General linear frame spaces have the Stiefel cohomology gap

The frame retraction. Fix $n\le N$ and let $A\in \mathrm{Inj}({\mathbb{C}}^n,{\mathbb{C}}^N)$ have columns $v_0,\dots ,v_{n-1}$, a linearly independent list. Applying the Gram-Schmidt recursion of [F1] to $(v_0,\dots ,v_{n-1})$ gives an orthonormal list $(e_0,\dots ,e_{n-1})$ with $\mathrm{span}(e_0,\dots ,e_{k-1})=\mathrm{span}(v_0,\dots ,v_{k-1})$ for every $k$, so the upper triangular matrix $R$ with $R_{jk}=⟨v_k,e_j⟩$ for $j<k$, $R_{kk}=\parallel u_k\parallel >0$ and $R_{jk}=0$ for $j>k$ satisfies $A=QR$ with $Q=(e_0,\dots ,e_{n-1})\in V_n({\mathbb{C}}^N)$. The Gram-Schmidt formulas are continuous rational expressions with positive denominators on the full-rank locus, so $A\mapsto (Q(A),R(A))$ is continuous. Writing $R=DU$ with $D$ the diagonal part and $U$ unit upper triangular, the family $t\mapsto Q(A)((1-t)D+tI)((1-t)U+tI)$ is continuous, has invertible upper triangular second factor with positive diagonal for every $t\in [0,1]$, and connects $A$ to $Q(A)$ inside $\mathrm{Inj}({\mathbb{C}}^n,{\mathbb{C}}^N)$; at $t=1$ it is $Q(A)$, so $A\mapsto Q(A)$ is a deformation retraction onto $V_n({\mathbb{C}}^N)$. This proves claim 1 and shows that $U_N$ is homotopy equivalent to ${\prod }_i{V}_{n_i}({\mathbb{C}}^{N_i})$.

Compact smooth frame manifolds. Let $f(A)=A^{\ast }A$ from the open full-rank locus of ${\mathbb{C}}^{N\times n}$ to the real vector space $\mathrm{Herm}(n)$. Its differential is $H\mapsto A^{\ast }H+H^{\ast }A$; for Hermitian $B$, taking $H=\frac{1}{2}A(A^{\ast }A{)}^{-1}B$ gives $A^{\ast }H=H^{\ast }A=B\mathrm{/}2$, hence differential $B$. Thus $I_n$ is a regular value and $V_n=f^{-1}(I_n)$ is a smooth submanifold by [F2]. It is closed in the ambient Euclidean space and bounded, since each frame column has norm one, so it is compact by [F3]. The subspace is Hausdorff and second countable, as required by the triangulation supplier. For $n=0$ it is a point.

The forgetting bundle. Fix $w\in V_{n-1}({\mathbb{C}}^N)$ and an orthonormal basis $c_1,\dots ,c_{N-n+1}$ of its orthogonal complement. For a nearby frame $w^{\mathrm{\prime }}$, put $P_{w^{\mathrm{\prime }}}=I-w^{\mathrm{\prime }}{w}^{\mathrm{\prime }\ast }$. The vectors $P_{w^{\mathrm{\prime }}}{c}_j$ remain linearly independent near $w$, since their Gram determinant is continuous and equals one at $w$. Gram-Schmidt gives a continuously varying orthonormal basis $c_j(w^{\mathrm{\prime }})$ of the complement of $w^{\mathrm{\prime }}$. The map $(w^{\mathrm{\prime }},z)\mapsto (w^{\mathrm{\prime }},{\sum }_j{z}_j{c}_j(w^{\mathrm{\prime }}))$ for $z\in S^{2(N-n)+1}\subset {\mathbb{C}}^{N-n+1}$ trivializes the forgetting map; its continuous inverse takes the inner products with $c_j(w^{\mathrm{\prime }})$. This also covers $n=1$ over the point $V_0$. By step 1.2 the base is compact Hausdorff, so [F4] supplies a finite subordinate partition of unity and makes the bundle numerable, hence Hurewicz and Serre.

Finite CW model and path-connectedness. By the triangulation theorem of [F3] applied to the compact smooth manifold $V_n({\mathbb{C}}^N)$, there is a finite simplicial complex $K_n$ and a homeomorphism $\mathrm{\mid }{K}_n\mathrm{\mid }\to V_n({\mathbb{C}}^N)$. A finite simplicial complex is a finite CW complex: its cells are the open simplices, the characteristic maps are the inclusions of the closed simplices, closure finiteness holds because there are finitely many simplices, and the weak topology agrees with the topology of the realization because the realization is the finite union of its closed simplices. Hence $V_n({\mathbb{C}}^N)$ is homeomorphic to a finite CW complex, as claimed in claim 2. For path-connectedness, induct on $n$. The space $V_0({\mathbb{C}}^N)$ is a point. For $n\ge 1$ the map $V_n({\mathbb{C}}^N)\to V_{n-1}({\mathbb{C}}^N)$ is the locally trivial bundle of step 2.1 with fiber the nonempty path-connected sphere $S^{2(N-n)+1}$ by [F6], and the base is path-connected by the induction hypothesis; the local connectedness argument of [F8] then makes $V_n({\mathbb{C}}^N)$ path-connected. This completes claim 2.

The gap for one factor. Induct on $n$, starting with $V_0$ a point. For $n\ge 1$ put $d=2(N-n)+1$. The forgetting bundle of step 2.1 has fibre $S^d$ and base homeomorphic to a path-connected finite CW complex by step 3.1. The induction hypothesis gives vanishing of the base cohomology in degrees $1$ through $2(N-n+1)=d+1$. In the Serre spectral sequence [F5], [F6] makes all fibre rows in degrees $1,\dots ,d-1$ zero; the degree-zero coefficient system is constant $k$, since the fibre is connected. Consequently all $E_2$ terms of total degree $1,\dots ,d-1$ are zero, while $E_2^{0,0}=k$. No differential can enter $(0,0)$ or leave it in this first-quadrant sequence. Strong convergence therefore gives $H^0(V_n;k)=k$ and $H^j(V_n;k)=0$ for $1\le j\le d-1=2(N-n)$, completing the induction. No constancy of the top-degree fibre coefficient system is needed.

The product. By step 1.1 it suffices to compute the cohomology of ${\prod }_{i=1}^s{V}_{n_i}({\mathbb{C}}^{N_i})$, and we induct on $s$. For $s=1$ this is step 4.1. For $s\ge 2$ let $P={\prod }_{i<s}{V}_{n_i}({\mathbb{C}}^{N_i})$ and consider the projection $P\times V_{n_s}({\mathbb{C}}^{N_s})\to P$, a trivial bundle with fiber $V_{n_s}({\mathbb{C}}^{N_s})$, hence numerable and a Serre fibration by [F4]. By step 3.1 each factor and, inductively, $P$ is homeomorphic to a finite CW complex, and a finite product of finite CW complexes is again a finite CW complex: identify $D^a\times D^b$ with $D^{a+b}$ by radial rescaling between their unit balls; the product characteristic maps into the product space then give cells $e\times e^{\mathrm{\prime }}$ and a finite cell decomposition, closure finiteness holds because there are finitely many cells, and the weak topology of the product cell structure agrees with the product topology for finite compact Hausdorff complexes, where the compactly generated product topology coincides with the ordinary one. So [F5] applies to the projection with constant coefficient system in fiber degree $0$. If $H^0(P;k)\cong k$ and $H^j(P;k)=0$ for $1\le j\le r_{s-1}={\min }_{i<s}2(N_i-n_i)$ by induction, and if $H^0(V_{n_s};k)\cong k$, $H^j(V_{n_s};k)=0$ for $1\le j\le 2(N_s-n_s)$ by step 4.1, then in total degree $m\le r_s=\min (r_{s-1},2(N_s-n_s))$ every nonzero $E_2^{a,b}$ with $a+b=m$ has $b=0$, for $b\ge 1$ would need fiber cohomology in degree $b\le r_s\le 2(N_s-n_s)$, which vanishes; the differential bookkeeping of step 4.1 then gives $H^m(P\times V_{n_s};k)\cong H^m(P;k)$, which is $k$ for $m=0$ and $0$ for $1\le m\le r_s$. This proves claim 3, $H^0(U_N;k)\cong k$ and $H^j(U_N;k)=0$ for $1\le j\le r_N$, by the homotopy equivalence of step 1.1.

The cohomology convention. The full-rank spaces are Euclidean open subsets and the Stiefel spaces and their products are smooth metric manifolds with contractible-open chart bases. Thus [F7] identifies their constant-sheaf cohomology with the singular cohomology computed in steps 4.1 and 5.1; naturality carries the frame deformation retraction of step 1.1 to these comparisons. In particular the same gap holds for the constant-sheaf fibre cohomology used by the later derived pullback lemmas.

Conclusion. Steps 1.1 and 5.1 give claims 1 and 3; steps 1.2, 2.1 and 3.1 give claim 2. AC is consumed only through the named compactness, triangulation, partition-of-unity, spectral-sequence and constant-sheaf comparison suppliers [F3]-[F7], as recorded in [A1].