Special-unitary column fibration

Fix $v_0\in S^{2N-1}$. Choose an orthonormal basis $w_1,\dots ,w_{N-1}$ of $v_0^{\perp }$ so the matrix $(w_1,\dots ,w_{N-1},v_0)$ has determinant $1$. For $v$ in a sufficiently small neighborhood $U$ of $v_0$, project each $w_i$ onto $v^{\perp }$ and apply Gram–Schmidt in that fixed order. The projected vectors remain linearly independent after shrinking $U$, because their Gram determinant is continuous and equals $1$ at $v_0$. The resulting orthonormal frame depends smoothly on $v$. Multiply its first vector by the inverse of the determinant of the frame if necessary; this scalar has modulus one and the adjustment is smooth, preserves orthonormality, and makes the determinant exactly one. Thus $s(v)\in SU(N)$ is a smooth local section of $p$ on $U$.

For $g\in p^{-1}(U)$ with last column $v$, the matrix $s(v{)}^{-1}g$ fixes the last basis vector. Unitarity makes its orthogonal complement invariant, so it is block diagonal $\mathrm{diag}(A,1)$; determinant one gives $A\in SU(N-1)$. The maps $(A,v)⟼s(v)\mathrm{diag}(A,1),g⟼(\mathrm{upperBlock}(s(p(g){)}^{-1}g),p(g))$ are smooth inverses between $SU(N-1)\times U$ and $p^{-1}(U)$. This proves local triviality with the asserted fiber.

The sphere is compact, so finitely many such neighborhoods cover it; [F2] gives a smooth partition of unity and hence a numeration. Applying [F1] proves the bundle is a Hurewicz fibration. For $N=2$, the stabilizer of the last column in $SU(2)$ has its upper block in $SU(1)=\{1\}$, so the fiber is a point.