A survey of the homology cobordism group. (English) Zbl 1529.57012
Summary: In this survey, we present the most recent highlights from the study of the homology cobordism group, with 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 \(\Theta_{\mathbb{Z}}^3\). Finally, we briefly discuss the knot concordance group \(\mathcal{C}\) and the rational homology cobordism group \(\Theta_{\mathbb{Q}}^3\), focusing on their algebraic structures, relating them to \(\Theta_{\mathbb{Z}}^3\), 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.
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 57K41 | Invariants of 4-manifolds (including Donaldson and Seiberg-Witten invariants) |
| 57R57 | Applications of global analysis to structures on manifolds |
| 57R58 | Floer homology |
| 57R90 | Other types of cobordism |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
Keywords:
homology cobordism group; smooth manifolds; low-dimensional topology; homology \(3\)-spheres; knot concordanceReferences:
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