Stiefel manifolds are connected in positive codimension and simply connected in codimension at least two

For $1\le m\le n$ the first-vector map $\pi :V_m({\mathbb{R}}^n)\to V_1({\mathbb{R}}^n)=S^{n-1}$, $\pi (v_1,\dots ,v_m)=v_1$, is a locally trivial fibre bundle with fibre $V_{m-1}({\mathbb{R}}^{n-1})$. The open sets $U_{+}=\{u\in S^{n-1}:⟨u,e_n⟩>-\frac{2}{3}\}$ and $U_{-}=\{u\in S^{n-1}:⟨u,e_n⟩<\frac{2}{3}\}$ cover $S^{n-1}$, and on $U_{\pm }$ the denominators $\mathrm{\parallel }{e}_n\pm u{\mathrm{\parallel }}^2=2(1\pm ⟨u,e_n⟩)$ are bounded below by $\frac{2}{3}$, so the Householder reflections $H_u^{\pm }v=v-\frac{2⟨v,e_n\pm u⟩}{\mathrm{\parallel }{e}_n\pm u{\mathrm{\parallel }}^2}(e_n\pm u)$ depend continuously on $u$ and are orthogonal; they satisfy $H_u^{+}(e_n)=-u$ and $H_u^{-}(e_n)=u$, hence map the hyperplane $e_n^{\perp }\cong {\mathbb{R}}^{n-1}$ isometrically onto $u^{\perp }$. Therefore ${\mathrm{\Theta }}_{\pm }(u,f_2,\dots ,f_m)=(u,H_u^{\pm }{f}_2,\dots ,H_u^{\pm }{f}_m)$ are homeomorphisms $U_{\pm }\times V_{m-1}({\mathbb{R}}^{n-1})\to {\pi }^{-1}(U_{\pm })$ over $U_{\pm }$, with inverses $(v_1,\dots ,v_m)\mapsto (v_1,(H_{v_1}^{\pm }{)}^{-1}{v}_2,\dots ,(H_{v_1}^{\pm }{)}^{-1}{v}_m)$.

Base case $m=1$: $V_1({\mathbb{R}}^n)=S^{n-1}$ by [F1], which is nonempty and path connected for $n\ge 2=m+1$ by [F8]; and for $n\ge 3=m+2$ every based loop $S^1\to S^{n-1}$ is nullhomotopic by [F7] with $k=1<r=n-1$, so ${\pi }_1(V_1({\mathbb{R}}^n),e_1)=0$.

The explicit functions ${\sigma }_{\pm }(u)=\max (0,\pm ⟨u,e_n⟩+\frac{1}{3})$ satisfy ${\sigma }_{+}+{\sigma }_{-}\ge \frac{1}{3}$ on $S^{n-1}$ and have closed supports $\mathrm{supp}{\sigma }_{\pm }=\{\pm ⟨u,e_n⟩\ge -\frac{1}{3}\}\subseteq U_{\pm }$, so ${\rho }_{\pm }={\sigma }_{\pm }\mathrm{/}({\sigma }_{+}+{\sigma }_{-})$ is a partition of unity subordinate to the two-element cover of step 1.1. With these charts the bundle of step 1.1 is numerable, so by [F3] it is a Hurewicz fibration and in particular a Serre fibration; the only choice-like step of the cited proof is the well-order of finite words over the chart alphabet, and for the two-element alphabet the words are explicitly enumerated by their binary digits, so this application uses no choice. The identification of the fibre over $u$ with $V_{m-1}({\mathbb{R}}^{n-1})$ is the homeomorphism $H_u^{\pm }$ restricted to $e_n^{\perp }$, so ${\pi }_0$ and ${\pi }_1$ of the fibre are those of $V_{m-1}({\mathbb{R}}^{n-1})$ by [F6].

Step (a) for $m\ge 2$: assume $V_{m-1}({\mathbb{R}}^{n-1})$ is path connected, which by the induction hypothesis holds because $n\ge m+1$ gives $n-1\ge (m-1)+1$. The bundle of steps 1.1 and 2.1 is a based Serre fibration with path-connected base $S^{n-1}$ and fibre $V_{m-1}({\mathbb{R}}^{n-1})$ over $e_1$, and its exact sequence in degree zero reads ${\pi }_0(F)\to {\pi }_0(V_m({\mathbb{R}}^n),e_0)\to {\pi }_0(S^{n-1},e_1)$, a sequence of pointed sets whose two outer terms are singletons; exactness makes the middle term a singleton as well, that is, $V_m({\mathbb{R}}^n)$ is path connected.

Step (b) for $m\ge 2$: assume ${\pi }_1(V_{m-1}({\mathbb{R}}^{n-1}),\cdot )=0$, which by the induction hypothesis holds because $n\ge m+2$ gives $n-1\ge (m-1)+2$. The exact sequence of the same based Serre fibration reads ${\pi }_1(F)\to {\pi }_1(V_m({\mathbb{R}}^n),e_0)\to {\pi }_1(S^{n-1},e_1)$, and the target is trivial by [F7] with $k=1<r=n-1$ because $n\ge 3$; exactness makes the first map surjective, so the triviality of ${\pi }_1(F)$ forces ${\pi }_1(V_m({\mathbb{R}}^n),e_0)=0$. Since $V_m({\mathbb{R}}^n)$ is path connected by step 3.1, triviality at one basepoint gives triviality at every basepoint.

For the final identifications: the map $\mathrm{\Phi }:V_m({\mathbb{R}}^{m+1})\to \mathrm{S}\mathrm{O}(m+1)$ that sends $(v_1,\dots ,v_m)$ to the matrix whose first $m$ columns are $v_1,\dots ,v_m$ and whose last column is the unique unit vector $w$ orthogonal to all $v_i$ with $\det (v_1,\dots ,v_m,w)=+1$ is a bijection onto $\mathrm{S}\mathrm{O}(m+1)$: the orthogonal complement of $\mathrm{span}\{v_1,\dots ,v_m\}$ is a line containing exactly two unit vectors, and exactly one of them gives determinant $+1$; the coordinates of $w$ are the $m\times m$ minors of the matrix $(v_1\mid \cdots \mid v_m)$, namely the coefficients of the Hodge dual, which are polynomial in the entries of the $v_i$, and the inverse is the continuous projection to the first $m$ columns, so $\mathrm{\Phi }$ is a homeomorphism. Hence by [F6] the homotopy invariants of $V_m({\mathbb{R}}^{m+1})$ and $\mathrm{S}\mathrm{O}(m+1)$ agree, and $V_m({\mathbb{R}}^{m+1})$ is connected by step 3.1 applied with $n=m+1$. The case $m=1$ gives $V_1({\mathbb{R}}^2)=S^1\cong \mathrm{S}\mathrm{O}(2)$, consistent with $V_1({\mathbb{R}}^k)=S^{k-1}$. Steps 1.2, 3.1 and 4.1 cover $m=1$ and all $m\ge 2$ in the stated ranges, and together with the definition of simple connectivity [F10] they give that $V_m({\mathbb{R}}^n)$ is simply connected whenever $n\ge m+2$.