in the interior of the surface bounded by $\alpha$. Then the surface bounded by $\alpha$ and $A$ must be an annulus. From here one can produce an arc connecting the base point 0 to a point in $A$. Together with $A$, this provides the lollipop.
Finally, we argue that $g$ is injective. Suppose $L_1$ and $L_2$ are two lollipops such that $g(L_1)$ and $g(L_2)$ are isotopic, denote the isotopy by $f$. By the isotopy extension theorem (see for example [FM11, Proposition 1.11]) there exists an isotopy $F: D_{d,r}^{\infty} \times [0, 1] \rightarrow D_{d,r}^{\infty}$ such that $F|{D{d,r}^{\infty} \times 0} = id_{D_{d,r}^{\infty}}$ and $F|{g(L_1) \times [0,1]} = f$. In particular $F|{D_{d,r}^{\infty} \times 1}$ maps the almost admissible loop $g(L_1)$ to the almost admissible loop $g(L_2)$. Hence $L_1$ is isotoped through $F$ to a lollipop which lies in a small neighborhood of $L_2$ and is bounded by the loop $g(L_2)$. Therefore, one can then isotope $L_1$ to $L_2$. $\square$
We now have the following definition of lollipop complex.
Definition 4.21. The lollipop complex $L_r^\infty(D, D_{d,1}^\infty)$ has vertices as lollipops, and $p+1$ lollipops, $L_0, L_1, \dots, L_p$, form a $p$-simplex if they are pairwise disjoint outside the base point 0 and there exists at least one admissible loop which does not lie inside the disks bounded by the $L_i$s.
The following lemma is immediate from Lemma 4.20.
Lemma 4.22. The complex $L_r^\infty(D, D_{d,1}^\infty)$ is isomorphic to $T_r^\infty(D, D_{d,1}^\infty)$ as a simplicial complex.
Lemma 4.23. Given a $p$-simplex $\sigma$ in $L_r^\infty(D, D_{d,1}^\infty)$, its link $Lk(\sigma)$ is isomorphic to $L_{r_\sigma}^\infty(D, D_{d,1}^\infty)$ for some $r_\sigma > 0$.
Proof. By Lemma 4.22, we can just prove the lemma for $T_r^\infty(D, D_{d,1}^\infty)$. Let $\alpha_0, \alpha_1, \dots, \alpha_p$ be the vertices of $\sigma$ which are almost admissible loops. Up to isotopy, we can assume they are pairwise disjoint except at the basepoint 0. Now let $C$ be the complement surface of $\sigma$, whose based boundary is the concatenation of $\alpha_p, \alpha_{p-1}, \dots, \alpha_0$ and $\partial D$. The surface $C$ has a naturally induced $d$-rigid structure. In particular, $C$ is asymptotically rigidly homeomorphic to $D_{d,r_\sigma}^\infty$ for some $r_\sigma > 0$. Thus link $Lk(\sigma)$ is isomorphic to $T_{r_\sigma}^\infty(D, D_{d,1}^\infty)$. $\square$
Let us summarize the relationships we have so far between our various complexes by the following diagram.
In Proposition 4.27, we will deduce the connectivity of $U_r^\infty(D, D_{d,1}^\infty)$ using the connectivity of the lollipop complex $L_r^\infty(D, D_{d,1}^\infty)$ by applying a bad simplices argument. Our goal now is to show that $L_r^\infty(D, D_{d,1}^\infty)$ is highly connected. Let us make some definitions first.
Definition 4.24. Given any lollipop $L: (A, 0) \rightarrow (D_{d,r}^\infty, 0)$, we define the free height $\mathfrak{h}L$ to be the minimal number $m$ such that $L([1, 2])$ is contained in $D{d,r,m}$ up to free isotopy. We also define the height of an admissible loop to be the minimal number $m$ such that it is contained in $D_{d,r,m}$.
To analyze the connectivity of $L_r^\infty(D, D_{d,1}^\infty)$, we need the following lemma which is a direct translation of [SW19, Lemma 3.8].
Lemma 4.25. For any $r,p,N \ge 1$, there exists a number $\mathfrak{h}{r,p,N} \ge 0$, such that for any $p$-simplex $\sigma$ in $L_r^\infty(D, D{d,1}^\infty)$, and any $\mathfrak{h} \ge \mathfrak{h}{r,p,N}$, there are at least $N$ lollipops of free height $\mathfrak{h}$ in $L_r^\infty(D, D{d,1}^\infty)$ that are in $Lk(\sigma)$.