Аннотация:
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 Nn. 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 Nn and its mapping to X which realises the class z with some multiplicity. Moreover, the obtained manifold Nn appears to be a finite-fold non-ramified covering over a very interesting special manifold Mn, which can be regarded either as an isospectral manifold of symmetric tridiagonal real (n+1)×(n+1)-matrices or as a small covering over a permutohedron.