ScalingStacks

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.