I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism (i.e if it is periodic)?
6
-
$\begingroup$ Your question is not clear to me. Do you mean a presentation of the group of diffeomorphisms? Or something else? $\endgroup$– MTSCommented Jul 1, 2012 at 22:25
-
$\begingroup$ Of course I mean generators of homeotopy group. $\endgroup$– AndrewCommented Jul 1, 2012 at 23:30
-
$\begingroup$ This is still not clear. Are you asking for an example of a surface with explicit generators and relations for the group of diffeomorphisms of the surface? Or, as your comment seems to indicate, are you asking for an example of a surface $S$ with presentations of the higher homotopy groups $\pi_n(S)$? Diffeomorphisms up to homotopy (i.e. the mapping class group)? There are too many possible interpretations for this question. You are more likely to get a good response if you make it clear exactly what you are after. $\endgroup$– MTSCommented Jul 2, 2012 at 0:00
-
1$\begingroup$ "a presentation of a diffeomorphisms" ? $\:$ $\endgroup$– user5810Commented Jul 2, 2012 at 2:15
-
$\begingroup$ From Andrew's comment to Igor's question below, it looks to me like the question should be something like this: given a particular element of the mapping class group, say the mapping class of a homeomorphism or diffeomorphism expressed in some concrete manner, is there a procedure which will produce a word in the standard generators of the mapping class group which represents the given mapping class? This is still not a very good question because it does not specify concretely how the mapping class is given. $\endgroup$– Lee MosherCommented Jul 2, 2012 at 15:32
|
Show 1 more comment
1 Answer
$\begingroup$
$\endgroup$
4
If your question is: can you present the homeotopy group in terms of generators and relations, the answer is "yes", following the work of Hatcher-Thurston, Wajnryb, and most recently M. Korkmaz, who gives a relatively civilized presentation. If you mean: given a homeomorphism, can you express its isotopy class in terms of the generators, I assume that the answer is yes, but it obviously depends on how the homeomorphism is given. For related work, see Brinkmann, Peter(1-UT) An implementation of the Bestvina-Handel algorithm for surface homeomorphisms. (English summary) Experiment. Math. 9 (2000), no. 2, 235–240.
answered Jul 2, 2012 at 0:00
-
$\begingroup$ Thank you for your response. I will comment your answer and upper questions here. I am considering a particular diffeomorphism (up to isotopy to identity) of a surface F given by a function f: F --> F. Can I write down a presentation of this f in terms of Dehn twists? For example, given an involution x |--> -x,can this be made? $\endgroup$– AndrewCommented Jul 2, 2012 at 12:43
-
$\begingroup$ Andrew, your use of the word 'presentation' is not standard, you should either use standard terminology or explain clearly what you mean. $\endgroup$ Commented Jul 2, 2012 at 14:47
-
$\begingroup$ @Andrew: unless you specify your involution more concretely, this is still not a good question. The formula $x \mapsto −x$ does not specify an involution of F. $\endgroup$ Commented Jul 2, 2012 at 15:32
-
$\begingroup$ Sorry for my terminology. I didn't take care of it hoping that if someone is familiar with this then he could help, without digging in details. $\endgroup$– AndrewCommented Jul 2, 2012 at 16:17