A survey of the homology cobordism group
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917560647155712 |
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| author | Şavk, Oğuz |
| author_facet | Şavk, Oğuz |
| contents | In this survey, we present most recent highlights from the study of the homology cobordism group, with a particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology $3$-spheres and the structure of $Θ^3_\mathbb{Z}$. Finally, we briefly discuss the knot concordance group $\mathcal{C}$ and the rational homology cobordism group $Θ^3_\mathbb{Q}$, focusing on their algebraic structures, relating them to $Θ^3_\mathbb{Z}$, and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology $3$-spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_10735 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A survey of the homology cobordism group Şavk, Oğuz Geometric Topology History and Overview In this survey, we present most recent highlights from the study of the homology cobordism group, with a particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology $3$-spheres and the structure of $Θ^3_\mathbb{Z}$. Finally, we briefly discuss the knot concordance group $\mathcal{C}$ and the rational homology cobordism group $Θ^3_\mathbb{Q}$, focusing on their algebraic structures, relating them to $Θ^3_\mathbb{Z}$, and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology $3$-spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen. |
| title | A survey of the homology cobordism group |
| topic | Geometric Topology History and Overview |
| url | https://arxiv.org/abs/2209.10735 |