Characteristic number associated to mass linear pairs

TL;DR

This paper links the concept of mass linear pairs in Delzant polytopes to characteristic numbers of associated symplectic fibrations, providing criteria for identifying infinite cyclic subgroups in Hamiltonian groups.

Contribution

It establishes a new equivalence between mass linear pairs and vanishing characteristic numbers, and identifies specific loops in Hamiltonian groups for certain classes of Delzant polytopes.

Findings

Equivalence between mass linear pairs and vanishing characteristic numbers.

Identification of loops generating infinite cyclic subgroups in Hamiltonian groups.

Application to specific polytopes like Hirzebruch surfaces and blow-ups.

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 ${\rm Ham}(M_{\Delta})$ 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 ${\mathbb C}P^n$.