Point pushing the last puncture

Assume the Axiom of Choice, let $n\ge 1$, and let $D^2$, $\mathrm{int}{D}^2$ and the base configuration $Q_n=(q_1,\dots ,q_n)$ be as in Boundary-fixed mapping class group of a punctured disk. Point pushing holds the first $n-1$ punctures fixed and moves the last one around them.

The puncture complement. Put

$Y_n:=\mathrm{int}{D}^2\setminus \{q_1,\dots ,q_{n-1}\};$

for $n=1$ the removed set is empty and $Y_1=\mathrm{int}{D}^2$. The domain $Y_n$ contains $q_n$, because $q_n$ is distinct from $q_1,\dots ,q_{n-1}$, and every point of $Y_n$ is distinct from the first $n-1$ marked points.

The ordered loop and its orbit. Let $\gamma :I\to Y_n$ be a based loop of $Y_n$ at $q_n$, that is, a continuous map with $\gamma (0)=q_n=\gamma (1)$. Its ordered lift is

$L_{\gamma }:I⟶F_n(\mathrm{int}{D}^2),L_{\gamma }(t):=(q_1,\dots ,q_{n-1},\gamma (t))\mathrm{.}$

The $n$ coordinates of $L_{\gamma }(t)$ are pairwise distinct because $\gamma (t)\mathrm{\ne }{q}_j$ for $j<n$ and the $q_j$ are pairwise distinct, so $L_{\gamma }$ takes values in the ordered configuration space (Ordered configuration spaces $F_n(X)$); it is continuous, being built from constant maps and $\gamma$, and $L_{\gamma }(0)=Q_n=L_{\gamma }(1)$. Composing with the quotient map $p_n:F_n(\mathrm{int}{D}^2)\to C_n(\mathrm{int}{D}^2)$ gives the based loop

$\bar{\gamma }:=p_n\circ L_{\gamma }:I⟶C_n(\mathrm{int}{D}^2),\bar{\gamma }(t)=[(q_1,\dots ,q_{n-1},\gamma (t))],$

of the unordered configuration space at the basepoint $[Q_n]$ (Unordered configuration spaces $C_n(X)$).

The point-pushing class. Let $\delta$ be the inverse-endpoint boundary map of Boundary map from point motions. The point-pushing homomorphism at the last puncture is defined on the path-homotopy class $[\gamma ]$ of a based loop $\gamma$ at $q_n$ by

${\mathrm{Push}}_n([\gamma ]):=\delta ([\bar{\gamma }])\in \mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)\mathrm{.}$

The class is pure. The tuple $L_{\gamma }$ is a pure geometric braid based at $Q_n$: it is a tuple of continuous paths in $\mathrm{int}{D}^2$ with pairwise distinct values and both the initial tuple $L_{\gamma }(0)$ and the terminal tuple $L_{\gamma }(1)$ equal to $Q_n$ (Geometric braids in the disc with setwise endpoints), and its raw slice is $S(L_{\gamma })=p_n\circ L_{\gamma }=\bar{\gamma }$ (Geometric braid classes and the unordered configuration fundamental group). Let $\mathrm{\Psi }$ be the braid-to-mapping-class isomorphism of Braid group as boundary-fixed punctured-disk mapping classes, so that $\mathrm{\Psi }=\delta \circ ({\iota }_{\ast }^C{)}^{-1}\circ \mathrm{\Phi }$ with $\mathrm{\Phi }([\beta ])=({\iota }_{\ast }^C[S(\beta )]{)}^{-1}$ precomposed with the inverse of the open-to-closed configuration isomorphism; then

$\mathrm{\Psi }([L_{\gamma }])=\delta ([S(L_{\gamma }){]}^{-1})=\delta ([\bar{\gamma }]{)}^{-1}={\mathrm{Push}}_n([\gamma ]{)}^{-1}\mathrm{.}$

Since $L_{\gamma }$ is pure, $\mathrm{\Psi }([L_{\gamma }])$ lies in $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ by Pure braids as pure mapping classes, and $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ is a subgroup of $\mathrm{Mod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ (Pure boundary-fixed mapping classes), so its inverse ${\mathrm{Push}}_n([\gamma ])$ lies in $\mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ as well. Thus the point push of the last puncture is a pure mapping class: it fixes the first $n-1$ punctures and acts trivially on the marked set. In particular the fixed coordinates do not merely preserve $Q_n$ setwise; they prevent any exchange of punctures.

Well-definedness. The value is independent of the representative of $[\gamma ]$: if $H:I\times I\to Y_n$ is a path homotopy relative to $\{0,1\}$ from $\gamma$ to a second based loop ${\gamma }^{\mathrm{\prime }}$, then $(t,u)\mapsto (q_1,\dots ,q_{n-1},H(t,u))$ is a path homotopy relative to $\{0,1\}$ in $F_n(\mathrm{int}{D}^2)$ from $L_{\gamma }$ to $L_{{\gamma }^{\mathrm{\prime }}}$, and composing it with the continuous map $p_n$ gives a path homotopy relative to $\{0,1\}$ from $\bar{\gamma }$ to ${\bar{\gamma }}^{\mathrm{\prime }}$: the composite of a continuous homotopy with a continuous map is continuous and it is constant on $\{0,1\}\times I$ because $H$ is. Since $\delta$ is well defined on path-homotopy classes (Point-motion boundary map is a homomorphism), the class ${\mathrm{Push}}_n([\gamma ])$ depends only on $[\gamma ]$.

Multiplicativity. Let $\gamma ,{\gamma }^{\mathrm{\prime }}$ be based loops at $q_n$ and let $\gamma \ast {\gamma }^{\mathrm{\prime }}$ be their concatenation, traversed first $\gamma$ then ${\gamma }^{\mathrm{\prime }}$ (Based loops and the fundamental group). Since concatenation is computed coordinatewise, $L_{\gamma \ast {\gamma }^{\mathrm{\prime }}}(t)=L_{\gamma }(2t)$ for $t\le \frac{1}{2}$ and $L_{\gamma \ast {\gamma }^{\mathrm{\prime }}}(t)=L_{{\gamma }^{\mathrm{\prime }}}(2t-1)$ for $t\ge \frac{1}{2}$, that is, $L_{\gamma \ast {\gamma }^{\mathrm{\prime }}}=L_{\gamma }\ast L_{{\gamma }^{\mathrm{\prime }}}$; applying the continuous map $p_n$ gives $\overline{\gamma \ast {\gamma }^{\mathrm{\prime }}}=\bar{\gamma }\ast {\bar{\gamma }}^{\mathrm{\prime }}$. Hence, by the multiplicativity of $\delta$ (Point-motion boundary map is a homomorphism) and the product convention $[\gamma ][{\gamma }^{\mathrm{\prime }}]=[\gamma \ast {\gamma }^{\mathrm{\prime }}]$ of Based loops and the fundamental group,

${\mathrm{Push}}_n([\gamma ][{\gamma }^{\mathrm{\prime }}])=\delta ([\overline{\gamma \ast {\gamma }^{\mathrm{\prime }}}])=\delta ([\bar{\gamma }][{\bar{\gamma }}^{\mathrm{\prime }}])=\delta ([\bar{\gamma }])\delta ([{\bar{\gamma }}^{\mathrm{\prime }}])={\mathrm{Push}}_n([\gamma ]){\mathrm{Push}}_n([{\gamma }^{\mathrm{\prime }}])\mathrm{.}$

So ${\mathrm{Push}}_n:{\pi }_1(Y_n,q_n)\to \mathrm{PMod}(D^2,Q_n;\mathrm{\partial }{D}^2)$ is a group homomorphism. No choice is made in the definition itself: the lift of $\bar{\gamma }$ used to evaluate $\delta$ is supplied by the evaluation fibration, and its endpoint component is independent of the lift by the cited well-definedness lemma. The Axiom of Choice enters only through the evaluation fibration and the isomorphisms $\mathrm{\Psi }$ and $\delta$ built from it.

No injectivity claim. The kernel of ${\mathrm{Push}}_n$ is not computed here. The Birman exact sequence and the resulting injectivity claim are deferred to the companion pure-braid page. This item only defines ${\mathrm{Push}}_n$ and proves that it is a homomorphism into the pure subgroup.

Elementary case. For $n=1$ the domain $Y_1=\mathrm{int}{D}^2$ carries no puncture, the first $n-1$ coordinates are absent, the construction above applies verbatim, and $\mathrm{PMod}(D^2,Q_1;\mathrm{\partial }{D}^2)=\mathrm{Mod}(D^2,Q_1;\mathrm{\partial }{D}^2)$ because the setwise and pointwise stabilisers of a one-point marked set coincide (Pure boundary-fixed mapping classes); no injectivity is claimed in this case either.

• The homomorphism pushes the $n$-th puncture along loops in the complement of the other $n-1$ punctures. The first $n-1$ points are frozen throughout, so the resulting ambient isotopy moves only the last point among the marked points and represents a pure class, even though the definition itself only records the unordered loop.
• The definition is the disk boundary-fixed version of the classical point-pushing construction: the evaluation boundary map plays the role of the connecting homomorphism, and the inverse-endpoint convention of the library is what makes ${\mathrm{Push}}_n$ a homomorphism rather than an anti-homomorphism.
• The deferred injectivity is exactly the content of Birman's exact sequence for the disk, and it is stated on the pure-braid page after the relevant higher homotopy group has been shown to vanish; the present item must not be used as if it already contained that theorem.