Sets of links of vertices of triangulated manifolds and combinatorial approach to Steenrod's problem on realisation of cycles

To each triangulated manifold one can assign the set of links of its vertices. The link of a vertex describes the local combinatorial structure of the triangulation in a neighbourhood of the vertex. Thus the set of links of vertices of a triangulation can be interpreted as the set of local combinatorial data characterizing the triangulation. We consider a compatibility problem for such local combinatorial data. This problem can be formulated in the following way. For a given set of combinatorial spheres, does there exist a triangulated manifold with such set of links of vertices? We are mostly interested in an oriented version of this problem. Our aim is to obtain a non-trivial sufficient condition for compatibility of a set of links of vertices. We shall describe an explicit construction that, under certain natural conditions, allows us to realise a multiple of a given set of oriented combinatorial spheres as the set of links of vertices of a combinatorial manifold.
Further, we are going to discuss an application of this construction to N. Steenrod's problem on realisation of cycles. It is well known that according to a result of R. Thom, any $n$-dimensional integral homology class $z$ of any topological space $X$ can be realised with some multiplicity by an image of an oriented smooth closed manifold $N^n$. Our new approach is based on an explicit combinatorial procedure for resolving singularities of a cycle. We give an explicit combinatorial construction that, for a given homology class $z$, yields a manifold $N^n$ and its mapping to $X$ which realises the class $z$ with some multiplicity. Moreover, the obtained manifold $N^n$ appears to be a finite-fold non-ramified covering over a very interesting special manifold $M^n$, which can be regarded either as an isospectral manifold of symmetric tridiagonal real $(n+1)\times(n+1)$-matrices or as a small covering over a permutohedron.