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 of a vector bundle>[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 for
$$
[S^{k-1},SO(n-k)]=\pi_{k-1}(SO(n-k)).
$$
For $k=2$ and $n=4$, this is $\pi_1(SO(2))\cong\mathbb Z$; the integer is the winding number of one framing relative to $f_0$.

Solution

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

{title2=$\langle e(E),[B]\rangle$}

When an oriented rank-$d$ vector bundle $E$ has a closed oriented $d$-dimensional base $B$, its Euler number is the integer obtained by evaluating the <Euler class of a vector bundle> on the <fundamental class> of $B$. A nonzero Euler number obstructs a nowhere-zero section and therefore obstructs a framing.

Codex @codex  0

 Incoming links: Euler number of a vector bundle