An irreducible real projective plane in the 4-sphere
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911679302860800 |
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| author | Hughes, Mark Kim, Seungwon Miller, Maggie Nahm, Gheehyun |
| author_facet | Hughes, Mark Kim, Seungwon Miller, Maggie Nahm, Gheehyun |
| contents | We construct an irreducible embedded projective plane in $S^4$. This gives a counterexample to the Kinoshita conjecture and answers Problem 4.37 of the K3 problem list. Moreover, we answer both Questions (i) and (ii) of Problem 4.37: (i) the connected sum $R\# R$ is a Klein bottle in $S^4$ with extremal normal Euler number that does not admit an unknotted projective plane summand, and (ii) we show that our projective plane $R$ is irreducible by showing that the peripheral map $π_1 (\partial (S^4\setminus\mathring{N}(R)))\to π_1 (S^4 \setminus \mathring{N}(R))$ has kernel of order $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12921 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An irreducible real projective plane in the 4-sphere Hughes, Mark Kim, Seungwon Miller, Maggie Nahm, Gheehyun Geometric Topology 57K45 We construct an irreducible embedded projective plane in $S^4$. This gives a counterexample to the Kinoshita conjecture and answers Problem 4.37 of the K3 problem list. Moreover, we answer both Questions (i) and (ii) of Problem 4.37: (i) the connected sum $R\# R$ is a Klein bottle in $S^4$ with extremal normal Euler number that does not admit an unknotted projective plane summand, and (ii) we show that our projective plane $R$ is irreducible by showing that the peripheral map $π_1 (\partial (S^4\setminus\mathring{N}(R)))\to π_1 (S^4 \setminus \mathring{N}(R))$ has kernel of order $2$. |
| title | An irreducible real projective plane in the 4-sphere |
| topic | Geometric Topology 57K45 |
| url | https://arxiv.org/abs/2605.12921 |