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.

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You know, Gompf has a trick:

Give it a really complicated name, and people will name it after you out of laziness.

Then again, you might be out of luck, as “Johnson kernel” is already taken. :)

Comment by Richard Kent — November 11, 2008 @ 10:11 am |

Right. That should make it a more interesting competition to rule out that sort of thing right away.

Comment by Jesse Johnson — November 11, 2008 @ 10:45 am |

Maybe first think up names for the various parts of the definition, then the final name for the whole thing might seem “obvious.”

Comment by Derek — November 11, 2008 @ 1:10 pm |

I’ll call H the splitting surface and S the abstract surface. So then I might call your group Isot(H, M) the isotopy group of H in M.

Question: Suppose Emb(S, M) is the space of smooth embeddings f: S \to M. Note that Diff(S) acts by precomposition. Mod out by this action. (So we forget about how f(S) \subset M is marked by S.) Hatcher calls this P(S, M), the space of “placements” of S in M [in his paper “Spaces of incompressible surfaces”]. We fix attention on P_H(S, M), the component of P(S, M) containing the Heegaard surface H. Is pi_1 of P_H(S, M) the same as Isot(H, M)? (Here the basepoint is the class of the given embedding realizing the Heegaard surface.) If that is right, then computing pi_i(P_H(S, M)), for i > 1, would be interesting. I’ll venture a guess that all of the higher homotopy groups are trivial when M is hyperbolic.

Well, this is all very complicated.

Hmm. Perhaps a better name would be the “Richard Kent IV kernel”? Of course, an advantage of the latter name is we also take care of the next three objects in the competition. :)

Comment by saul schleimer — November 11, 2008 @ 1:42 pm |

So a transformation in the group sends to itself, but can’t be deformed along itself to the identity. That is, it’s been

wrappedaround something in . Thus, the “wrapping class group”.Comment by John Armstrong — November 11, 2008 @ 1:42 pm |

I think “Richard Kent IV kernel” is sufficiently complicated to merit calling it the Johnson kernel.

Comment by Richard Kent — November 11, 2008 @ 2:21 pm |

Saul: This group should be isomorphic to the fundamental group of the placement space in most cases. If the diffeomorphism group of the ambient manifold is non-trivial (for example in Seifert Fibered spaces) then you get some sort of exact sequence. In fact, for genus at least two, the placement should be a free quotient by this unnamed group of a space homotopy equivalent to the diffeomorphism group of the 3-manifold. Daryl McCullough and I are working on writing up a theorem to this effect.

But I don’t think that helps much with finding a name.

Comment by Jesse Johnson — November 11, 2008 @ 2:34 pm |

This smells like the Torelli subgroup of the MCG of surfaces. Why not “Torelli subgroup”?

Comment by JeffE — November 11, 2008 @ 4:52 pm |

My suggestion: Phony(H)

Comment by Richard Kent — November 11, 2008 @ 5:10 pm |

It seems to me that an automorphism in this subgroup is “ambiently trivial”. So it’s the “ambiently trivial subgroup”.

Comment by Henry Wilton — November 11, 2008 @ 10:03 pm |

Normally speaking, a nontrivial element in the MCG is not isotopic to the identity. It’s not included in a one-parameter family. It’s a radical change, a breakthrough. With your definition, elements in this kernel are nontrivial in a “MCG”, but they are isotopic to the identity. If we looks at them closely, they looked like a radical change, but they are not. It’s an ersatz of change. Why don’t you call this kernel “Obama’s subgroup?”

More seriously, I’m rather confident that we can turn this metaphor of “ersatz of change” in a lovely, poetic name…

Sincerely yours.

Comment by McCain — November 12, 2008 @ 7:15 pm |

Since at first glance, when you forget to look at the Heegaard surface, these automorphisms are trivial, but a closer look (taking into account the Heegaard surface) shows that they aren’t quite so trivial, I suggest calling the subgroup the “superficially trivial subgroup”. Or, since they are only non-trivial “on the surface” maybe it should be called the “superficially non-trivial subgroup”.

Comment by Scott Taylor — November 12, 2008 @ 8:48 pm |

Most math names are confusing reuses of similar concepts from related domains. So why stop the tradition now!

It should be called the “central base subgroup”.

The slogan is “So important it’s generic”.

Comment by Matt — November 13, 2008 @ 4:15 pm |

Good morning;

I hace a question that perhaps is out of context. Now I am translating an article from english to spanish which uses the term “space of placements”. It is a sociological paper, yet the expression is used in a mathematical context (topology). Could you help me with this please? Does anyboy knows how to translate it to spanish? I would really appreciate your help,

Alejandro

Comment by Alejandro — January 19, 2009 @ 11:59 am |

A asked Fabiola Manjarrez-Gutierrez (one of my former fellow grad. students at UC Davis) and she suggested the following:

“espacio de posiciones” or “espacio de arreglos” or “espacio de colocaciones”

But I don’t know if there is a standard way to translate it.

Comment by Jesse Johnson — January 20, 2009 @ 10:57 am |