Immersed but not embedded homology classes

TL;DR

The paper resolves Liu's question by constructing explicit examples of immersions that are not embeddings and by separating Steenrod representability from immersion. It develops an obstruction-theoretic framework using Thom spaces $MSO_k$ and the infinite-loopspace variant $QMSO_k$, along with Whitney's self-intersection formula and Postnikov-tower analysis, to distinguish immersion from embedding. Core results show that the generators of $H_7(Sp_2)$ and $H_{13}(N)$ are immersed but not embedded, while there exist Steenrod representable integral classes not represented by immersions (Theorem D). Together, the results clarify the relationships among Steenrod representability, immersion, and embedding, and provide explicit formal immersions representing the studied classes.

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.

Theorems & Definitions (25)

• Theorem A
• Corollary 1.2
• proof
• Remark 1.3
• Theorem B
• Theorem C
• Theorem D
• Proposition 2.1
• proof
• Theorem 3.1