TITLE: Classification of homology 3-spheres?
QUESTION [18 upvotes]: Is there some general description of all homology 3-spheres?

REPLY [11 votes]: Another way to represent homology spheres is to take a Heegaard splitting
for $S^3$, cut and reglue by an element of the Torelli group. This is
not canonical, but any two Heegaard splittings are equivalent after some
number of stabilizations. If you wanted to enumerate every homology sphere, you could list elements of the Torelli group, and construct 3-manifolds, then throw away repeats by using some solution to the homeomorphism problem for 3-manifolds. This is not really feasible to carry out in practice, but is one way to give a "general description" of homology spheres at least in theory, by giving a recursive enumeration of them.