Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 112 / 5 / a

A framing of an embedded sphere $C:S^{k-1}\hookrightarrow N^{n-1}$ is a trivialization of its rank-$(n-k)$ normal bundle. The standard complex line ${\mathbb{CP}}^1\subset {\mathbb{CP}}^2$ has normal bundle of Euler number $+1$, so this embedded $S^2$ has no framing.

If one framing $f_0$ exists, every other orientation-compatible framing is obtained from it by a map $S^{k-1}\to SO(n-k)$. Consequently the set of homotopy classes is a torsor forFor $k=2$ and $n=4$, this is ${\pi }_1(SO(2))\cong \mathbb{Z}$; the integer is the winding number of one framing relative to $f_0$.