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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)?

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Your question is not clear to me. Do you mean a presentation of the group of diffeomorphisms? Or something else? – MTS Jul 1 '12 at 22:25
Of course I mean generators of homeotopy group. – Andrew Jul 1 '12 at 23:30
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. – MTS Jul 2 '12 at 0:00
"a presentation of a diffeomorphisms" ? $\:$ – Ricky Demer Jul 2 '12 at 2:15
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. – Lee Mosher Jul 2 '12 at 15:32

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.

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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? – Andrew Jul 2 '12 at 12:43
Andrew, your use of the word 'presentation' is not standard, you should either use standard terminology or explain clearly what you mean. – Fernando Muro Jul 2 '12 at 14:47
@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. – Lee Mosher Jul 2 '12 at 15:32
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. – Andrew Jul 2 '12 at 16:17

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