A characteristic number of bundles determined by mass linear pairs

Abstract

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 $(\Delta ,\mathbf{b})$. 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, part I.} {\tt arXiv:0807.0900 [math.SG]}) and the vanishing of a characteristic number of $E$ in the following cases: When $\Delta$ is a ${\Delta }_{n-1}$ bundle over ${\Delta }_1$; when $\Delta$ is the polytope associated with the one point blow up of $\mathbb{C}{P}^n$; and when $\Delta$ is the polytope associated with a Hirzebruch surface.

17 pages. Section 3 and Introduction have been rewritten

References in corpus (1)

• Polytopes with mass linear functions, part I

Cited by in corpus (2)

• Remarks on invariants of hamiltonian loops
• Characteristic number associated to mass linear pairs