Immersed but not embedded homology classes

TL;DR

This paper presents the first known examples of immersions of closed oriented manifolds that are not homologous to embeddings, highlighting new distinctions in immersion and embedding homology classes.

Contribution

It provides the first documented examples of such immersions and explores the homological implications of double points and Steenrod representability.

Findings

Double points of immersions represent non-trivial homology classes.

Examples of Steenrod representable classes not realized by immersions.

First known examples answering Liu's question.

Abstract

We provide the first documented examples of immersions of closed oriented manifolds which are not homologous to embeddings, thus answering a question posed by Zhenhua Liu. In these examples we show that for any representing self-transverse immersion the double points must represent a non-trivial homology class in the source manifold. We also provide examples of Steenrod representable integral homology classes which are not represented by immersions.