Explicit construction of manifolds realizing the prescribed homology classes

Details Explicit construction of manifolds realizing the prescribed homology classes Explicit construction of manifolds realizing the prescribed homology classes

2007-12-11

arxiv.org/abs/0712.1709v1

Mathematics

Geometric Topology

3 pages

Scientific paper

We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinatorial construction that for a given integral cycle provides a combinatorial manifold realizing a multiple of the homology class of this cycle. The construction is based on a local procedure for resolving singularities of a pseudo-manifold. We give an application of our result to the problem of constructing a combinatorial manifold with the prescribed set of links of vertices.

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