Euclidean Spin sphere bundle and explicit rotor sections

The group $SO(n)$ acts transitively on $S^{n-1}$: for $a,v$ unit, a rotation in their plane sends $a$ to $v$; when $v=-a$, a half-turn in a plane containing $a$ does so, which exists because $n\ge 2$. By [F1], each such rotation has a Spin lift, so $p$ is onto.

If $x\in {\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(n)$ fixes $e_n$, then ${\rho }_x$ restricts to an element of $SO(e_n^{\perp })$. By [F3] it is a product of an even number of reflections with normals in $e_n^{\perp }$; lifting these gives $y\in {\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(n-1)$ with ${\rho }_y={\rho }_x$. Then $x{y}^{-1}$ lies in the scalar kernel [F2], so $x=\pm y$. Since $n\ge 2$, choose a unit vector $u\in e_n^{\perp }$; the even product $u(-u)=-1$ lies in the embedded ${\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(n-1)$. Therefore both $y$ and $-y$ belong to that subgroup. Conversely each vector in $e_n^{\perp }$ fixes $e_n$ under its reflection, so each even product from that subspace fixes $e_n$. This identifies the stabilizer.

For unit $a,v$ with $B(v,a)>-1$, put $u=(v+a)\mathrm{/}\sqrt{2(1+B(v,a))}$. Then $Q(u)=1$ and $s_a(v)=ua=(1+va)\mathrm{/}\sqrt{2(1+B(v,a))}$ is a product of two unit vectors, so lies in Spin. Directly, $s_a(v)a=(a+v)\mathrm{/}\sqrt{2(1+B(v,a))}=v{s}_a(v)$, giving ${\rho }_{s_a(v)}(a)=v$; the formula is continuous on $U_a$. For the bundle map $p=p_{e_n}$, this gives the section $s_{e_n}$ on $U_{e_n}$. For any unit $a$, choose $g_a\in Spin(n)$ with ${\rho }_{g_a}(e_n)=a$, which exists by step 1.1 and [F1]. Since ${\rho }_{g_a^{-1}}v\in U_{e_n}$ exactly when $v\in U_a$, the translated formula ${\sigma }_a(v)=g_a{s}_{e_n}({\rho }_{g_a^{-1}}v)$ is defined continuously on $U_a$ and satisfies $p({\sigma }_a(v))=v$. Thus every cap $U_a$ is a chart domain for $p$.

For a section $s:U\to Spin(n)$, the map $p^{-1}(U)\to U\times Spin(n-1)$ given by $x\mapsto (p(x),s(p(x){)}^{-1}x)$ is continuous, and its second coordinate fixes $e_n$ by the group action. Its inverse is $(v,h)\mapsto s(v)h$, so the chart is a homeomorphism commuting with projection. Right multiplication by the stabilizer is free and transitive on each fiber, proving the asserted local bundle structure.

The caps $U_a$ cover the sphere. For each point choose a smaller metric ball of radius $r$ whose closed ball of radius $2r$ lies in one such cap; compactness of $S^{n-1}$ gives finitely many smaller balls covering it. For each selected center $c_j$, define ${\psi }_j(v)=\max (0,1-d(v,c_j)\mathrm{/}(2r_j))$. These continuous functions have supports in their chart caps and their sum is positive because the smaller balls cover. Then ${\varphi }_j={\psi }_j\mathrm{/}({\sum }_i{\psi }_i)$ form a finite partition of unity with closed supports inside the corresponding charts. This is the numerability data used in the universal-cover proof.