I thought I would try something new for this blog: a competition to choose a name for a mathematical object that so far doesn’t have one. I will describe it below and ask you, gentle readers, to nominate potential names for it in the comments. Once there are a few suggestions, I’ll post another entry listing the contestants and ask for votes (also in the form of comments) on which should be the name. Whichever name wins, I will use in the future whenever speaking or writing about the object and will encourage others to do the same. If the contest is a success, we can try it again with something else.
So here’s the object: Start with a Heegaard splitting of a 3-manifold M, i.e. a decomposition of M into two handlebodies that intersect in a surface S, called the Heegaard surface. The mapping class group of the Heegaard splitting is the group of homeomorphisms from M to itself that take S onto itself, modulo isotopies in which each intermediate homeomorphism takes S onto itself. You can also think of this as the group of connected components of the group of homeomorphisms of the pair (M,S) onto itself. There is a subgroup of the mapping class group consisting of automorphisms that are isotopy trivial as automorphisms of M (i.e. if you forget about the Heegaard surface.) This subgroup is the kernel of the canonical homomorphism from the mapping class group of the Heegaard splitting to the mapping class group of M. This subgroup also doesn’t have a name. So, now’s your chance: suggest the best name for this subgroup in the comments below. I have one or two papers on the way that will use this group extensively. If your name wins, not only will I use it in these papers, but I’ll give you credit as the namer of the subgroup.
Also, if there are any unnamed mathematical objects that you would like to get a similar treatment in a future post, please feel free to suggest them.