I know that the self homeomorphisms of a closed torus is disconnected, with connected component the group of translations, and component group the mapping class group, isomorphic to $SL(2,\mathbb{Z})$ (here everything is oriented/orientation-preserving). The self-homeomorphisms of a punctured torus should correspond to the self-homeomorphisms of the closed torus which fix the origin. By lifting to universal covers, any such self-homeomorphism lifts to an $\mathbb{R}$-linear self map of $\mathbb{R}^2$ which stabilizes the lattice, and hence again must be an element of $SL(2,\mathbb{Z})$.
Assuming that restriction gives a bijection between the group of self-homeomorphisms of a torus fixing the origin with the group of self-homeomorphisms of the torus with the origin removed, this seems to suggest that the group of self-homeomorphisms of a punctured torus is discrete and isomorphic to $SL(2,\mathbb{Z})$. Is this correct?
If this is not true, can someone give an example of a nontrivial homeomorphism of the punctured torus which is homotopic to the identity?
Best Answer
Your conclusion isn't true. Here is an example of a map homotopic to the identity:
Pick a disk around the punctured point and declare the map outside this disk to be the identity. Pick a chart so this is the unit disk. Then define the map inside the disk to rotate the points by $1-r$ where r is the distance to the origin.
A homotopy to the identity is given by $H(x,t)$ is the identity if $x$ is outside the disk and $H(x,t)$ rotates the points by $(1-t)(1-r)$.
I think the issue with your reasoning is your statement that it is $\mathbb{R}$-linear and stabilize the lattice. I can't see any reason either of these would be true.