Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908916453998592 |
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| author | Chow, Bennett Freedman, Michael H. Shin, Henry Zhang, Yongjia |
| author_facet | Chow, Bennett Freedman, Michael H. Shin, Henry Zhang, Yongjia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models Chow, Bennett Freedman, Michael H. Shin, Henry Zhang, Yongjia Differential Geometry Algebraic Topology Geometric Topology 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. |
| title | Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models |
| topic | Differential Geometry Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2505.03823 |