A survey of the homology cobordism group

TL;DR

This survey reviews recent developments in the study of the homology cobordism group, emphasizing its algebraic structure, historical context, and open problems in low-dimensional topology and related groups.

Contribution

It compiles recent results, discusses algebraic structures, and highlights open problems in the homology cobordism group and related topological groups.

Findings

Summarizes recent advances in the algebraic understanding of the homology cobordism group.

Identifies key open problems in the behavior of homology 3-spheres.

Discusses relationships between various cobordism and knot concordance groups.

Abstract

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 $\Theta^3_\mathbb{Z}$. Finally, we briefly discuss the knot concordance group $\mathcal{C}$ and the rational homology cobordism group $\Theta^3_\mathbb{Q}$, focusing on their algebraic structures, relating them to $\Theta^3_\mathbb{Z}$, 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.