Lemma 2.8 (0N33) · ScalingStacks

0N33

Lemma 2.8. A $B$ -framing $g$ of $\mathbb{R}^{n}$ determines a homotopy equivalence $\Emb^{B}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega_{g}B$ of topological monoids, where $\Omega_{g}B$ is the loop space of $B$ based at the homotopy point $g:\mathbb{R}^{n}\rightarrow B$ .

0N34

Proof. By definition, the space $\Emb^{B}(\mathbb{R}^{n},\mathbb{R}^{n})$ sits in a homotopy pullback square:

There are evident equivalences of spaces $\Map_{/\BTop(n)}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega\BTop(n)\simeq\Top(n)$ and likewise $\Map_{/\negthinspace B}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega_{g}B$ . It is standard that the the composite map of spaces

\Top(n)\longrightarrow\Emb(\mathbb{R}^{n},\mathbb{R}^{n})\longrightarrow\Map_{/\BTop(n)}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega\BTop(n)

is an equivalence. By Kister–Mazur, Theorem 2.3, the first map including $\Top(n)$ into $\Emb(\mathbb{R}^{n},\mathbb{R}^{n})$ is a homotopy equivalence. It follows that the middle map in the above display is also an equivalence of spaces. We conclude that the bottom horizontal map in the above homotopy pullback square is an equivalence of spaces. This implies the top horizontal map too is an equivalence of spaces.

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