Skeletons and Toric Extensions of Maximally Short Complexity One Spaces

Complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,\omega,\Phi)$ such that $\frac{1}{2}\dim M -\dim T=1$. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we prove that the moment image of the skeleton of a compact, connected maximally short complexity one $T$-space, which is in fact a GKM space, is connected. The proof relies on the well-known fact that each connected component of regular values of a proper moment map is a convex locally polyhedral set. We also gave an elementary proof of that fact along the way. Then we use the connectedness result to estimate the number of symplectic toric $(T \times S^1)$-manifolds whose underlying complexity one $T$-space is the same as the given maximally short complexity one $T$-space.