Realisation of cycles by aspherical manifolds

TL;DR

This paper introduces a combinatorial method to realize all homology classes in topological spaces via images of finite covers of a specific aspherical manifold, expanding understanding of cycle realization.

Contribution

It presents a new combinatorial approach to Steenrod's problem, demonstrating that every homology class can be realized by images of finite covers of a particular aspherical manifold.

Findings

Every n-dimensional homology class can be realized by a finite cover of the isospectral manifold.

All homology classes of arcwise connected spaces can be realized by images of aspherical manifolds.

The approach applies to arbitrary topological spaces, broadening cycle realization techniques.

Abstract

We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real symmetric tridiagonal (n=1)x(n+1) matrices. In particular, every homology class of every arcwise connected topological space can be realised by a continuous image of an aspherical manifold.