论文标题
同源性共同体小组的调查
A survey of the homology cobordism group
论文作者
论文摘要
在这项调查中，我们介绍了同源性共同体小组研究的最新亮点，并特别强调了其在光滑歧管背景下的长期悠久历史。此外，我们列出了有关其代数结构的各种结果，并讨论了其在低维拓扑的发展中的关键作用。此外，我们分享了一系列关于同源性$ 3 $ -SPHERES的行为以及$θ^3_ \ Mathbb {Z} $的结构的开放问题。最后，我们简要讨论了结的一致性组$ \ MATHCAL {C} $和理性同源性COBORDISM组$θ^3_ \ Mathbb {Q} $，重点关注其代数结构，将它们与$θ^3_ \ Mathbb {Z} $相关，并凸显了几个开放的问题。附录是由Brieskorn，Dehn，Gordon，Seifert，Siebenmann和Waldhausen引入的几种同源性$ 3 $ SPHERES的几种结构和演示的汇编。
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.
