Adjoint reductions of tangent bundles of spheres

Abstract

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 $r=\mathrm{rank}G$, we show that the adjoint map $\mathrm{Ad}:G⟶\mathrm{SO}(d)$ 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 $r\le \rho (d+1)$ for an adjoint reduction of $T{S}^d$. In particular, no such reduction exists for a non-trivial compact connected group, simple or not, of dimension $d>1$ with $d\not\equiv 3(\mathrm{mod}4)$. As an application, this excludes the adjoint $E_7$-reduction of $T{S}^{133}$ that is the exceptional branch in the structure-group argument of Bor, Hernández-Lamoneda, Jiménez-Desantiago and Montejano for Banach's isometric subspace problem.

8 pages