Virtual $3$-manifolds of complexity $1$ and $2$

Matveev in 2009 introduced the notion of virtual $3$-manifold, which generalizes the classical notion of $3$-manifold. A virtual manifold is an equivalence class of so-called special polyhedra. Each virtual manifold determines a $3$-manifold with nonempty boundary and $\mathbb{R}P^2$-singularities. Many invariants of manifolds, such as Turaev–Viro invariants, can be extended to virtual manifolds. The complexity of a virtual $3$-manifold is $k$ if its equivalence class contains a special polyhedron with $k$ true vertices and contains no special polyhedra with a smaller number of true vertices. In this paper we give a complete list of virtual $3$-manifolds of complexity $1$ and present two-sided bounds for the number of virtual $3$-manifolds of complexity $2$. The question of the complete classification for virtual $3$-manifolds of complexity $2$ remains open.