The two-point configuration space of a punctured disk

Let $D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\},\mathrm{int}D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\},$ let $n\ge 1$, and let $P=\{p_1,\dots ,p_n\}\subset \mathrm{int}D$ be a set of $n$ distinct points, called punctures. Throughout the Lawrence-Krammer-Bigelow page the punctures are normally taken on the real axis with $-1<p_1<\cdots <p_n<1$, as the standard configuration.

The two-point configuration space of the punctured disk is the unordered configuration space $C:=C_2(D\setminus P)=\{\{x,y\}:x,y\in D\setminus P,x\mathrm{\ne }y\}$ of Unordered configuration spaces $C_n(X)$. Its elements are written $\{x,y\}$ for the orbit of the ordered pair $(x,y)$. The basepoint is $c_0:=\{d_1,d_2\},$ where $d_1,d_2\in \mathrm{\partial }D$ are two distinct points of the lower arc of $\mathrm{\partial }D$, specified once and for all together with the standard configuration and with the following convention: $d_1$ lies to the left of $d_2$ and the arc of $\mathrm{\partial }D$ from $d_1$ to $d_2$ passing through $-i$ is the lower arc used in every noodle construction on this page. Thus $c_0\in C$ is a genuine basepoint.

Action of the boundary-fixed mapping class group. Let $\mathrm{Mod}(D,P;\mathrm{\partial }D)$ be the boundary-fixed mapping class group of Boundary-fixed mapping class group of a punctured disk: isotopy classes relative to $\mathrm{\partial }D\cup P$ of homeomorphisms $h:D\to D$ with $h{\mathrm{\mid }}_{\mathrm{\partial }D}=\mathrm{id}$ and $h(P)=P$. Every such homeomorphism sends $D\setminus P$ to itself, commutes with the interchange $(x,y)\mapsto (y,x)$, and therefore induces a homeomorphism of $C$. Since $h$ fixes $\mathrm{\partial }D$ pointwise, it fixes $d_1$ and $d_2$ and hence fixes $c_0$. These representative homeomorphisms act literally on $C$. An isotopy $h_t$ relative to the outer boundary and the marked set gives the continuous based homotopy $\{x,y\}\mapsto \{h_t(x),h_t(y)\}$, fixing $c_0$ throughout. Thus mapping classes act on based homotopy classes of self-maps, and induce well-defined actions on ${\pi }_1(C,c_0)$ and homology; no literal action of a mapping class on the individual points of $C$ is asserted.

Elementary properties. $C$ is path-connected and locally path-connected. Indeed, $D\setminus P$ has disk neighborhoods at interior points and relative half-disk neighborhoods at boundary points; choose them small enough to miss $P$. It is not an open subset of the plane. Polygonal paths with finite puncture detours give path connectivity. To join two ordered configurations, choose two distinct interior buffer points different from the four prescribed mobile positions and $P$. Move the first point to its buffer avoiding the stationary second point, then the second to its buffer avoiding the first; move the first to its target and finally the second to its target, with the same finite-point detours. The distinct buffer choices prevent an occupied target during each stage. Passing to unordered pairs gives path connectivity. Local path-connectedness is inherited from $(D\setminus P{)}^2$ through the two-to-one quotient map $(x,y)\mapsto \{x,y\}$: a small product neighbourhood of $(x,y)$ maps onto a neighbourhood of $\{x,y\}$, and the only non-injectivity is the interchange of the two coordinates. These conventions fix the space, basepoint and action used by the covering homomorphism and the LKB pairing below.