A survey of the homology cobordism group

Şavk, Oğuz

doi:10.1090/bull/1806

A survey of the homology cobordism group
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by Oğuz Şavk

Bull. Amer. Math. Soc. 61 (2024), 119-157

DOI: https://doi.org/10.1090/bull/1806

Published electronically: October 6, 2023

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Abstract:

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.

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