Does every topological manifold admit a triangulation? In 1990, Casson showed that the answer in dimension four is no, and in 2013, Manolescu showed that the answer is also no in every dimension greater than four. Manolescu's proof relied on using an equivariant version of Seiberg-Witten Floer homology to prove a statement about the 3-dimensional homology cobordism group. Two of the lasting impacts of this work are the development of other equivariant Floer theories and a renewed interested in the homology cobordism group. We will survey some of the recent work in this area.