Currently I am reading a paper titled "On the Group of Homeomorphisms of an Arc" by N.J Fine and G.E. Schweigert, that was published by Annals of Mathematics in 1955. This paper talks about the group of homeomorphisms of the closed unit interval $[0,1]$, the different properties of this group in some detail and then briefly talk about the group of homeomorphisms of $S^{1}$ (circle).

My question is that why studying these group of homeomorphism of topological spaces are important? What kind of topological spaces can we take such that these corresponding groups has "nice properties". So, basically, one question is that what other papers or books can be read in this direction?

I know that these groups of homeomorphism of topological spaces can be given a topology such that they become topological groups provided the topological space concerned has some "nice properties". But since I don't know any theory regarding topological groups, it is difficult for me to ask question in that direction or understand what role these groups play in the theory of topological groups. But since these are infinite groups, I would like to ask what kind of role does these groups play in infinite group theory. Does these groups gives large class of examples of infinite simple groups(If the topological space is taken to be "nice" in some sense!)?

In summary what broad subject or area of mathematics does this paper and subsequent publications lead to? What I want is reference to books where I can find such queries that I have posted(if possible), or some motivating examples, or a certain small list of papers related to this paper that can be read.

Thank you!

  • $\begingroup$ Whoever gave a downvote can you please tell me what possible reason is there to give a downvote. Am I asking something that is not suitable for this site? If this is the case, do suggest me what can I change or do, to make it more suitable! $\endgroup$ – Riju Nov 29 '17 at 20:44
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    $\begingroup$ Voted to close as "too broad". Basically, you are asking for a survey. If you make the question more specific, it may be okay for MO. $\endgroup$ – Igor Belegradek Nov 29 '17 at 22:01
  • $\begingroup$ I made certain changes, I hope it's okay!! $\endgroup$ – Riju Nov 29 '17 at 22:08
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    $\begingroup$ @Riju: I did not find the question improved. Anyway, title search in mathscinet on "homeomorphism group", without quotation marks, returns 388 hits, and many of them are relevant to your inquiries. For example, look at arxiv.org/abs/1411.2868 which is now published as a book [R. Bieri and R. Strebel, On groups of PL-homeomorphisms of the real line. Mathematical Surveys and Monographs, 215. American Mathematical Society, Providence, RI, 2016, 174 pp]. $\endgroup$ – Igor Belegradek Nov 30 '17 at 0:10
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    $\begingroup$ Thompson's groups $T$ and $F$ can both be defined as groups of homeomorphisms (of $S^1$ and $[0,1]$, respectively). Both groups, but particularly the latter, are the source of a great deal of mystery and are studied widely by many people, especially with regard to proving (non-)amenability. This should give plenty of motivation as to why groups of homeomorphisms are worth studying... $\endgroup$ – Nick Gill Nov 30 '17 at 14:12