Realizing homology classes up to cobordism

TL;DR

This paper introduces a new concept of realizing homology classes up to cobordism, showing that immersions suffice for this purpose while certain multisingular maps do not.

Contribution

It defines the notion of realization up to cobordism and demonstrates the sufficiency of immersions for this weaker realization.

Findings

Immersions are sufficient for realizing homology classes up to cobordism.

Maps with fixed finite multisingularities remain insufficient for this realization.

The paper clarifies the limitations of multisingular maps in homology class realization.

Abstract

It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod 2 homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient.