Universal Realisators for Homology Classes

TL;DR

This paper introduces the URC property for manifolds, showing many small covers and some hyperbolic manifolds possess it, enabling mappings of non-zero degree to any 4-manifold, confirming a conjecture.

Contribution

It identifies a broad class of manifolds with the URC property, including hyperbolic 4-manifolds, and proves the existence of non-zero degree maps to any 4-manifold.

Findings

Many small covers of simple polytopes have the URC property.

Existence of hyperbolic 4-manifolds with URC property.

Hyperbolic 4-manifolds can map to any 4-manifold with non-zero degree.

Abstract

We study oriented closed manifolds M^n possessing the following Universal Realisation of Cycles (URC) Property: For each topological space X and each integral homology class z of it, there exist a finite-sheeted covering \hM^n of M^n and a continuous mapping f of \hM^n to X such that f takes the fundamental class [\hM^n] to kz for a non-zero integer k. We find wide class of examples of such manifolds M^n among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we prove that for each 4-dimensional oriented closed manifold N^4, there exists a mapping of non-zero degree of a hyperbolic manifold M^4 to N^4. This was conjectured by Kotschick and Loeh.