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!
