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

TL;DR

This note corrects a previous proof and establishes a new result relating the embedding properties of 4-manifolds to the algebraic structure of their boundary's homology, extending known theorems.

Contribution

It corrects an earlier proof and proves a more general theorem linking embedding properties of 4-manifolds to the structure of their boundary's torsion homology.

Findings

If many copies of a 4-manifold embed in another, the boundary's torsion homology splits as a direct double.

The linking pairing on the boundary's homology vanishes on one summand, indicating a split metabolic form.

Generalizes Hantzsche's theorem about embeddings of 3-manifolds in 4-spheres.

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 connected oriented $4$-manifold with connected boundary $\partial M$, and if an unbounded number of disjoint copies of $M$ embed topologically and locally flatly in the interior of a compact $4$-manifold $N,$ then $\operatorname{Tor}H_1(\partial M;\mathbb{Z})$ is a direct double, i.e., $\operatorname{Tor}H_1(\partial M;\mathbb{Z})\cong A \oplus A$, with the linking pairing vanishing identically on the first summand, i.e., the linking pairing is split metabolic. This partially generalizes Hantzsche's theorem stating that the linking pairing for a closed $3$-manifold that embeds in $S^4$ is hyperbolic.