Corollary 2.13 (0N3A) · ScalingStacks

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. ∎