(Reprinted from my old blog)
Here’s an interesting fact: The abelianizations of the mapping class groups of two surfaces are isomorphic if both surfaces have sufficiently high genus. In other words, if you take a high genus surface and add a handle to it, the abelianization of its mapping class group doesn’t change. Now, the abelianization of a group is just the first homology group of its classifying space (a space with the given group as its fundamental group and whose universal cover is contractible) and for each dimension, we can define the homology of the original group as the homology of its classifying space. Well, it turns out for these higher homology groups don’t change either, if you start with a higher genus surface.
This phenomenon is called homological stability and similar results have been proved for other sequences of groups, mostly automorphism groups of objects that become progressively more complicated. Hatcher and Wahl  recently proved the following: Let M and N be three manifolds and let M_k be the connect sum or boundary connect sum of M and k copies of N. Then the nth homology mapping class groups of M_k is stable for sufficiently large k.
As you might have guessed if you’ve read many of my previous posts, I’m now going to suggest a connection to Heegaard splittings. Consider a sequence of stabilizations of a fixed Heegaard splitting, i.e. the nth splitting is the result of attaching n unknotted handles to the original. The question is: are the mapping class groups of these Heegaard splittings homologically stable? Hatcher and Wahl’s result implies that mapping class groups of handlebodies are homologically stable (a handlebody is a boundary connect sum of a number of solid tori) but that isn’t enough to prove it for a Heegaard splitting.