A survey of the homology cobordism group

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 3 -spheres and the structure of Θ Z 3 \Theta _{\mathbb {Z}}^3 . Finally, we briefly discuss the knot concordance group C \mathcal {C} and the rational homology cobordism group Θ Q 3 \Theta _{\mathbb {Q}}^3 , focusing on their algebraic structures, relating them to Θ Z 3 \Theta _{\mathbb {Z}}^3 , and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology 3 3 -spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.