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Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

Bennett Chow, Michael H. Freedman, Henry Shin, Yongjia Zhang

TL;DR

The note corrects an error in a previous proof of Proposition 13 and proves a more general statement: if a compact $4$-manifold $M$ with boundary diffeomorphic to a spherical space form $S^3/\Gamma$ is embedded in a compact $4$-manifold $N$ with arbitrarily many disjoint copies, then $H_1(\partial;\mathbb{Z})$ is a direct double $A \oplus A$ for some finite abelian group $A$. The authors extend the result to the broader setting where $\partial$ is only assumed to be a rational homology $3$-sphere, using a Mayer–Vietoris framework on unions of copies of $M$ and a defect lemma to force the splitting. This yields a partial generalization of Hantzsche's theorem for rational homology $3$-spheres and clarifies the torsion structure in $H_1(\partial)$ arising from embeddings. The work strengthens the connection between 4-dimensional embedding obstructions and 3-manifold torsion phenomena, and fixes a gap in the previous literature.

Abstract

This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if $M$ is a compact 4-manifold with boundary $\partial M$ a rational homology 3-sphere, and if an unbounded number of disjoint copies of $M$ embed in a compact 4-manifold $N$, then $H_1(\partial M;\mathbb{Z})$ is a direct double. This partially generalizes a theorem of Hantzsche in the case of rational homology 3-spheres.

Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

TL;DR

The note corrects an error in a previous proof of Proposition 13 and proves a more general statement: if a compact -manifold with boundary diffeomorphic to a spherical space form is embedded in a compact -manifold with arbitrarily many disjoint copies, then is a direct double for some finite abelian group . The authors extend the result to the broader setting where is only assumed to be a rational homology -sphere, using a Mayer–Vietoris framework on unions of copies of and a defect lemma to force the splitting. This yields a partial generalization of Hantzsche's theorem for rational homology -spheres and clarifies the torsion structure in arising from embeddings. The work strengthens the connection between 4-dimensional embedding obstructions and 3-manifold torsion phenomena, and fixes a gap in the previous literature.

Abstract

This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if is a compact 4-manifold with boundary a rational homology 3-sphere, and if an unbounded number of disjoint copies of embed in a compact 4-manifold , then is a direct double. This partially generalizes a theorem of Hantzsche in the case of rational homology 3-spheres.
Paper Structure (2 sections, 4 theorems, 11 equations)

This paper contains 2 sections, 4 theorems, 11 equations.

Table of Contents

  1. A partial generalization of Hantzsche's theorem
  2. Proof of Proposition \ref{['prop:1']}

Key Result

Proposition 1

Let $M$ be a compact $4$-manifold with connected boundary $\partial := \partial M$ diffeomorphic to a spherical space form $S^3/\Gamma$. If there exists a compact $4$-manifold $N$ containing an unbounded number of disjoint copies of $M$, then $H_1(S^3/\Gamma;\mathbb{Z} )$ is a direct double, i.e., i

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Remark 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem: the new lemma 17']}
  • Lemma 5
  • proof