Characteristic number associated to 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 $(\Delta ,\mathbf{b})$. 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 $\mathrm{Ham}(M_{\Delta })$ the Hamiltonian group of the symplectic manifold defined by $\Delta$, we determine loops in $\mathrm{Ham}(M_{\Delta })$ that define infinite cyclic subgroups in ${\pi }_1(\mathrm{Ham}(M_{\Delta }))$, 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 $\mathbb{C}{P}^n$.

Revised version which will appear in ISRN Geometry

References in corpus (1)

• A characteristic number of bundles determined by mass linear pairs