Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological group with respect to the compact open topology $\tau_C$.

What can be said about the connectedness of $(H(X),\tau_C)$?

I would really like to get some references regarding this question and/or the more general one: for a connected space $X$, what topologies (eg. compact-open,pointwise, uniform) make $C(X)$ connected?

These questions arose from this specific one:

How can I prove that the group $H_+(S^1)$ (orientation preserving homeomorphisms of the circle) is connected?

  • $\begingroup$ What is $C(X)$? $\endgroup$ – John Pardon Mar 9 '14 at 19:02
  • $\begingroup$ The specific question about connectedness of $H_+(S^1)$ is easy to treat explicitly: you can just write down a path between any homeo and the identity. $\endgroup$ – John Pardon Mar 9 '14 at 19:03
  • 2
    $\begingroup$ @Ludolila. Have you read about the mapping class group? Essentially the mapping class group of a space is the group of autohomeomorphisms modulo the subgroup of autohomeomorphisms isotopic to the identity. In particular, the mapping class group is the collection of all path components in $Aut(X)$. $\endgroup$ – Joseph Van Name Mar 9 '14 at 19:12
  • $\begingroup$ @JohnPardon $C(X)$-the space of all continuous functions on $X$. $\endgroup$ – Ludolila Mar 9 '14 at 19:43
  • $\begingroup$ @Ludolila: well then $C(X)$ is a vector space and thus contractible. $\endgroup$ – John Pardon Mar 9 '14 at 19:45

Regarding the last part of your question:

Let $f_{0}$ and $f_{1}$ be two orientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f_{0}$ and $f_{1}$ to homeomorphisms $F_{0}$ and $F_{1}$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number.

Then consider the quotient of the homotopy $F_{t}=tF_{0}+(1-t)F_{1}$ as a path of homeomorphism on $S^{1}$.

More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G_{t}=P \circ F_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G_{t}(x+1)=G_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f_{0}$ to $f_{1}$.

  • $\begingroup$ Thank you! The truth is, I am not very familiar with these methods, though I do encounter the notion of rotation number every once in a while. I thought that maybe there is a more set-theoretic topological approach? $\endgroup$ – Ludolila Mar 10 '14 at 8:34
  • $\begingroup$ @Ludolila In fact we do not need to "rotation number". We need only to lift a homeomorphism $f$ of $S^{1}$ to a homeomorphism $F$ of $R$. This statment is written in the first line after the definition of the above link $\endgroup$ – Ali Taghavi Mar 12 '14 at 4:49

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