Torus actions of complexity 1 and their local properties

We consider an effective action of a compact $(n-1)$-torus on a smooth $2n$-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than $n-1$ has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold $G_{4,2}$, the complete flag manifold $F_3$, and quasitoric manifolds with an induced action of a subtorus of complexity $1$.