A survey of the homology cobordism group

Date

2023-10-6

Author

Şavk, Oğuz

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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 Θ3ℤ. Finally, we briefly discuss the knot concordance group C and the rational homology cobordism group Θ3ℚ, focusing on their algebraic structures, relating them to Θ3ℤ, 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.

URI

https://hdl.handle.net/11511/116172

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Department of Mathematics, Article