Explicit construction of manifolds realizing the prescribed homology classes

TL;DR

This paper presents an explicit combinatorial method to construct manifolds that realize given homology classes, resolving singularities of pseudo-manifolds to achieve precise topological realizations.

Contribution

It introduces a new combinatorial construction that explicitly realizes homology classes by manifolds, advancing the classical problem of homology realization.

Findings

Provides a local procedure for resolving singularities of pseudo-manifolds.

Constructs manifolds that realize prescribed homology classes with explicit combinatorial methods.

Applications include constructing manifolds with specified vertex link sets.

Abstract

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.