Realisation of cycles and small covers over graph-associahedra

A classical question by Steenrod (late 1940s) was whether it is possible to realize an integral homology class of a topological space by a continuous image of the fundamental class of an oriented smooth closed manifold. (Homology classes satisfying this condition are called realizable.) This question was answered by Thom (1954) who showed that there exist non-realizable homology classes but a certain multiple of any homology class is realizable.
In 2007 the speaker found an explicit combinatorial procedure that, for a given singular cycle in a topological space, constructs a manifold realizing a multiple of the homology class representing by this cycle. Moreover, this construction allowed us to prove that, for every $n$, there exists an oriented smooth closed manifold $M^n$ that satisfy the following Universal Realization of Cycles property (or the URC-property): A multiple of any $n$-dimensional integral homology class of any topological space can be realized by an image of the fundamental class of a non-ramified finite-sheeted covering over $M^n$. Several series of examples of URC-manifolds (i.e. manifolds satisfying the URC-property) were found by the speaker in 2013. The simplest of them was the so-called Tomei manifold, which is a small cover of a special simple polytope called the permutohedron.
In the talk we shall present a modification of the explicit procedure for the realization of cycles that will allow us to find URC-manifolds that are even simpler than the Tomei manifolds. In particular, for an important class of simple polytopes called graph-associahedra, we shall show that small covers over them are also URC-manifolds. In particalar, all small covers of a well known Stacheff associahedra are URC-manifolds.