Locally flat simple spheres in $\mathbb{C} P^2$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929545447211008 |
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| author | Conway, Anthony Orson, Patrick |
| author_facet | Conway, Anthony Orson, Patrick |
| contents | The fundamental group of the complement of a locally flat surface in a $4$-manifold is called the knot group of the surface. In this article we prove that two locally flat $2$-spheres in $\mathbb{C} P^2$ with knot group $\mathbb{Z}_2$ are ambiently isotopic if they are homologous. This combines with work of Tristram and Lee-Wilczyński, as well as the classification of $\mathbb{Z}$-surfaces, to complete a proof of the statement: a class $d \in H_2(\mathbb{C} P^2) \cong \mathbb{Z}$ is represented by a locally flat $2$-sphere with abelian knot group if and only if $|d| \in \lbrace 0,1,2\rbrace$; and this sphere is unique up to ambient isotopy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10546 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Locally flat simple spheres in $\mathbb{C} P^2$ Conway, Anthony Orson, Patrick Geometric Topology The fundamental group of the complement of a locally flat surface in a $4$-manifold is called the knot group of the surface. In this article we prove that two locally flat $2$-spheres in $\mathbb{C} P^2$ with knot group $\mathbb{Z}_2$ are ambiently isotopic if they are homologous. This combines with work of Tristram and Lee-Wilczyński, as well as the classification of $\mathbb{Z}$-surfaces, to complete a proof of the statement: a class $d \in H_2(\mathbb{C} P^2) \cong \mathbb{Z}$ is represented by a locally flat $2$-sphere with abelian knot group if and only if $|d| \in \lbrace 0,1,2\rbrace$; and this sphere is unique up to ambient isotopy. |
| title | Locally flat simple spheres in $\mathbb{C} P^2$ |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2312.10546 |