Pure boundary-fixed mapping classes

Let $n\in \mathbb{N}$ and let $D^2$, the base configuration $Q_n=(q_1,\dots ,q_n)$ and the boundary-fixed mapping class group $\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ be as in Boundary-fixed mapping class group of a punctured disk. While that group allows its elements to permute the marked points, the present definition records the classes that fix them.

The pointwise stabilizer. Write

${\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2;{\widehat{Q}}_n):=\{f\in {\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2):f(q_i)=q_i\text{ for every }i\}$

for the set of homeomorphisms of $D^2$ that fix $\mathrm{\partial }{D}^2$ pointwise and fix every marked point. It is a subgroup of ${\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2)$: the identity fixes every $q_i$, the composite of two such homeomorphisms fixes every $q_i$, and the inverse of such a homeomorphism fixes every $q_i$; it is also contained in the setwise stabilizer of $Q_n$, because fixing each point preserves the set. It carries the subspace topology of the compact-open topology on ${\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2)$, and its group operations are continuous there.

The pure mapping class group. The pure boundary-fixed mapping class group of the punctured disc is

$\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2):={\pi }_0({\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2;{\widehat{Q}}_n)),$

the set of path components of the pointwise stabilizer. A path $s\mapsto f_s$ in the pointwise stabilizer is exactly a continuous $H:D^2\times I\to D^2$ with every $H(-,s)$ a homeomorphism fixing $\mathrm{\partial }{D}^2$ and every marked point, that is, an isotopy rel $\mathrm{\partial }{D}^2$ that fixes each $q_i$ for all times; two elements of the subgroup are isotopic in this sense exactly when they lie in the same component. Composition of representatives descends to $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$, since the pointwise stabilizer is a topological group, so $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ is a group with the same product $[f][g]=[f\circ g]$ and identity $[{\mathrm{id}}_{D^2}]$ as $\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$.

Comparison with the setwise group. The inclusion ${\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2;{\widehat{Q}}_n)\hookrightarrow {\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2;Q_n)$ induces a map $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)\to \mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$, and this map is injective: if $f,g$ both fix every $q_i$ and a path in the setwise stabilizer joins them, then the permutation of the finite set $\{q_1,\dots ,q_n\}$ induced by the time-$s$ homeomorphism is a locally constant function of $s$ (the permutation is a discrete-valued continuous function of $s$ because each strand $s\mapsto f_s(q_i)$ is continuous and lands in the finite discrete set $\{q_1,\dots ,q_n\}$), hence constant, so the path lies in the pointwise stabilizer. Thus $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ is identified with the subgroup of $\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ consisting of the classes with trivial permutation of $Q_n$, and this identification is used throughout the pair. In particular, for $n=0$ and $n=1$ every boundary-fixed class is pure, because the setwise and pointwise stabilizers coincide.

Relation to the punctured disc. A homeomorphism fixing $\mathrm{\partial }{D}^2$ pointwise and every $q_i$ restricts to a homeomorphism of $D^2\setminus Q_n$. In this convention the punctured-disc isotopies are restrictions of continuous ambient isotopies fixing $\mathrm{\partial }{D}^2$ pointwise and every marked point at every time. Thus their ambient extensions are paths in the pointwise stabilizer, and conversely every such path restricts to an isotopy with these conditions. Allowing boundary rotation gives a different isotopy relation and is excluded. This is the convention for $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ throughout the pair.

• The notation ${\widehat{Q}}_n$ is a reminder that each marked point is fixed individually; the setwise stabilizer is written with plain $Q_n$.
• Fixing every marked point throughout the isotopy is a strictly stronger requirement than fixing the set $\{q_1,\dots ,q_n\}$ throughout; the companion page's setwise-puncture counterexample exhibits the difference at the level of classes, and the comparison just recorded says that even at the level of paths the two conventions differ exactly by the permutation.
• No orientation condition is imposed separately: every element of the boundary-fixed group lies in the identity component of the full homeomorphism group of the disc, by Alexander contraction of the boundary-fixed disk homeomorphism group.