ScalingStacks

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.