Timeline for Presentations of mapping class groups in dimension $3$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24, 2021 at 23:57 | comment | added | Ryan Budney | Stated another way, you might ask, why are people so concerned with presentations for 2-dimensional mapping class groups? It stems from the type of group you are dealing with. 2-dimensional mapping class groups are generally infinite and non-abelian, moreover, they were not otherwise studied. So presentations were a way to get some insight into the group structure. For 3-dimensional mapping class groups, many of them are familiar groups for which presentations might be viewed as a distraction. | |
| May 24, 2021 at 19:08 | answer | added | rpotrie | timeline score: 2 | |
| May 24, 2021 at 19:04 | comment | added | Ryan Budney | Lots of presentations are known. By and large, viewing things through the lens of presentations might be limiting. Often times you have nice models for $Diff(M)$ or $BDiff(M)$. If you are particularly motivated you can derive presentations for the mapping class group from that. | |
| May 24, 2021 at 18:00 | comment | added | Andy Putman | In general, it's very natural to want to know that a group is finitely presentable. Specific relations are also often useful. But actually writing down an explicit finite presentation for a very complicated group is often not a good way to understand it (and I say this as someone whose PhD thesis was all about finding a presentation for a specific subgroup of the mapping class group of a surface!). | |
| May 24, 2021 at 17:08 | history | edited | HJRW |
Added gt.geometric-topology tag.
|
|
| May 24, 2021 at 10:48 | comment | added | HJRW | By the way, the "second viewpoint" isn't really useful for computing the mapping class group -- you have to first translate to the "first viewpoint" by computing the Kneser--Milnor and JSJ decompositions, and then the geometric structures of the pieces. (Algorithms are known for all of these things.) I suspect there are examples of 3-manifolds with very similar link presentations but with radically different JSJ decompositions (and hence mapping class groups). | |
| May 24, 2021 at 10:46 | comment | added | HJRW | ... and how to "reassemble" them into the mapping class group of the whole thing. If you are also interested in the reducible case (e.g. doubles of handlebodies, as @SamNead mentions) then the answer is more complicated still. So it would help a great deal if you were interested in some more specific class of 3-manifolds. If you really want the answer in full generality, a better question might be to try to write down a roadmap to understanding all the different components. | |
| May 24, 2021 at 10:42 | comment | added | HJRW | It is certainly true that there is an algorithm to compute a presentation of a mapping class group of a given 3-manifold, but I don't think this is a reasonable thing to describe in practice; certainly not in an answer to an MO question, and possibly not ever. As indicated in @SamNead's answer (see my comment below), even if you restrict attention to closed hyperbolic manifolds, every finite group arises. If you really want to understand all irreducible 3-manifolds, you need to learn about the JSJ decomposition, how to compute the mapping class groups of the pieces... | |
| May 22, 2021 at 7:37 | answer | added | Sam Nead | timeline score: 4 | |
| May 21, 2021 at 22:25 | history | asked | Student | CC BY-SA 4.0 |