Abstract:
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.
