The universal oriented sphere-bundle total space has the homotopy type of BSO(n-1)

Local charts and admissibility. Finite Stiefel spaces are closed bounded subsets of finite matrix spaces and hence compact Hausdorff; quotienting orthonormal frames by the compact orthogonal or special orthogonal group gives compact Hausdorff Grassmannians, with closed coordinate inclusions. Their stable weak topologies satisfy [F4]. The graph charts of [F1] lift to the two oriented sheets; orthonormalizing the graph frame gives continuous oriented isometric bundle charts, compatible in the finite stages. They trivialize the sphere bundle with fiber $S^{n-1}$ and the orientation cover with fiber two points. Their numerations exist by [F4]. The ordinary Grassmannian is a CW complex by [F1], hence is compactly generated and CGWH. Applying the compact-fiber result [F3] to the orientation cover makes $B_n$ paracompact Hausdorff, CGWH, and of CW type. Applying it again to $p$ makes $S_n$ paracompact Hausdorff, CGWH, and of CW type. The numerable sphere bundle is a Hurewicz fibration by [F3]. Thus both bases needed below lie in the Euler-class scope of [F5].

Adapted frames and the complement. The continuous map from $V_n({\mathbb{R}}^{\mathrm{\infty }})$ sending an oriented orthonormal frame $(v,w_1,\dots ,w_{n-1})$ to $(V,o,v)$ identifies $S_n$ with the quotient by the subgroup $\mathrm{diag}(1,\mathrm{SO}(n-1))$. To check the topology and local sections, in any oriented isometric bundle chart and near a fixed unit vector, project a fixed basis of its perpendicular plane onto the varying perpendicular plane and apply finite Gram orthonormalization; independence persists on an open neighborhood, and the orientation sign is constant there. This supplies adapted-frame charts and proves the quotient identification. Dropping the first vector therefore induces the continuous complement map $r:S_n\to B_{n-1}$ with exactly the displayed orientation.

Define a candidate inverse. Fix the first standard vector $e_1$, orthogonal to the image of $P$. For an oriented $(n-1)$-plane $(W,o_W)$ put $s(W,o_W)=(\mathbb{R}{e}_1\oplus PW,(e_1,P{o}_W),e_1)\mathrm{.}$ On frames this is the continuous map $(w_1,\dots ,w_{n-1})\mapsto (e_1,P{w}_1,\dots ,P{w}_{n-1})$, equivariant for the frame changes defining the quotients, so it induces a continuous map $s:B_{n-1}\to S_n$. Its composite $rs$ is the even-coordinate embedding on oriented planes.

Descent of the parity homotopies. Write a frame as a column matrix $A$ and the homotopy in [F2] as $K_t(A)=L_{t}A(A^{\mathsf{T}}{L}_t^{\mathsf{T}}{L}_{t}A{)}^{-1\mathrm{/}2}$. If $Q$ is orthogonal, the positive Gram matrix for $AQ$ is $Q^{\mathsf{T}}(A^{\mathsf{T}}{L}_t^{\mathsf{T}}{L}_{t}A)Q$, whose inverse square root is its conjugate by $Q$; this follows from uniqueness of the positive square root. Hence $K_t(AQ)=K_t(A)Q$. In rank $n-1$, the homotopy descends to oriented planes and gives ${\mathrm{id}}_{B_{n-1}}\simeq rs$. In rank $n$, it descends by the subgroup in step 1.2 to a homotopy from ${\mathrm{id}}_{S_n}$ to $(V,o,v)\mapsto (PV,Po,Pv)$. There is no continuity inference merely from pointwise formulas: [F2] gives continuity on stable frames times the interval, and the orbit quotient is open (the saturation of an open set is a union of translates). Its product with the identity of the interval is therefore also an open quotient, so both equivariant homotopies descend continuously.

Complete the homotopy on $S_n$. For a point represented by the adapted frame $(v,w_1,\dots ,w_{n-1})$, rotate its even-coordinate frame through $(\cos \theta Pv+\sin \theta e_1,P{w}_1,\dots ,P{w}_{n-1}),0\le \theta \le \pi \mathrm{/}2.$ These vectors are orthonormal: $e_1$ is perpendicular to all even coordinates and $Pv\perp P{w}_j$. The formula is equivariant for $\mathrm{SO}(n-1)$ on the last columns and is continuous on the stable frame space (addition and scalar multiplication here are the finite-stage operations used in [F2]); the same open-quotient argument as in step 3.1 gives a continuous homotopy on $S_n$. At the final endpoint this is $(\mathbb{R}{e}_1\oplus PW,(e_1,P{o}_W),e_1)=sr(V,o,v)$. Concatenate with step 3.1 to get ${\mathrm{id}}_{S_n}\simeq sr$. Together with $rs\simeq \mathrm{id}$ this proves the homotopy equivalence, with an explicit inverse.

Splitting and Euler vanishing. Over $(V,o,v)$ the map $(a,w)\mapsto av+w$ is a linear isometric isomorphism $\mathbb{R}\oplus (v^{\perp }\cap V)\to V$; its inverse sends $z$ to $(⟨z,v⟩,z-⟨z,v⟩v)$. Both vary continuously in the charts of step 1.1. The chosen complement orientation makes this isomorphism orientation preserving with the trivial line first. The factor-swap sign is $(-1{)}^{n-1}$ by [F5]. The section $(V,o,v)\mapsto v$ never vanishes. The pullback bundle is numerable by pulling back the numeration of ${\gamma }_n^{+}$, and both its base and the original base are paracompact Hausdorff CGWH spaces of CW type by step 1.1. Thus [F5] gives $p^{\ast }e({\gamma }_n^{+})=e(p^{\ast }{\gamma }_n^{+})=0$ with integral coefficients.

Boundary and model conventions. For $n=2$, the complement has rank one and $B_1={\mathrm{Gr}}_1^{+}({\mathbb{R}}^{\mathrm{\infty }})=V_1({\mathbb{R}}^{\mathrm{\infty }})$, since $\mathrm{SO}(1)$ is trivial. It is the contractible infinite unit sphere by [F2], not literally a point. Step 4.1 consequently makes $S_2$ contractible; the actual fibration retains its circle fiber. The bases are nonempty and ranks zero and one are outside the assertion's n-range. The coefficient ring is $\mathbb{Z}$ throughout. No assertion identifies the fixed Grassmannian total-space model homeomorphically with $S_n$; the fibration and splitting are on $S_n$, and $r$ is the explicit homotopy equivalence carrying its complement bundle.