Stable Clifford–Hopf decomposition over sphere bundles

Represent the bundle by a triple $(E_{+},\sigma ,E_{-})$ over the two disk halves, using [F4]. The volume $\omega =c(e_1)\cdots c(e_n)$ is global because $V$ is oriented. For an ungraded module it defines the canonical grading, so [F3] applies without changing the ungraded module category. On its upper half the even generators $-c(e_1)c(e_{j+1})$ form a positive $C_{n-1}$ family, by multiplying their Clifford relations and using $\omega =I$ there. Locally normalize $\sigma$ at a chosen $v_0=e_1$ by $\sigma (v_0{)}^{-1}\sigma (v)$. The determinant is $+1$ because the sphere is connected and its normalized value at $v_0$ is $I$. Its stable winding $\eta$ is constant on each connected component of $X$ and independent of these charts by [F2].

One regular Hopf summand has upper rank $h=2^{n-1}$ and positive winding $h\mathrm{/}2$. Add finitely many to make $\eta \ge s_{n-1}(\dim X+1)$. Add trivial real summands to make $p\ge 4\eta$ and satisfy the finite size inequalities in [F2]. Next embed $E_{+}$ into the positive half $R^{+}$ of a sufficiently large regular module $R=C(V{)}^N$: [F1] embeds it into a trivial bundle, while each regular positive half contains the global nonzero vector $(1+\omega )\mathrm{/}2$, so enough copies contain that trivial bundle. Add the pulled-back complement $F=R^{+}\ominus E_{+}$ to the triple. This only increases $p$ and leaves $\eta$ fixed. Enlarge $N$ so the size inequalities still hold, with $p$ now divisible by $s_{n-1}$. At the end add an ordinary complement of $F$ to turn this additional stabilization into a trivial real stabilization.

We prove the relative statement with $E_{+}=R^{+}$. Suppose over a subcomplex $X^{\mathrm{\prime }}$ we already have an orthogonal module splitting $R=R_0\oplus R_1$ and an isometry $f:E_{-}{\mathrm{\mid }}_{D(V{\mathrm{\mid }}_{X^{\mathrm{\prime }}})}\to R_0^{+}\oplus R_1^{-}$ such that $f\sigma =\alpha$, where $\alpha (v)=I_{R_0^{+}}\oplus c(v){\mathrm{\mid }}_{R_1^{+}}$. Attach one cell $D$. Trivialize $V$ and $R$ compatibly on its characteristic disk, so $R^{+}$ becomes a fixed positive representation space. Normalize the source lower trivialization by $\sigma (v_0{)}^{-1}$ and the target boundary lower trivialization by $\alpha (v_0{)}^{-1}$. In these coordinates $\widehat{f}=\alpha (v_0{)}^{-1}f\sigma (v_0),\widehat{\sigma }(v)=\sigma (v_0{)}^{-1}\sigma (v),\widehat{\alpha }(v)=\alpha (v_0{)}^{-1}\alpha (v)\mathrm{.}$ On $\mathrm{\partial }D\times D^n$, $\widehat{f}(x,v_0)=I$, and on $\mathrm{\partial }D\times S^{n-1}$, $\widehat{f}\widehat{\sigma }=\widehat{\alpha }$, which is affine Hopf.

The boundary lower gauge has the explicit extension $\widehat{F}(rx,v)=\widehat{f}(x,v_0+r(v-v_0)),\widehat{F}(0,v)=I\mathrm{.}$ Here $x\in \mathrm{\partial }D$, $0<r\le 1$, and $v\in D^n$. Compactness and $\widehat{f}(x,v_0)=I$ give continuity at $r=0$ uniformly in $x,v$. It equals $\widehat{f}$ on the cell boundary and equals $I$ on $D\times \{v_0\}$. Thus $\tau =\widehat{F}\widehat{\sigma }$ is a based sphere-map family over $D$, affine on its boundary. Since each $\widehat{F}$ sphere map extends over the disk, it is nullhomotopic and leaves winding unchanged. Apply [F2] relatively to $\mathrm{\partial }D$ to obtain ${\tau }_t$, with ${\tau }_0=\tau$ and ${\tau }_1$ affine. Its recovered invariant projection extends the given splitting. Set $R_1^{-}=c(v_0)R_1^{+}$ and similarly for $R_0$. Invariance under the even Clifford algebra and odd multiplication makes $R_0,R_1$ full ungraded submodules.

An equatorial gauge alone is insufficient. Put $q_t={\tau }_t{\tau }_0^{-1}$. It has $q_0=I$ and is $I$ over $\mathrm{\partial }D$ for every $t$. Cone it over the lower disk: $Q(x,rv)=q_r(x,v)(r>0),Q(x,0)=I\mathrm{.}$ Uniform continuity at $r=0$ proves continuity; it is $I$ on $\mathrm{\partial }D\times D^n$. Therefore $G=Q\widehat{F}$ extends the normalized lower gauge and satisfies $G\widehat{\sigma }={\tau }_1$ on the equator. With the final normalization ${\mathrm{\Theta }}_1={\alpha }_1(v_0{)}^{-1}$, the actual lower gauge is ${\mathrm{\Theta }}_1^{-1}G\sigma (v_0{)}^{-1}$. On the cell boundary it equals the old $f$. Thus this is a genuine isomorphism over both disk halves with upper gauge $I$, not merely an equatorial identity.

Induct over the finitely many cells, starting with empty relative data for existence. The splittings and lower gauges agree along the attaching maps by step 4.1 and step 5.1. We obtain $R=R_0\oplus R_1$ and a triple isomorphism to ${\pi }^{\ast }{R}_0^{+}\oplus H(R_1)$. The earlier regular and trivial shifts give exactly the claimed stable decomposition. The same induction works relative to any given subcomplex with affine splitting and lower gauge; use connectivity at least the relative cell dimension. All ranks, winding labels, and size inequalities were arranged before induction and remain unchanged during it.