球面切丛的伴随约化
Adjoint reductions of tangent bundles of spheres
AI总结：
该研究通过紧连通李群G的伴随表示探讨球面切丛的约化，证明伴随映射同伦性质，得到线性无关截面存在条件，给出TS^d伴随约化的必要条件并排除特定情形下的约化，还应用于Banach等距子空间问题。
AI中文摘要：
我们通过紧连通李群G的伴随表示研究S^dim G的切丛的约化。设d=dim G，r=rank G，我们证明伴随映射Ad∶G→SO(d)作为普通映射同伦于取值在SO(d−r+1)中的映射。由此可得，通过伴随表示与主G丛关联的球面上的每个向量丛都容许r−1个线性无关截面。结合Adams关于球面上向量场的定理，这给出TS^d的伴随约化的必要条件r≤ρ(d+1)。特别地，对于维数d>1且d≢3 mod4的非平凡紧连通单或非单群，不存在此类约化。作为应用，这排除了Bor、Hernández-Lamoneda、Jiménez-Desantiago和Montejano在Banach等距子空间问题的结构群论证中作为例外分支的TS^{133}的伴随E_7约化。
英文摘要：
We study reductions of the tangent bundle of $S^{\dim G}$ through the adjoint representation of a compact connected Lie group $G$. If $d=\dim G$ and $r=\operatorname{rank} G$, we show that the adjoint map $\operatorname{Ad}\colon G\longrightarrow \mathrm{SO}(d)$ is homotopic, as an ordinary map, to one with values in $\mathrm{SO}(d-r+1)$. It follows that every vector bundle over a sphere associated to a principal $G$-bundle via the adjoint representation admits $r-1$ linearly independent sections. Combined with Adams's theorem on vector fields on spheres, this gives the necessary condition $r\le ρ(d+1)$ for an adjoint reduction of $TS^d$. In particular, no such reduction exists for a non-trivial compact connected group, simple or not, of dimension $d>1$ with $d\not\equiv3\pmod4$. As an application, this excludes the adjoint $E_7$-reduction of $TS^{133}$ that is the exceptional branch in the structure-group argument of Bor, Hernández-Lamoneda, Jiménez-Desantiago and Montejano for Banach's isometric subspace problem.
