ScalingStacks

Example 2.2. Consider the composite continuous homomorphism

$${\sf Top}(n)\xrightarrow{~(-)^{+}~}{\sf Aut}_{\ast}(S^{n})\xrightarrow{~\pi_{0}~}\mathbb{Z}/2\mathbb{Z}$$

given by applying 1-point compactification to obtain based homotopy automorphisms of a sphere followed by taking path components. For ${\sf STop}(n)\subset{\sf Top}(n)$ the kernel of this homomorphism, a ${\sf BSTop}(n)$-framing on a topological $n$-manifold is precisely an orientation.