Toric one-skeletons for complexity-one spaces

A complexity-one space is a compact symplectic manifold $(M, \omega)$ endowed with an effective Hamiltonian action of a torus $T$ of dimension $\frac{1}{2}\dim(M)-1$. In this note we prove that for a certain class of complexity-one spaces the Poincar\'e dual of the Chern class $c_{n-1}$ can be represented by a collection of $\frac{n}{2}\chi(M)$ symplectic embedded $2$-spheres, where $\chi(M)$ is the Euler characteristic of $M$ and $\dim(M)=2n$. We call such a collection a toric one-skeleton. The classification of complexity-one spaces is an important subject in symplectic geometry. A nice subcategory of those spaces are the ones which are monotone. The existence of a toric one-skeleton is a useful tool to understand six-dimensional monotone complexity-one spaces. In particular, we will show that the existence of a toric one-skeleton for such a space implies that the second Betti number of $M$ is at most seven. This is a simple application of results by Sabatini-Sepe and Lindsay-Panov.