Real Stiefel spaces with complement rank at least two are simply connected

The constraint map $\mathrm{\Phi }(F)=F^{T}F-I_k$ takes values in symmetric $k$-by-$k$ matrices. Its derivative at an orthonormal frame is $A\mapsto F^{T}A+A^{T}F$, which is onto: for symmetric $B$, choose $A=FB\mathrm{/}2$. Thus [F2] makes $V_k({\mathbb{R}}^N)$ an embedded smooth manifold. Every based continuous loop can be represented by a smooth based loop: first reparametrize it to be constant on an arc about the basepoint, using a degree-one circle reparametrization based-homotopic to the identity, then use [F3] relative to a closed smaller arc. A continuous disk filling a smooth boundary loop can likewise be made smooth while keeping its boundary values: first compress the original filling radially into a smaller disk and use the boundary loop, constant in the radial variable, on the remaining annulus. Extend this map outside the unit circle by the same radial-constant formula and apply [F3] on ${\mathbb{R}}^2$ relative to a closed exterior annulus contained in its smooth region. Restrict to the disk. The formula is smooth near the unit circle, so no manifold-with-boundary version of approximation is needed. These operations use [A1] only through [F3].

We record the needed explicit complement construction. Let $P(z)$ be any smooth family of orthogonal projections of fixed rank on a disk, and set $P_z(t)=P(tz)$ for $0\le t\le 1$. Define $A_z(t)={\dot{P}}_z(t)P_z(t)-P_z(t){\dot{P}}_z(t)$, and solve $U_z^{\mathrm{\prime }}=A_z{U}_z$, $U_z(0)=I_N$ using [F4]. The coefficient is skew symmetric, so differentiating $U_z^T{U}_z$ makes it constant, equal to $I_N$. Differentiating $P_z^2=P_z$ gives $P_z{\dot{P}}_z{P}_z=0$ and ${\dot{P}}_z{P}_z+P_z{\dot{P}}_z={\dot{P}}_z$, hence $[A_z,P_z]={\dot{P}}_z$. Therefore $Q_z=U_{z}P(0)U_z^T$ and $P_z$ solve the same linear matrix equation $Q_z^{\mathrm{\prime }}=[A_z,Q_z]$ with the same initial value. Uniqueness, applying [F4] in the coordinate array of matrices, gives $Q_z=P_z$. Consequently transporting an orthonormal basis of $\mathrm{im}P(0)$ by $U_z(1)$ gives an orthonormal basis of $\mathrm{im}P(z)$. These transported vectors depend smoothly on $z$ by [F4], including at $z=0$ since $P(tz)$ is jointly smooth there. Smoothness up to the disk boundary follows by extending the smooth projection a little beyond that boundary. This construction uses unique solutions, not a choice of a solution for each parameter.

For path connectivity, take two frames and extend each to an orthonormal basis of ${\mathbb{R}}^N$, choosing the last complementary vector so the full matrix has determinant one. Such a finite completion exists by ordinary finite-dimensional orthogonal-complement algebra. Every matrix in $SO(N)$ is joined to $I_N$ by plane rotations: rotate its first column to $e_1$ within the plane it spans with $e_1$, using an auxiliary coordinate direction in the antipodal case, then restrict to the orthogonal complement of $e_1$ and repeat. Each rotation has a continuous path from the identity, fixes the previously aligned columns, and has determinant one; at the final one-dimensional stage determinant one forces the last entry to be $1$. Concatenating these finitely many rotation paths joins the two full matrices, and their first $k$ columns give a path between the original frames. The standard frame proves nonemptiness.

Induct on $k$, simultaneously for all $N\ge k+2$. For $k=1$, the constraint space is $S^{N-1}$, so [F1] proves simple connectivity because $N-1\ge 2$. Suppose $k\ge 2$ and the result holds for $k-1$ in every allowed ambient dimension. By step 1.1 it suffices to fill a smooth loop $\gamma =(v_1,\dots ,v_k)$ in $V_k({\mathbb{R}}^N)$. Its first column is a smooth sphere loop, which bounds a continuous disk by [F1]. Make this filling smooth, with the same boundary values, by step 1.1, and denote it $f_1:D^2\to S^{N-1}$. Apply step 1.2 to $P(z)=I_N-f_1(z)f_1(z{)}^T$. It gives a smooth isometric frame $C(z):{\mathbb{R}}^{N-1}\to f_1(z{)}^{\perp }$ throughout the disk. On the boundary the remaining columns have coordinates $w_i=C^T{v}_i$, $2\le i\le k$, giving a loop in $V_{k-1}({\mathbb{R}}^{N-1})$. Since $(N-1)-(k-1)=N-k\ge 2$, induction supplies a continuous disk filling $({\tilde{w}}_2,\dots ,{\tilde{w}}_k)$ of that loop. Then $(f_1,C{\tilde{w}}_2,\dots ,C{\tilde{w}}_k)$ is a continuous disk of orthonormal $k$-frames with boundary exactly $\gamma$. Thus every loop extends over a disk and is nullhomotopic. For the original continuous based loop, attach its basepoint-fixed smoothing homotopy to this disk filling. Disk extension is the usual loop nullhomotopy criterion; contracting the disk towards its chosen boundary basepoint gives a based nullhomotopy.

Step 2.1 applies at every basepoint and proves that each fundamental group is trivial. Together with step 1.3 this establishes simple connectivity for all $N\ge k+2$. For $N>r\ge 2$, take $k=N-r\ge 1$; then $N-k=r\ge 2$, which is exactly the permitted range. No fibration, numerability theorem, general bundle classification, or arbitrary choice is used.