ScalingStacks

Proof. Lemma 2.5 gives an equivalence $\disk_{n/M}^{B}\simeq\disk_{n/M}$. So it suffices to assume the case of an equality $B=\BTop(n)$.

The maximal $\infty$-subgroupoid of $\disk_{n}$ necessarily lies over the maximal $\infty$-subgroupoid of $\fin$, which is $\underset{i\geq 0}{\coprod}\mathsf{B}\Sigma_{i}$. The first assertion will be implied upon verifying, for each $i\geq 0$, that the map

$${\Sigma_{i}\wr{\sf Top}(n)}\longrightarrow\Emb(\underset{i}{\sqcup}\mathbb{R}^{n},\underset{i}{\sqcup}\mathbb{R}^{n})$$

is weakly homotopy equivalent to an inclusion of connected components. This is an immediate consequence of the Kister–Mazur Theorem 2.3. Via translation, the inclusion of the subgroup of origin preserving homeomorphisms $\Top_{0}(n)\xrightarrow{\simeq}\Top(n)$ is a homotopy equivalence, and so we recognize further the identification

$$\underset{i\geq 0}{\coprod}\mathsf{B}(\Sigma_{i}\wr\Top_{0}(n))\xrightarrow{~\simeq~}(\disk_{n})^{\sim}~.$$

Because the projection $\disk_{n/M}\to\disk_{n}$ is a right fibration, we recognize the maximal $\infty$-subgroupoid of $\disk_{n/M}$ as

$$\underset{i\geq 0}{\coprod}\Emb\bigl(\underset{i}{\sqcup}\mathbb{R}^{n},M\bigr)_{\Sigma_{i}\wr\Top_{0}(n)}\xrightarrow{~\simeq~}(\disk_{n/M})^{\sim}~.$$

Therefore, the second assertion follows upon showing that the $\Sigma_{i}$-equivariant continuous map

$${\sf ev}_{0}\colon\Emb\bigl(\underset{i}{\sqcup}\mathbb{R}^{n},M\bigr)_{\Top_{0}(n)^{i}}\longrightarrow\conf_{i}(M)$$

is a weak homotopy equivalence. This is implied upon showing the homotopy fiber of the continuous map

$${\sf ev}_{0}\colon\Emb(\underset{i}{\sqcup}\mathbb{R}^{n},M)\longrightarrow\conf_{i}(M)$$

is weakly homotopy equivalent to $\Top_{0}(n)^{i}$. In a standard manner, this map is a Serre fibration, and the fiber over $c\colon\{1,\dots,i\}\hookrightarrow M$ is the space $\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$ of embeddings under $c$. Fix such a based embedding $e_{0}\colon\underset{i}{\sqcup}\mathbb{R}^{n}\hookrightarrow M$. So we must show that the composite inclusion

$$\Top_{0}(n)^{i}\hookrightarrow\Emb((0\in\mathbb{R}^{n}),(0\in\mathbb{R}^{n}))^{i}\cong\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},\underset{i}{\sqcup}\mathbb{R}^{n})\xrightarrow{-\circ e_{0}}\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$$

is a weak homotopy equivalence.

The Kister–Mazur Theorem gives that the first of these maps is a homotopy equivalence. The problem is thus reduced to showing that the second of these maps is a weak homotopy equivalence. This is the problem of showing that each solid diagram among topological spaces

admits a filler with respect to which the diagram commutes up to homotopy. Choose a continuous map $\phi\colon(0,1]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $\phi_{t}$ is an origin preserving open embedding for each $t$, $\phi_{1}={\sf id}_{\mathbb{R}^{n}}$, the closure $\overline{\phi_{s}(\mathbb{R}^{n})}\subset\phi_{t}(\mathbb{R}^{n})$ whenever $s<t$, and the collection of images $\{\phi_{t}(\mathbb{R}^{n})\mid 0<t\leq 1\}$ is a basis for the topology about $0\in\mathbb{R}^{n}$. Choose a continuous map $\mathbb{D}^{k}\xrightarrow{\epsilon}(0,1]$ for which the restriction $\epsilon_{|S^{k-1}}\equiv 1$ is identically one, and the composition

$$\mathbb{D}^{k}\xrightarrow{~f~}\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)\xrightarrow{~(\underset{i}{\sqcup}\phi_{\epsilon})^{\ast}~}\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$$

factors through $\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},\underset{i}{\sqcup}\mathbb{R}^{n})$. Define $\widetilde{f}$ to be this factorization. By construction, the restriction $\widetilde{f}_{|S^{k-1}}=f_{0}$. The map $[0,1]\times\mathbb{D}^{k}\to\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$ given by $(t,p)\mapsto f\circ(\underset{i}{\sqcup}\phi_{t\epsilon+(1-t)})^{\ast}(p)$ demonstrates a homotopy making the lower triangle commute.

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