A Characteristic Number of Bundles Determined by Mass Linear Pairs

Abstract. Let ∆ be a Delzant polytope in Rn and b ∈ Zn. Let E denote the symplectic fibration over S determined by the pair (∆, b). We prove the equivalence between the fact that (∆, b) is a mass linear pair (D. McDuff, S. Tolman, Polytopes with mass linear functions, part I. arXiv:0807.0900 [math.SG]) and the vanishing of a characteristic number of E in the following cases: When ∆ is a ∆n−1 bundle over ∆1; when ∆ is the polytope associated with the one point blow up of CP n; and when ∆ is the polytope associated with a Hirzebruch surface.