Proposition 2.16 (0N3E) · ScalingStacks

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 $s=0$ .

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