Mapping class group

Remarks

For a surface with marked points or punctures, the homeomorphisms may permute the marked points, and the isotopies must keep them fixed. Some authors also allow orientation-reversing homeomorphisms; this gives the extended mapping class group, which contains $\mathrm{Mod}(S)$ with index $2$ when $S$ admits an orientation-reversing homeomorphism fixing $\partial S$.

Example: for the torus $T^2 = \R^2 / \Z^2$, the map $\mathrm{Mod}(T^2) \to SL(2, \Z)$ that records the action on the first homology group $H_1(T^2; \Z) \cong \Z^2$ is an isomorphism. A matrix $A \in SL(2, \Z)$ maps $\Z^2$ onto itself, so it induces a homeomorphism of $T^2$, and since $\det A = 1$ it preserves orientation.

Non-example: let $A = S^1 \times [0, 1]$ be the annulus, with points written $(\theta, t)$, and let $T(\theta, t) = (\theta + 2\pi t, t)$, a Dehn twist. The maps $H_s(\theta, t) = (\theta + 2\pi s t, t)$ for $s \in [0, 1]$ deform the identity into $T$, but for $0 < s < 1$ they rotate the boundary circle $S^1 \times \set{1}$, so they fail clause 1. In fact $T$ is not isotopic to the identity in the sense above, and $\mathrm{Mod}(A) \cong \Z$, generated by the class of $T$. For the closed disk, the Alexander lemma gives $\mathrm{Mod}(D^2) = 1$.