AI 中文总结
该研究从维数为2、4、8的实合成可除代数出发，构造实射影空间上的显式实向量丛，实现了熟知丛的配边缺陷，还给出八元数相关丛的半代数实现并描述了普菲斯特二次丛。
AI 中文摘要
设\boldsymbol{D}\boldsymbol{\u2208}\u007b\boldsymbol{\u2102},\boldsymbol{\u2108},\boldsymbol{\u210a}\u007d是维数\boldsymbol{d}\u2208\u007b2,4,8\u007d的实合成可除代数。我们通过在每个标准仿射图上删除一个齐次坐标，将剩余坐标视为\boldsymbol{D}的元素，并将对应的左乘矩阵作为转移数据，构造了一个显式的秩\boldsymbol{d}实向量丛\boldsymbol{E_D}\u2192\boldsymbol{\u2119^d}。该丛在\boldsymbol{d+1}个坐标点外具有矩阵确定的平凡化，相对于此平凡化，每个删除点的局部 clutching 映射为\boldsymbol{S^{d-1}\u2192\boldsymbol{SO(d)},\boldsymbol{u}\u21a6\boldsymbol{L_u}}，分别为复数、四元数或八元数霍普夫 clutching 映射。由同一矩阵生成的截面恰好以坐标点为非退化零点，因此\boldsymbol{w(E_D)=1+x^d}，结合实\boldsymbol{K}-理论与欧拉类消去可得\boldsymbol{E_D}\u2245\boldsymbol{\u03b3^{\u2295 d}}。该构造为这些熟知的丛提供了显式的配边缺陷实现，尤其在\boldsymbol{\u2119^8}上给出了\boldsymbol{\u03b3^{\u22958}}的半代数八元数实现，带有9个指定的局部霍普夫缺陷。我们还描述了任意域上相关的普菲斯特二次丛。
英文摘要
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 $E_D\congγ^{\oplus d}$. 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.
