C}P^n$.


Introduction
Let (N, Ω) be a closed connected symplectic 2n-manifold. By Ham(N, Ω), we denote the Hamiltonian group of (N, Ω) [6,8]. Associated with a loop ψ in Ham(N, Ω), there exist characteristic numbers which are invariant under deformation of ψ. These invariants are defined in terms of characteristic classes of fibre bundles and their explicit values are not easy to calculate, in general. Here, we will consider a particular invariant I, whose definition we will be recall below. By proving the non-vanishing of I for certain loops, we will deduce the existence of infinity cyclic subgroups of π 1 (Ham(N, Ω)), when N is a toric manifold. The vanishing of the invariant I on particular loops in Ham(N, Ω) is related with the concept of mass linear pair, which has been developed in [7]. In this introduction, we will state the main results of the paper and will give a schematic exposition of the concepts involved in these statements.
A loop ψ in Ham(N, Ω) determines a Hamiltonian fibre bundle E → S 2 with standard fibre N , via the clutching construction. Various characteristic numbers for the fibre bundle E have been defined in [4]. These numbers give rise to topological invariants of the loop ψ. In this article, we will consider only the following characteristic number is an R-valued group homomorphism [4].
Our purpose is to study this characteristic number when N is a toric manifold and ψ is a 1-parameter subgroup of Ham(N ) defined by the toric action. The referred 1-parameter subgroup is determined by an element b in the integer lattice of the Lie algebra of the corresponding torus. On the other hand, a toric symplectic manifold is determined by its moment polytope. For a general polytope, a mass linear function on it is a linear function "whose value on the center of mass of the polytope depends linearly on the positions of the supporting hyperplanes" [7]. In this article, we will relate the vanishing of the number I(ψ) with the fact that b defines a mass linear function on the polytope associated with the toric manifold. In the following paragraphs, we provide a more detailed exposition of this relation.
Let T be the torus (U (1)) n and ∆ = ∆(n, k) the polytope in t * with m facets defined by where k j ∈ R and the n j ∈ t are the outward conormals to the facets. The facet defined by the equation x, n j = k j will be denoted F j , and we put Cm(∆) for the mass center of the polytope ∆. In [7] the chamber C ∆ of ∆ is defined as the set of k ′ ∈ R m such that the polytope ∆ ′ := ∆(n, k ′ ) is analogous to ∆; that is, the intersection ∩ j∈J F j is nonempty iff ∩ j∈J F ′ j = ∅ for any J ⊂ {1, . . . , m}. When we consider only polytopes which belong to the chamber of a fixed polytope we delete the n in the notation introduced in (1.3).
Further, McDuff and Tolman introduced the concept of mass linear pair: Given the polytope ∆ and b ∈ t, the pair (∆, b) is mass linear if the map where R j and C are constant. Let us assume that ∆ is a Delzant polytope [1]. We shall denote by (M ∆ , ω ∆ , µ ∆ ) the toric manifold determined by ∆ (µ ∆ : M → t * being the corresponding moment map). Given b, an element in the integer lattice of t, we shall write ψ b for the loop of Hamiltonian diffeomorphisms of (M ∆ , ω ∆ ) defined by b through the toric action. We will let I(∆; b) for the characteristic number I(ψ b ). When we consider only polytopes in the chamber of a given polytope, we will write I(k; b) instead of I(∆(k); b) for k in this chamber.
The group G of the translations defined by the elements of t * acts freely on C ∆ . We put r := m−n for the dimension of the quotient C ∆ /G. Thus, r is the number of effective parameters which characterize the polytopes in C ∆ considered as "physical bodies".
We will prove the following theorem: be a pair consisting of a Delzant polytope in t * and an element in the integer lattice of t. If r ≤ 2, the following statements are equivalent (a) is a mass linear pair as in (1.5), with j R j = 0.
In [12], by direct computation, we proved the equivalence between the vanishing of I(k; b) on C ∆ and the fact that (∆, b) is a mass linear pair, when ∆ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii) it is a ∆ p bundle over ∆ 1 [7], (iii) ∆ is the truncated simplex associated with the one point blow up of CP n . On the other hand, when ∆ is any of these polytopes (i)-(iii), the number r is equal to 2; thus, from Theorem 1 and the result of [12], it follows that condition j R j = 0 is satisfied by all the mass linear pairs (∆, b). This fact can also be proved by direct calculation (Propositions 14, 18 and 21). So, Theorem 1, together with these Propositions, generalize the result proved in [12].
Although the homotopy type of the Hamiltonian groups Ham(N, Ω) is known only for some symplectic manifolds [5], the invariant I allows us to identify nontrivial elements in π 1 (Ham(N, Ω)). As I is a group homomorphism, from Theorem 1, we deduce that a sufficient condition for ψ b to generate an infinite cyclic subgroup in π 1 (Ham(M ∆ , ω ∆ )) is that the above condition (b) does not hold for (∆, b). More precisely, we have the following consequence of Theorem 1: Theorem 2. Given the Delzant polytope ∆ and b an element in the integer lattice of t. If r ≤ 2 and (∆, b) is not mass linear, then ψ b generates an infinite cyclic subgroup in π 1 (Ham(M ∆(k) , ω ∆(k) )), for all k ∈ C ∆ .
In the proof of Theorem 1, a formula for the characteristic number I(ψ b ) obtained in [11] plays a crucial role. This formula gives I(ψ b ) in terms of the integrals, on the facets of the polytope, of the normalized Hamiltonian function corresponding to the loop ψ b (see (2.9)). From this expression for I(ψ b ), we will deduce a relation between the directional derivative of map (1.4) along the vector (1, . . . , 1) of R m , the Euclidean volume of ∆(k) and I(k; b) (see (3.1)). From this relation, it is easy to complete the proof of Theorem 1.
This article is organized as follows: In Section 2, we study the characteristic number I(k; b), when (∆, b) is a linear pair and k varies in the chamber of ∆; we prove that I(k; b) is a homogeneous polynomial of the k j (Proposition 6).
In Section 3, we prove Theorem 1. In Proposition 11, a sufficient geometric condition for the Delzant polytope ∆ to admit a mass linear pair (∆, b) is given. For a Delzant polytope ∆, Proposition 12 gives a necessary condition for the vanishing of I(k; b) on C ∆ . We also express j R j in terms of the displacement of the center of mass Cm(∆(k)) produced by the change k j → k j + 1 (Proposition 13). Section 4 concerns the form which Theorem 2 adopts, when ∆ is a Delzant polytope of the particular types (i)-(iii) mentioned above (see Corollary 15, Theorems 17 and 20). We also prove that, in these particular cases, if (∆, b) is a mass linear pair, then j R j = 0.

A characteristic number
Let us suppose that the polytope ∆ defined in (1.3) is a Delzant polytope in t * . Following [2], we recall some points of the construction of (M ∆ , ω ∆ , µ ∆ ) from the polytope ∆. We putT := (S 1 ) m−n . The n i determine weights w j ∈t * , j = 1, . . . , m for aT -action on C m . Then moment map for this action is The k i define a regular value σ for J, and the manifold M ∆ is the following orbit space where the relation defined byT is iff there is y ∈t such that z ′ j = z j e 2πi wj ,y for j = 1, . . . , m. Identifyingt * with R r , σ = (σ 1 , . . . , σ r ) and each σ a is a linear combination of the k j .
Given a facet F of ∆, we choose a vertex p of F . After a possible change in numeration of the facets, we can assume that F 1 , . . . , F n intersect at p. In this numeration F = F j , for some j ∈ {1, . . . , n}.
If we write z a = ρ a e iθa , then the symplectic form can be written on {[z] ∈ M : z a = 0, ∀a} The facet F = F j of ∆ is the image by µ ∆ of the submanifold We write x i := πρ 2 i , then Let b be an element in the integer lattice of t. The normalized Hamiltonian of the circle action generated by b is the function f determined by, Moreover, An expression for the value of the invariant I(ψ b ) in terms of integrals of the Hamiltonian function has been obtained in Section 4 of [11] (see also [10] and [9]) where the contribution N (F ) of the above facet F = F j (with j = 1, . . . , n) is Given ∆ = ∆(n, k), we consider the polytope ∆ ′ = ∆(n, k ′ ) obtained from ∆ by the translation defined by a vector a of t * . As we said, we write I(k; b) and I(k ′ ; b) for the corresponding characteristic numbers. According to the construction of the respective toric manifolds, But the normalized Hamiltonians f and f ′ corresponding to the action of b on M ∆ and M ∆ ′ are equal. Thus, it follows from (2.9) that I(k; b) = I(k ′ ; b). More precisely, we have the evident proposition: Proposition 3. If a is an arbitrary vector of t * , then I(k; b) = I(k ′ ; b), for k ′ j = k j + a, n j , j = 1, . . . , m. By Proposition 3, we can assume that all d j in (2.4) are zero for the determination of I(k; b).
The following lemma is elementary: Thus, in the particular case that ∆ = S n (c, τ ), the integral M∆ (ω ∆ ) n is a monomial of degree n in τ , and M∆ µ ∆ , b (ω ∆ ) n is a monomial of degree n + 1.
We return to the general case in which ∆ is the polytope defined in (1.3). Its vertices are the solutions to (2.12) x, n ja = k ja , a = 1, . . . , n; hence, the coordinates of any vertex of ∆ are linear combinations of the k j . A hyperplane in R n through a vertex (x 0 1 , . . . , x 0 n ) of ∆ is given by an equation of the form (2.13) x, n = x 0 , n =: κ.
Thus, the independent term κ is a linear combination (l. c.) of the k j . Moreover, the coordinates of the common point of n hyperplanes with κ i l. c. of the k j are also l. c. of the k j . By drawing hyperplanes through vertices of ∆ (or more generally, through points which are the intersection of n hyperplanes as (2.14)), we can obtain a family { β S} of subsets of ∆ such that: a) Each β S is the transformed of a simplex S n (b, τ ) by an element of the group of Euclidean motions in R n . b) For α = β, α S ∩ β S is a subset of the border of α S. c) β β S = ∆. Thus, by construction, each facet of β S is contained in a hyperplane π of the form x, n = κ, with κ l. c. of the k j .
On the other hand, the hyperplane π is transformed by an element of SO(n) in an hyperplane x, n ′ = κ. If T is a translation in R n which applies S n (b, τ ) onto β S, then this transformation maps (0, . . . , 0) in a vertex a = (a 1 , . . . , a n ) of β S. So, the translation T transforms π in x, n = κ + a, n =: κ ′ . As each a j is a l. c. of the k j , so is κ ′ . Hence, any element of the group of Euclidean motions in R n which maps S n (b, τ ) onto β S transforms the hyperplane π Let assume that (RT a )(S(b, τ )) = β S, with R ∈ SO(n) and T a the translation defined by a. Then the oblique facet of S(b, τ ), contained in the hyperplane b i x i = τ , is the image by T −a R −1 of a facet of β S, which in turn is contained in a hyperplane of equation (2.15) (κ ′ being a l. c. of the k j ). The argument of the preceding paragraph applied to R −1 and T −a proves that τ is a l. c. of the k j . Hence, by (2.11) the integral is a monomial of degree n of a l. c. of the k j . Thus, is a homogeneous polynomial of degree n of the k j . Similarly, is a homogeneous polynomial of degree n + 1 of the k j . Analogous results hold for From formulas (2.6)-(2.10) together with the preceding argument, it follows the following proposition: Analogously, we have We will use the following simple lemma in the proof of Theorem 1.
Proof. The vertices of ∆(n, k) are the solutions of (2.12) and the vertices of ∆(n,k) are the solutions of x, n ja = sk ja , with a = 1, . . . , n. Thus, the vertices of ∆(n,k) are those of ∆(n, k) multiplied by s.
The Lemma also follows from the fact that (2.16) and (2.17) are homogeneous polynomials of degree n and n + 1, respectively.

Proof of Theorem 1
Let us assume that the polytope ∆ defined by (1.3) is Delzant and let k be an element of C ∆ . We denote by M (k) , ω (k) and µ (k) , the manifold, the symplectic structure and the moment map (resp.) determined by ∆(k). The facets of ∆(k) will be denoted by by F (k)j .
Let b be an element in the integer lattice of t. We put By (2.6), 1 n! B (k) is the Euclidean volume of the polytope ∆(k). Given a facet F (k)j , we can assume that j ∈ {1, . . . , n} (see third paragraph of Section 2). So, F (k)j is defined by the equation x j = 0. If we make an infinitesimal variation of the facet F (k)j , by means of the translation defined by k j → k j + ǫ (keeping unchanged the other k i ), then the volume of ∆(k) changes according to We write dX j for dx 1 . . .d x j . . . dx n . Thus, So, by (2.7), From (2.9) and (2.10), it follows Thus, we have proved the following proposition: Next, we will parametrize the quotient C ∆ /G (of classes of polytopes in C ∆ module translation) defined in the Introduction.
After a possible renumbering, we may assume that the intersection of facets F 1 , . . . , F n is a vertex of ∆. Thus, the conormals n 1 , . . . , n n are linearly independent in t. So, given k ∈ C ∆ , there is a unique v ∈ t * , such that, (Expressing the n i in terms of a basis of t and v in the dual basis, (3.2) is a compatible and determined system of linear equations for the coordinates of v.) Moreover v = v(k) depends linearly of the k i ; that is, v(k), c is a linear function of k 1 , . . . , k n , for all c ∈ t.
If m − n = 2, we write where v(k) the element in t * defined by (3.2). From the linearity of v with respect to the k i , it follows that λ and τ are linear combinations of k 1 , . . . , k m . The polytope in C ∆ defined by (k ′ 1 = 0, . . . , k ′ n = 0, λ, τ ) will be denoted by ∆ 0 (λ, τ ). It is the result of the translation of ∆(k) by the vector v(k); i. e., Let b an element in the integer lattice of t, we define the function g by The function g is defined on the pairs (λ, τ ) such that (0, . . . , 0, λ, τ ) ∈ C ∆ . By Lemma 7, it follows g(sλ, sτ ) = sg(λ, τ ), for any real number s such that (sλ, sτ ) belongs to the domain of g. This property implies that (3.4) g = λ ∂g ∂λ + τ ∂g ∂τ .
Theorem 9. If I(k; b) = 0, for all k ∈ C ∆ and r = 2, then Cm(∆(k), b = j R j k j , with R j constant (that is, (∆, b) is a mass linear pair) and j R j = 0.
Proof. We set f (k 1 , . . . , k m ) := Cm(∆(k), b . It follows from (3.3) that By the hypothesis and Proposition 8, where p, q, t stand for the following constants Since qλ − pτ and tτ − qg are first integrals of (3.7), the general solution of this equation is where Φ is a derivable function of one variable. It follows from (3.4) and (3.8) that (3.9) Φ(u) = uΦ ′ (u).
We have for f In other words, f is a linear function of the k j ; i.e., f (k) = j R j k j , with R j constant. From (3.6), it follows j R j = 0.
On the other hand, the proof of this theorem does not admit an adaptation to the case r > 2. In fact, the corresponding function Φ would be a function of r − 1 variables Φ(u 1 , . . . , u r−1 ). The equation which corresponds to (3.9) in this case would be But this condition does not implies the linearity of Φ.
When (∆, b) is a mass linear pair as in (1.5), by (3.1) for all k ∈ C ∆ . From (3.10), we deduce the following proposition: Proof of Theorem 1. It is a direct consequence of Proposition 10, Theorem 9 and the Remark above.
Proposition 11. If all pointsĊ m(∆(k)), for k ∈ C ∆ , belong to a hyperplane of (R n ) * with a conormal vector in Z n and r ≤ 2, then ∆ admits a mass linear function.
Taking the limit as k → 0, Next, we will describe a geometric interpretation of the number j R j . Given an arbitrary Delzant polytope ∆. If a is a vector of t * , then (3.11) Cm(∆(k ′ )) = Cm(∆(k)) + a, if k ′ j = k j + a, n j . We will denote by d the element of t * defined by the following relation (3.12) Cm(∆(k)) = Cm(∆(k)) + d, withk j = k j + 1 for all j.
Now, we assume that (∆, b) is a mass linear pair. From (1.5), it follows These formulas allow us to state the following proposition, that gives an interpretation of the sum j R j in terms of the variation of Cm(∆(k)) with the k j .
Proposition 13. Let (∆, b) be a mass linear pair as in (1.5). Then, d being the element of t * defined by (3.12).

Examples
In this Section, we will deduce the particular form which adopts Theorem 2, when ∆ is a polytope of the types (i)-(iii) mentioned in the Introduction. For each case, we will determine the center of mass of the corresponding polytope ∆(k) and the condition for (∆, b) to be a mass linear pair. We will dedicate a subsection to each type.
That is, N is the toric manifold M ∆ determined by the trapezium ∆.

Remark.
We denote by φ t the following isotopy of M ∆ φ is a loop in the Hamiltonian group of M ∆ . By φ ′ we denote the Hamiltonian loop In Theorem 8 of [10] we proved that I(φ ′ ) = (−2/r)I(φ). If b = (b 1 , b 2 ) ∈ Z 2 , then That is, I(ψ b ) = 0 iff rb 1 = 2b 2 , which is in agreement with Proposition 14 and Theorem 1.
where a ·b = p j=1 a j b j and B = p j=1 b j . By Proposition 12, we have: Theorem 17. Let ∆ be the ∆ p bundle over ∆ 1 defined by (4.8) and (4.9).
It is straightforward to check that is also a sufficient condition for (∆, b) to be a mass linear pair. Since if (4.19) holds, using (4.11) and (4.12), one obtains By (4.18), for k ∈ C ∆ , with λ and τ given by (4.17). If b =b, the condition (4.19) reduces to (p + 1)a ·b = AB and Cm(∆(k)), b = (a ·b + AB) Hence, Cm(∆(k)), b = R j k j , where A similar calculation for the case b =ḃ shows that the corresponding j R j vanishes. That is, Proposition 18. Let ∆ be a ∆ p bundle over ∆ 1 . If (∆, b) is a mass linear pair, then j R j = 0.
For p = 2, let b be the following linear combination of the conormal vectors b = γ 1 n 1 + γ 2 n 2 + γ 3 n 3 with γ 1 + γ 2 + γ 3 = 0. By (4.8) In this case condition (4.19) reduces to Or in terms of the γ i (4.20) This is a necessary and sufficient condition for (∆, b) to be mass linear. This result is the statement of Lemma 4.8 in [7].
If (∆, b) is a mass linear pair, by (4.23) and Proposition 19, we have Cm(∆ 0 (λ, τ )), b = b n τ. Thus, where R j = b n − b j , for j = 1, . . . , n and R n+1 = b n . Hence, we have the following proposition: Proposition 21. Let ∆ be the polytope obtained by truncating the standard nsimplex S n (τ ) by a horizontal hyperplane. If (∆, b) es a mass linear pair, then j R j = 0.