Enhanced Hantzsche Theorem

TL;DR

This paper introduces an enhanced version of Hantzsche's theorem by relating the smooth embeddability of closed 3-manifolds in  to a new property of their Heegaard diagrams called doubly unlinked, providing a refined obstruction criterion.

Contribution

The paper develops an improved embedding obstruction for 3-manifolds by extending Hantzsche's theorem through the concept of doubly unlinked diagrams.

Findings

Doubly unlinked property characterizes embeddability in .

Enhanced obstruction refines previous criteria for embedding.

Heegaard diagram analysis provides new insights into 3-manifold embeddings.

Abstract

A closed 3-manifold $M$ may be described up to some indeterminacy by a Heegaard diagram $\mathcal{D}$. The question "Does $M$ smoothly embed in $\mathbb{R}^4$?'' is equivalent to a property of $\mathcal{D}$ which we call $\textit{doubly unlinked}$ (DU). This perspective leads to an enhancement of Hantzsche's embedding obstruction.