Volume chirality and Clifford–Hopf clutching

In an oriented orthonormal frame the generators anticommute and each has square $-1$. Reversing the product order takes $n(n-1)\mathrm{/}2=2m(4m-1)$ swaps, an even number; multiplying the $n$ generator squares contributes $(-1{)}^{4m}=1$. Thus ${\omega }^2=1$. For any orthonormal frame, pairwise anticommutation makes the product equal to the alternating product $\mathrm{Alt}(v_1,\dots ,v_n)=\frac{1}{n!}{\sum }_{\sigma }\mathrm{sgn}(\sigma )c(v_{\sigma (1)})\cdots c(v_{\sigma (n)})$. This is an alternating multilinear function of the frame vectors. An oriented orthogonal change has determinant $+1$, so it leaves this top alternating product unchanged. Hence $\omega$ is frame-independent and, in local frames, varies continuously.

Moving any unit vector $v$ past the other $n-1$ orthogonal generators gives $\omega c(v)=(-1{)}^{n-1}c(v)\omega =-c(v)\omega$. If the module also has a parity grading, $\omega$ has even parity because $n$ is even, so it preserves ${\mathrm{\Lambda }}^0$ and ${\mathrm{\Lambda }}^1$. The continuous maps $p_{\pm }=(I\pm \omega )\mathrm{/}2$ are idempotents with $p_{+}{p}_{-}=0$ and $p_{+}+p_{-}=I$. In any local frame their matrix entries vary continuously, and the trace of an idempotent equals its rank, so each image rank is locally constant. At a point choose a basis of the image; applying $p_{\pm }$ to the same frame vectors remains independent on a neighborhood and spans the image there. Thus their images are complementary subbundles (graded when a parity grading is supplied). The anticommutation identity makes $c(v)$ interchange their images, and $c(v{)}^{-1}=-c(v)$ for $\parallel v\parallel =1$, so the interchange is an isomorphism.

Pull the two eigenbundles back to the two closed disk halves of $S(\mathbb{R}\oplus V)$. Thicken the halves slightly so their overlap is an equatorial collar; radial projection in each fiber identifies that collar continuously with $S(V)$, so extend the transition $c(v)$ constantly along the radial coordinate. The opposite transition is $-c(v)$, its inverse, and the two-chart cocycle is therefore satisfied. The quotient gluing theorem [F4] gives a vector bundle, and its relative disk/sphere triple is exactly $({\mathrm{\Lambda }}^{+},c(v),{\mathrm{\Lambda }}^{-})$. The volume eigenspaces are defined by $\omega$, whereas an additional parity grading is the grading operator in [F1]; these are distinct constructions, and no identification between them is assumed.