A partial generalization of Hantzsche's theorem and a correction

This note corrects an error in the proof of Proposition 13 in [1] and simultaneously establishes a more general result. We prove that if M is a compact connected oriented 4-manifold with connected boundary ∂ 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 T o r ⁢ 𝐻 1 ⁡ ( ∂ 𝑀 ; ℤ ) is a direct double, i.e., T o r ⁢ 𝐻 1 ⁡ ( ∂ 𝑀 ; ℤ ) ≅ 𝐴 ⊕ 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.