Subsection 2.3. (0N5V) · ScalingStacks

0N3C

Definition 2.14. $\mfld_{n}^{\partial}$ is the symmetric monoidal topological category of topological $n$ -manifolds, possibly with boundary, which have finite good covers by Euclidean spaces $\mathbb{R}^{n}$ and upper half spaces $\mathbb{R}_{\geq 0}\times\mathbb{R}^{n-1}$ . Morphisms are open embeddings which map boundary to boundary. $\disk_{n}^{\partial}$ is the full symmetric monoidal topological subcategory of $\mfld_{n}^{\partial}$ consisting of finite disjoint unions of $\mathbb{R}^{n}$ and $\mathbb{R}_{\geq 0}\times\mathbb{R}^{n-1}$ .

0N3D

Remark 2.15. The category $\disk_{n}^{\partial}$ is designed to be minimal with respect to the condition that any finite subset of an $n$ -manifold with boundary has an open neighborhood homeomorphic to an object of $\disk_{n}^{\partial}$ . In particular, the closed $n$ -disk $\mathbb{D}^{n}$ is consequently not an object of $\disk_{n}^{\partial}$ .

0N3E

Proposition 2.16. The functor

\mathbb{R}_{\geq 0}\times-:\mfld_{n-1}\longrightarrow\mfld_{n}^{\partial}

is homotopically fully faithful. That is, for every pair of $(n-1)$ -manifolds $M$ and $N$ , the map

\Emb(M,N)\longrightarrow\Emb\bigl(\mathbb{R}_{\geq 0}\times M,\mathbb{R}_{\geq 0}\times N\bigr)

is a homotopy equivalence.

0N3F

Proof. This follows by the standard method of pushing off to infinity in the $\mathbb{R}_{\geq 0}$ direction (as in the Alexander trick or the contractibility of foliations on $\mathbb{R}^{n}$ up to integrable homotopy). That is, define a deformation retraction onto the subspace $\Emb(M,N)$ by defining for each $t\in[0,1]$ the map

h_{t}:\Emb\bigl(\mathbb{R}_{\geq 0}\times M,\mathbb{R}_{\geq 0}\times N\bigr)\longrightarrow\Emb\bigl(\mathbb{R}_{\geq 0}\times M,\mathbb{R}_{\geq 0}\times N\bigr)

h_{t}(g)(s,x)=\left\{\begin{array}[]{l l}(s,g_{0}(x))&\quad\text{for $s<(1-t)^{-1}-1$}\\ g(s+1-(1-t)^{-1},x)&\quad\text{for $s\geq(1-t)^{-1}-1$}\end{array}\right.

where $g_{0}:M\hookrightarrow N$ is the restriction of $g$ at the value $s=0$ .

∎

0N3G

Remark 2.17. Together with the Kister–Mazur Theorem [Ki], the previous proposition implies that the map

\Top(n-1)\hookrightarrow\Emb(\mathbb{R}_{\geq 0}\times\mathbb{R}^{n-1},\mathbb{R}_{\geq 0}\times\mathbb{R}^{n-1})

is a homotopy equivalence. Likewise, $\disk^{\partial}_{n}$ is an unoriented variant of the Swiss cheese operad of Voronov [Vo]. Namely, the framed variant $\disk_{n}^{\partial,\fr}$ is homotopy equivalent to the PROP associated to the Swiss cheese operad.

2.3. Manifolds with boundary