A normal summand of rank at least two realizes every framing-loop obstruction

Let $\gamma$ be an arbitrary based continuous loop in $SO(N)$. By relative Whitney approximation after a basepoint-preserving reparametrization constant on a basepoint arc, represent its class by a smooth based loop, still denoted $\gamma$. Taking the first $k$ columns gives a smooth loop $F_{\mathrm{\partial }}$ in $V_k({\mathbb{R}}^N)$. The earlier local Stiefel lemma supplies a continuous disk filling, since $N-k=r\ge 2$. Its proof also gives the explicit relative smoothing procedure: make the filling radial-constant on an outer annulus, extend it beyond the unit disk, and apply relative Whitney approximation in the embedded Stiefel manifold while fixing a closed exterior annulus. We thus obtain a smooth filling $F:D^2\to V_k({\mathbb{R}}^N)$ with exactly the prescribed boundary columns.

Put $P(z)=I_N-F(z)F(z{)}^T$. This is a smooth rank-$r$ orthogonal projection. Along $tz$ solve $U_z^{\mathrm{\prime }}=[{\dot{P}}_z,P_z]U_z$, $U_z(0)=I_N$. The linear ODE existence and parameter-dependence suppliers give a unique solution smooth in $z$ on the whole interval. The commutator is skew symmetric, so $U_z$ is orthogonal; the identities $[[{\dot{P}}_z,P_z],P_z]={\dot{P}}_z$ and uniqueness give $U_{z}P(0)U_z^T=P_z$. Transport an orthonormal basis of $\mathrm{im}P(0)$ to obtain a smooth complementary frame $C(z)$. Choose its initial orientation so $S(z)=(F(z),C(z))$ has determinant one at the centre. Its determinant is continuous and takes values in $\{1,-1\}$, so is one throughout the disk. At the selected boundary basepoint $z_{\ast }$, $F(z_{\ast })$ is the standard first $k$ columns and $S(z_{\ast })=\mathrm{diag}(I_k,B)$ for some $B\in SO(r)$. Replace $S$ by $S\mathrm{diag}(I_k,B^{-1})$; it remains a disk completion and now equals $I_N$ at $z_{\ast }$.

On the boundary write $\gamma =S\mathrm{diag}(I_k,h)$, where $h=C^T(\text{last }r\text{ columns of }\gamma )\in SO(r)$ uses the normalized complementary frame. Orthogonality and determinant one ensure $h\in SO(r)$, and normalization ensures $h(z_{\ast })=I_r$. The based loop $S{\mathrm{\mid }}_{\mathrm{\partial }{D}^2}$ extends over the disk by $S$, so is based-nullhomotopic: compose the disk extension with the contraction of the disk to $z_{\ast }$, which fixes $z_{\ast }$. Multiplication of based loops in a topological group gives their fundamental-group product, as seen from the square $(s,t)\mapsto g(s)h(t)$; equivalently multiply this based nullhomotopy by the fixed loop $\mathrm{diag}(I_k,h)$. Thus $[\gamma ]$ is the image of $[h]$, proving surjectivity. To kill a framing discrepancy, choose a loop in the summand representing its inverse class and reparametrize it to be constant outside the interior of one chosen boundary arc. Acting by this loop preserves the summand subspace, the other summand, and the corner frame values; the corrected loop is nullhomotopic and extends over the disk. This corrects an adjustable admissible frame and asserts no extension of every previously prescribed frame.