A SURVEY OF THE HOMOLOGY COBORDISM GROUP

In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its longstanding 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 3Z. Finally, we briefly discuss the knot concordance group C and the rational homology cobordism group Theta 3 Q, focusing on their algebraic structures, relating them to Theta 3Z, 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.