Subsection 2.2. (0N5U) · ScalingStacks

2.2. Disks

We now consider $B$ -framed $n$ -disks. In terms of configuration spaces, we identify the maximal $\infty$ -subgroupoid of $\disk^{B}_{n/M}$ , $B$ -framed $n$ -disks embedding into a $B$ -framed $n$ -manifold.

0N35

Definition 2.9. The symmetric monoidal $\oo$ -category $\disk^{B}_{n}$ is the full $\oo$ -subcategory of $\mfld^{B}_{n}$ whose objects are disjoint unions of $B$ -framed $n$ -dimensional Euclidean spaces.

0N36

Remark 2.10. Consider $\ast\rightarrow\BTop(n)$ , the basepoint of $\BTop(n)$ . A $\ast$ -structure on an $n$ -manifold $M$ is then equivalent to a topological framing of the tangent microbundle $\tau_{M}$ of $M$ ,⁴⁴ 4 By smoothing theory, framed topological manifolds are essentially equivalent to framed smooth manifolds except in dimension 4. and we denote the associated $\oo$ -category of framed $n$ -disks as $\disk^{\fr}_{n}$ . This symmetric monoidal $\infty$ -category $\disk_{n}^{\fr}$ is homotopy equivalent to the PROP associated to the $\mathcal{E}_{n}$ operad of Boardman-Vogt [BV]. This follows because the inclusion of rectilinear embeddings as framed embeddings determines a homotopy equivalence $\mathcal{E}_{n}(I)\xrightarrow{\sim}\Emb^{\sf fr}(\bigsqcup_{I}\mathbb{R}^{n},\mathbb{R}^{n})$ from the $I$ -ary space of the $\mathcal{E}_{n}$ operad.

0N37

Example 2.11. For $B=\BO(n)$ , with the usual map $\BO(n)\rightarrow\BTop(n)$ , the $\oo$ -category of topological $n$ -disks with $\BO(n)$ -framings is equivalent to the $\oo$ -category of smooth $n$ -disks and smooth embeddings, $\disk_{n}^{\sm}\simeq\disk_{n}^{{\sf BO}(n)}$ . These are both equivalent to the PROP associated to the unoriented version of the ribbon, or “framed,” $\mathcal{E}_{n}$ operad; see [SW] for a treatment of this operad.⁵⁵ 5 The historical use of “framed” here is potentially misleading, since in the “framed” $\mathcal{E}_{n}$ operad the embeddings do not preserve the framing, while in the usual $\mathcal{E}_{n}$ operad the embeddings do preserve the framing (up to scale). It might lead to less confusion to replace the term “framed $\mathcal{E}_{n}$ operad” with “unoriented $\mathcal{E}_{n}$ operad.” To see these equivalences it is enough to explain why each of the natural maps $\mathsf{O}(n)\rightarrow\Emb^{\sm}(\mathbb{R}^{n},\mathbb{R}^{n})\rightarrow\Emb^{{\sf BO}(n)}(\mathbb{R}^{n},\mathbb{R}^{n})$ is an equivalence. Smoothing theory ([KS]) gives the equivalence of the second map. Via Gram–Schmidt, the inclusion $\mathsf{O}(n)\xrightarrow{\simeq}{\sf GL}(\mathbb{R}^{n})$ is a deformation retraction. Conjugation by scaling and translation, $(f,t)\mapsto\bigl(x\mapsto\frac{f(tx)-f(0)}{t}+f(0)\bigr)$ , demonstrates the inclusion ${\sf GL}(\mathbb{R}^{n})\xrightarrow{\simeq}\Emb^{\sf sm}(\mathbb{R}^{n},\mathbb{R}^{n})$ as a deformation retraction.

Given a topological space $X$ and a finite cardinality $i$ , we let $\conf_{i}(X)\subset X^{i}$ denote the subspace of those maps $\{1,\dots,i\}\to X$ which are injective. This configuration space has an evident free action of the symmetric group $\Sigma_{i}$ .

In the next result, for $M$ a $B$ -framed $n$ -manifold, we consider the over $\infty$ -category

\disk_{n/M}^{B}~:=~\disk^{B}_{n}\underset{\mfld_{n}^{B}}{\times}\mfld^{B}_{n/M}~.

Informally, an object is an embedding $\underset{i}{\sqcup}\mathbb{R}^{n}\hookrightarrow M$ for some $i$ .

0N38

Lemma 2.12. The maximal $\infty$ -subgroupoid of $\disk_{n}^{B}$ is canonically identified as the space

\underset{i\geq 0}{\coprod}{B}^{i}_{\Sigma_{i}}~\simeq~\bigl(\disk_{n}^{B}\bigr)^{\sim}

where the coproduct is indexed by finite cardinalities and each cofactor is the $\Sigma_{i}$ -homotopy coinvariants of the $i$ -fold product of the space $B$ . In particular, the symmetric monoidal functor $[-]\colon\disk_{n}^{B}\to\fin$ , given by taking sets of connected components of underlying manifolds, is conservative.

For $M$ a $B$ -framed $n$ -manifold, the maximal $\infty$ -subgroupoid of $\disk^{B}_{n/M}$ is canonically identified as the space

\underset{i\geq 0}{\coprod}\conf_{i}(M)_{\Sigma_{i}}~\simeq~\bigl(\disk^{B}_{n/M}\bigr)^{\sim}

where the coproduct is indexed by finite cardinalities, and each cofactor is an unordered configuration space.

0N39

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.

∎

We conclude this section by justifying the term tangent classifier for the functor $\mfld_{n}\xrightarrow{\tau}\spaces_{/\BTop(n)}$ of (1).

0N3A

Corollary 2.13. The value of the tangent classifier (1) on a topological $n$ -manifold $M$ is the map of spaces $M\xrightarrow{\tau_{M}}\BTop(n)$ classifying the tangent microbundle.

0N3B

Proof. Recognize $\mathcal{E}{\sf uc}_{n}\subset\disk_{n}$ as the full $\infty$ -subcategory consisting of the connected $n$ -manifolds. Specialize the second statement of Lemma 2.12 to $i=1$ to obtain an identification

M~\simeq~\mathcal{E}{\sf uc}_{n/M}~\simeq~\Emb(\mathbb{R}^{n},M)_{{\sf Top}(n)}\longrightarrow\BTop(n)

involving the homotopy ${\sf Top}(n)$ -coinvariants. Manifestly, this map of spaces agrees with the tangent classifier in the case $M=\mathbb{R}^{n}$ . By construction, this map is functorial in the argument $M\in\mfld_{n}$ . The general case follows. ∎

David Ayala, John Francis

Original source: arXiv:1206.5522v6