Boundary map from point motions

Assume the Axiom of Choice. Let $D^2\subseteq {\mathbb{R}}^2$ be the closed unit disc, let $Q_n=(q_1,\dots ,q_n)$ be the fixed base configuration of Boundary-fixed mapping class group of a punctured disk, and let

$E:={\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2),B:=C_n(\mathrm{int}{D}^2),F:={\mathrm{Homeo}}^{+}(D^2,\mathrm{\partial }{D}^2;Q_n),$

with the compact-open topology on both homeomorphism groups and the quotient topology on the unordered configuration space. By Evaluation is a numerable bundle and Hurewicz fibration the evaluation map

$\mathrm{ev}:E⟶B,\mathrm{ev}(h):=[h(q_1),\dots ,h(q_n)],$

is a Hurewicz fibration whose fibre over the basepoint $[Q_n]$ is exactly $F$; by A fibration has path lifting and homotopy lifting relative to a subspace every path in $B$ with a prescribed initial point has a lift in $E$, and the fibre components of $F$ are the elements of

${\pi }_0(F)=\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2),$

by Boundary-fixed mapping class group of a punctured disk. Recall from Based loops and the fundamental group that a based loop is a continuous $\alpha :I\to B$ with $\alpha (0)=\alpha (1)=[Q_n]$, that $[\alpha ]$ denotes its path-homotopy class, and that the fundamental-group product is first loop then second.

The boundary map. Let $\alpha :I\to B$ be a based loop at $[Q_n]$. Choose a lift $\tilde{\alpha }:I\to E$ with $\tilde{\alpha }(0)=\mathrm{id}$ and $\mathrm{ev}\circ \tilde{\alpha }=\alpha$, and define the class

$\delta ([\alpha ]):=[\tilde{\alpha }(1{)}^{-1}]\in {\pi }_0(F)=\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)\mathrm{.}$

The element $\tilde{\alpha }(1)$ is the time-one homeomorphism of the lifted point motion: its inverse is what makes the assignment compatible with the library's first-loop-then-second product. The independence of the choice of the lift, the independence of the representative loop, and the multiplicativity of the resulting map are not assumed here; they are proved in the lemma lem-the-point-motion-boundary-map-is-a-well-defined-homomorphism, which follows this definition and whose statement is the precise well-definedness claim for $\delta$.

Why the inverse endpoint. The published exact sequence of a fibration Long exact sequence of homotopy groups of a fibration defines its boundary by

${\mathrm{\partial }}_p[\gamma ]=[e_0]\cdot [\gamma {]}^{-1},$

where $[e_0]\cdot [\gamma ]$ is the endpoint component of a lift of $\gamma$ starting at $e_0$ and products of loops are traversed left-to-right. For the evaluation fibration this is precisely $[\tilde{\alpha }(1{)}^{-1}]$, so $\delta$ is the connecting map of the fibration in the library's convention, and the inverse is not a convention that may be dropped: the raw endpoint assignment $[\alpha ]\mapsto [\tilde{\alpha }(1)]$ reverses the order of the first-then-second product, whereas $\delta$ preserves it.

Elementary cases. For $n=0$ the base $B$ is a single point, the only based loop is constant, and the formula gives the identity class of $\mathrm{Mod}(D^2,Q_0;\mathrm{\partial }{D}^2)$; for $n=1$ the same construction applies without a collision condition. Nothing in the definition selects among lifts, representatives or enumerations of a configuration: the lift is exhibited in the following lemma, and the class computed by $\delta$ is proved there to be independent of these choices.

• The definition uses the total space of all boundary-fixing homeomorphisms of the closed disc, not only the homeomorphisms supported away from $\mathrm{\partial }{D}^2$ near a fixed collar; the boundary circle is fixed pointwise, so every lift is an ambient isotopy rel $\mathrm{\partial }{D}^2$.
• The target is the setwise mapping class group $\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$: the formula produces the class of the inverse of the evaluated endpoint, and that class lies in the pure subgroup $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ exactly when the lift's endpoint permutes the marked points trivially. Purity is a property of the particular endpoint, not of the definition of $\delta$.