Coordinate-Deletion Bundles from Composition Algebras: Hopf Defects, $KO$-Classes, and Real Projective Space

Abstract

Let \(D\in\{\C,\Hh,\Oct\}\) be a real composition division algebra of dimension \(d\in\{2,4,8\}\). We construct an explicit rank-\(d\) real vector bundle $ E_D\longrightarrow\RP^d$ by deleting one homogeneous coordinate on each standard affine chart, interpreting the remaining coordinates as an element of \(D\), and using the corresponding left-multiplication matrices as transition data. The bundle admits a matrix-determined trivialization away from the \(d+1\) coordinate points. Relative to this trivialization, each deleted point has local clutching map $S^{d-1}\longrightarrow\SO(d),\; u\longmapsto L_u$, which is respectively the complex, quaternionic, or octonionic Hopf clutching map. A section arising from the same matrices has exactly the coordinate points as nondegenerate zeros. Consequently, $w(E_D)=1+x^d$, and real \(K\)-theory together with Euler-class cancellation gives $E_D\cong {\gamma }^{\oplus d}$. Thus the construction supplies explicit framed-defect realizations of these familiar bundles. In particular, it gives a semialgebraic octonionic realization of \(γ^{\oplus8}\) on \(\RP^8\) with nine specified local Hopf defects. We also describe the associated Pfister quadratic bundle over arbitrary fields.