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This question already has an answer here:

What is the classification of the self-homeomorphisms of the product of the $n$-sphere by the circle, $S^n\times S^1$, when $n>3$, up to pseudo-isotopy (that is, concordance)?

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marked as duplicate by Francois Ziegler, Marco Golla, Stefan Kohl, Jan-Christoph Schlage-Puchta, Tony Huynh Feb 18 '17 at 13:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ You might want to consider making the title more specific $\endgroup$ – Saal Hardali Feb 10 '17 at 21:37
  • $\begingroup$ Is this a homework problem? $\endgroup$ – Noah Schweber Feb 10 '17 at 22:14
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    $\begingroup$ Welcome to MathOverflow Professor Laudenbach. $\endgroup$ – Lee Mosher Feb 10 '17 at 22:50
  • $\begingroup$ I do not understand the 1st question: Are you asking for the description of the group of self-homeomorphisms of the triangulated surface? Then it is a finite split extension of the direct product of finitely many copies of $Homeo(S^1;T)$ where $T\subset S^1$ is a 3-element subset. (The types of finite extensions are given by finite subgroups of $O(3)$.) I do not think one can get much beyond this description. $\endgroup$ – Misha Feb 11 '17 at 4:28
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    $\begingroup$ Prof Laudenbach: I might be wrong, but it looks like these are two independent questions. Might I suggest to ask them separately, and maybe provide a bit more context for the first one? $\endgroup$ – Marco Golla Feb 11 '17 at 14:23