Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But ...
From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ...
Casson's invariant is an invariant of a homology 3-sphere, obtained by ``counting" representations of the fundamental group into $SU(2)$. I was wondering if there is an analogous invariant counting ...