A survey of the homology cobordism group

TL;DR

This survey traces the rich structure of the homology cobordism group $\Theta_{\mathbb{Z}}^3$, surveying foundational history, key invariants from gauge theory and Floer homology, and the dramatic impact of modern techniques such as involutive Floer homology on understanding its infinite-rank subgroups and potential generators. It highlights how Rokhlin's invariant $\mu$ and its integral lifts (notably $\beta$) interact with Seifert-fibered and Brieskorn spheres to produce obstructions and to generate large free pieces, while newer invariants $\vec{f}$, $\kappa$, and related tools reveal deeper substructure and interconnections with knot concordance $\mathcal{C}$ and rational homology cobordism $\Theta_{\mathbb{Q}}^3$. The paper catalogs a spectrum of open problems on torsion, generators, and the full structure of $\Theta_{\mathbb{Z}}^3$, and discusses how contemporary methods illuminate how these invariants behave under surgeries, splices, and branched coverings. Overall, the work situates $\Theta_{\mathbb{Z}}^3$ at the crossroads of smooth 4-manifold topology, low-dimensional gauge theory, knot theory, and 3-manifold topology, underscoring both deep results and salient open questions with broad implications for topology and geometry.

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

Figures (1)

• Figure 1: The plumbing graphs of $X(n)$, $X'(n)$, and $W(n)$.

Theorems & Definitions (26)

• Theorem A: Theorem 1.2, KM63
• Theorem B: P1P3P2
• Theorem C: Theorem I.2, GA70
• Theorem D: Theorem 3, K69
• Theorem E: R52; Section I.5, GA70
• Theorem F: Theorem 1.2, FS85
• Theorem G: Theorem 2.1, F90
• Theorem H: Theorem 3, Fro02
• Theorem I: Theorem 1.1, DHST18
• Theorem J