The basepoint evaluation of the Stiefel section space is a fibration

Evaluation on smooth sections is locally trivial. The group $O(n)$ acts on every section by target multiplication. For any frame $a\in F$, complete it to an orthonormal basis; Gram-Schmidt applied to a nearby frame $b$ followed by the remaining fixed basis vectors gives a smooth orthogonal matrix $R_a(b)$ with $R_a(a)=I$ and $R_a(b)a=b$. Thus $s\mapsto (\mathrm{ev}(s),R_a(\mathrm{ev}(s){)}^{-1}s)$ identifies ${\mathrm{ev}}^{-1}(U_a)$ with $U_a\times {\mathrm{\Gamma }}_a$, including its inverse $(b,s)\mapsto R_a(b)s$. These maps are continuous for the weak $C^{\mathrm{\infty }}$ topology because target multiplication multiplies each derivative by the same finite matrix. The same argument works for continuous sections. If either section space is nonempty the transitive $O(n)$ action makes evaluation surjective; otherwise its fibres are all empty. If a section space is empty, its evaluation has the homotopy lifting property vacuously, so it is a Hurewicz fibration with empty fibres without invoking a surjective bundle convention. For a nonempty section space, compactness of $F$ gives finitely many such neighbourhoods, with a subordinate continuous partition obtained from finitely many ambient bump functions. Order these finitely many charts once, and well-order their finite words first by length and then lexicographically. Substituting this explicit well-order in the well-order construction of the proof of [F3] supplies its sole use of AC; all remaining constructions there apply to the supplied finite numeration without choice. Thus both evaluations are Hurewicz fibrations; their fibres at $s_0(x_0)$ are the prescribed fixed-value section spaces. This argument applies also when $m=n$ and $F$ is disconnected.

Polar normalization is well defined for any fibrewise injection $B$: in the global matrix presentation put $T=B^{\ast }B+x{x}^{\ast }$, which is positive definite, and normalize by $B{T}^{-1\mathrm{/}2}$. By [F11] this operation is smooth on smooth sections and continuous on continuous sections. The fibrewise path $B((1-t)I+t{T}^{-1\mathrm{/}2})$ remains injective and fixes orthonormal sections, giving the deformation retraction to the Stiefel model used in [F1]. We now compare its smooth and continuous fixed-value section spaces constructively. Represent an orthonormal section by matrices $A(x):{\mathbb{R}}^{m+1}\to {\mathbb{R}}^n$ with $A(x)=A(x)P_x$ and $A(x{)}^{\ast }A(x)=P_x$, where $P_x=I-x{x}^{\ast }$, and put $a=A(x_0)$. Extend $A$ radially to an annulus, multiply by a fixed radial cutoff equal to one near the unit sphere, and convolve with a fixed smooth compactly supported Euclidean kernel of radius $\epsilon <1\mathrm{/}8$. Restriction to the sphere and right multiplication by $P_x$ gives a smooth bundle map $B_{\epsilon }(A)$, converging uniformly to $A$. Pin its value by adding $\chi (x)(a-B_{\epsilon }(A)(x_0))P_x$, for a fixed smooth bump $\chi$ with $\chi (x_0)=1$; denote the result by $C_{\epsilon }(A)$. It equals $a$ at $x_0$, converges uniformly to $A$, depends continuously on $(A,\epsilon )$ into the smooth topology for $\epsilon >0$, and these operators have a common uniform norm bound.

Choose an orthonormal basis $e_1,\dots ,e_m$ of $x_0^{\perp }$. For the closed unit disk $D^m$, write $y=ru$ and define $q(y)=\sin (\pi r){\sum }_i{u}_i{e}_i-\cos (\pi r)x_0$, with $q(0)=-x_0$. The functions $\sin (\pi r)\mathrm{/}r$ and $\cos (\pi r)$ are smooth at $r=0$ by their power series, so $q$ is smooth on the closed disk. Its boundary maps to $x_0$, its interior maps homeomorphically to $S^m\setminus \{x_0\}$, and the induced map $D^m\mathrm{/}\mathrm{\partial }{D}^m\to S^m$ is a homeomorphism, since it is a continuous bijection from a compact space to a Hausdorff space. Put $P(t,y)=I-q(ty)q(ty{)}^{\ast }$ and $K=[{\mathrm{\partial }}_{t}P,P]$. Solve ${\mathrm{\partial }}_{t}O=KO$, $O(0,y)=I$. By [F10] the solution exists throughout $[0,1]$ and is jointly smooth in $(t,y)$: the local smooth solution maps patch along the compact time interval by uniqueness. Since $K^{\ast }=-K$, $O^{\ast }O=I$; differentiating $P^2=P$ gives ${\mathrm{\partial }}_{t}P=[K,P]$, and therefore $PO-OP(0)$ solves the linear equation with zero initial value and is zero. Consequently $O(1,y)e_1,\dots ,O(1,y)e_m$ is a smooth orthonormal frame of $q^{\ast }T{S}^m$ on the whole closed disk, including its boundary. By [F1] it trivialises $q^{\ast }E$.

There is a continuous positive smoothing radius on the entire metric space ${\mathrm{\Gamma }}_{\ast }^0$. For every integer $k\ge 1$, put $f_k(A)={\sup }_{0<\epsilon \le 1\mathrm{/}(8k)}\mathrm{\parallel }{C}_{\epsilon }(A)-A{\mathrm{\parallel }}_{\mathrm{\infty }}$ is continuous, since the uniformly bounded operators make these functions uniformly Lipschitz, and $f_k(A)\to 0$. Put $w_k(A)=2^{-k}\max (0,1-8f_k(A))$ and $\epsilon (A)=({\sum }_{k=1}^{\mathrm{\infty }}{w}_k(A)\mathrm{/}(8k))\mathrm{/}{\sum }_{k=1}^{\mathrm{\infty }}{w}_k(A)$. Both series converge uniformly, the denominator is positive, and if $k_0$ is the first positive weight then $\epsilon (A)\le 1\mathrm{/}(8k_0)$ and $\mathrm{\parallel }{C}_{\epsilon (A)}(A)-A{\mathrm{\parallel }}_{\mathrm{\infty }}<1\mathrm{/}8$. Every convex combination $L_t=(1-t)A+t{C}_{\epsilon (A)}(A)$ is therefore injective on the tangent fibres. Normalize it by $L_t(L_t^{\ast }{L}_t{)}^{-1\mathrm{/}2}$ on those fibres; the inverse square root is given by the convergent binomial series near the identity, since $\mathrm{\parallel }{L}_t^{\ast }{L}_t-I\mathrm{\parallel }<17\mathrm{/}64$. This normalization is continuous, smooth in $x$ when $A$ is smooth, and fixes $a$ at $x_0$. It gives a homotopy from the identity to a continuous smoothing map $S:{\mathrm{\Gamma }}_{\ast }^0\to {\mathrm{\Gamma }}_{\ast }$; restricted to smooth sections it is continuous in the weak $C^{\mathrm{\infty }}$ topology as well. Thus inclusion and $S$ are homotopy inverses. The same construction without the pinning correction compares the unrestricted section spaces. No family of charts or approximation choices is selected: the kernel, cutoffs and series are fixed.

In this frame the prescribed fibre value $s_0(x_0)$ determines a boundary map $g:\mathrm{\partial }{D}^m\to F$, generally nonconstant. A continuous fixed-value section pulls back to a map $\gamma :D^m\to F$ with $\gamma {\mathrm{\mid }}_{\mathrm{\partial }{D}^m}=g$. Conversely such a map gives a section of $q^{\ast }E$ whose images in $E$ all equal $s_0(x_0)$ on the collapsed boundary, so it descends to a unique continuous section of $E$. This bijection is a homeomorphism ${\mathrm{\Gamma }}_{\ast }^0\cong {\mathcal{M}}_g:=\{\gamma :\gamma {\mathrm{\mid }}_{\mathrm{\partial }{D}^m}=g\}$. For compact-open continuity, pullback is continuous, and if $K\subseteq S^m$ is compact then $q^{-1}(K)$ is compact, so the inverse image of the section neighbourhood $s(K)\subseteq U$ is the corresponding pullback neighbourhood over $q^{-1}(K)$; composing with the bundle frame and its inverse is continuous by the exponential correspondence. The same quotient reasoning applies to parametrized homotopies.

If $\mathrm{\Gamma }\mathrm{\ne }\mathrm{\varnothing }$ and $n\ge m+1$, [F2] makes $F$ path connected. A path from any evaluated value to $s_0(x_0)$ lifts under the smooth evaluation fibration of step 1.1, producing a section in ${\mathrm{\Gamma }}_{\ast }$. The same lifting moves a representative of every component of $\mathrm{\Gamma }$ into ${\mathrm{\Gamma }}_{\ast }$, so the map of component sets ${\pi }_0({\mathrm{\Gamma }}_{\ast })\to {\pi }_0(\mathrm{\Gamma })$ is surjective.

If ${\mathrm{\Gamma }}_{\ast }\mathrm{\ne }\mathrm{\varnothing }$, choose its disk model ${\gamma }_0\in {\mathcal{M}}_g$ and one $y_0\in \mathrm{\partial }{D}^m$, and put $e_0=g(y_0)$. The formula $g_t(u)={\gamma }_0((1-t)u+t{y}_0)$ contracts $g$ to the constant map $e_0$ while fixing $y_0$. Restriction $C(D^m,F)\to C(\mathrm{\partial }{D}^m,F)$ is a Hurewicz fibration: $\mathrm{\partial }{D}^m\subset D^m$ is a closed cofibration, as its radial collar gives the usual homotopy extension retraction of $D^m\times I$ onto $D^m\times \{0\}\cup \mathrm{\partial }{D}^m\times I$; for every test space, compose that retraction with the prescribed disk map and boundary homotopy and transpose by [F6]. Lifting the path $g_t$, with all points of its starting fibre as parameters, gives transport ${\mathcal{M}}_g\to {\mathcal{M}}_{e_0}$. Transport along the reversed path is a homotopy inverse: the two concatenations retrace the same path and contract to constant paths by shortening their excursion; relative homotopy lifting [F4] lifts these contractions to fibre homotopies, with prescribed initial maps. Thus ${\mathcal{M}}_g\simeq {\mathcal{M}}_{e_0}$.

The constant-boundary maps descend to the based mapping space $C_{\ast }((D^m\mathrm{/}\mathrm{\partial }{D}^m,\ast ),(F,e_0))$, again homeomorphically for compact-open topologies. Combining steps 2.1, 2.2 and 3.1 gives ${\mathrm{\Gamma }}_{\ast }\simeq C_{\ast }(S^m,F)$ and hence ${\pi }_0({\mathrm{\Gamma }}_{\ast })\cong {\pi }_m(F)$ by [F8]. A homotopy of sections fixed at $x_0$ gives a path in ${\mathcal{M}}_g$ and, under transport, a based homotopy in ${\mathcal{M}}_{e_0}$. Conversely a based homotopy can be transported back, and the fibre homotopies between the composites and the identities provide a fixed-value section homotopy; the smoothing comparison makes it a homotopy of smooth sections when the endpoints are smooth. These are both directions of the difference-class criterion. The identification depends on the disk frame, reference section and contraction; it is not canonical.

The homotopy exact sequence of evaluation, based at a chosen section in ${\mathrm{\Gamma }}_{\ast }$, is the displayed sequence of [F7]. When $F$ is simply connected, two fibre components that become connected in $\mathrm{\Gamma }$ differ by transport around a loop in $F$; a nullhomotopy of that loop and relative lifting show they were already connected in the fibre. Thus the component map is injective, and step 2.3 makes it surjective, giving ${\pi }_0(\mathrm{\Gamma })\cong {\pi }_m(F)$ by step 4.1. If instead ${\pi }_m(F)=0$, $n\ge m+1$ and $\mathrm{\Gamma }\mathrm{\ne }\mathrm{\varnothing }$, steps 4.1 and 2.3 show that ${\pi }_0(\mathrm{\Gamma })$ is a quotient of a singleton and hence itself a singleton. The connected-fibre hypothesis cannot be omitted: for $m=n=1$ the two orientations of an everywhere nonzero circle field give two section components even though ${\pi }_1(O(1))=0$.