The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\R^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is "stable under tangential homotopy equivalence": if it has an affirmative answer for all vector bundles over a certain manifold \(M\), then the same is true for any manifold tangentially homotopy equivalent to~\(M\). Our main tool is topological K-theory. Over \(B^{13}\), there is essentially one stable class of real vector bundles for which our method fails.

中文翻译：

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正弯曲均匀空间的稳定逆灵魂问题

稳定逆灵魂问题 (SCSQ) 询问，给定一个实向量丛 \(E\) 在紧流形上，一些稳定性 \(E\times\R^k\) 是否承认具有非负（截面）曲率的度量. 我们扩展了先前的结果以表明 SCSQ 对任何具有正曲率的简单连接齐次流形上的所有实向量丛都有一个肯定的答案，除了 Berger 空间 \(B^{13}\)。一路走来，我们证明了对于所有最多为 7 维的简单连通齐次空间、对于简单连通紧致 1 级对称空间的维度为 4 的倍数的任意乘积以及某些球体乘积也是如此。此外，我们观察到 SCSQ “在切向同伦等价下是稳定的”：如果它对某个流形 \(M\) 上的所有向量丛都有一个肯定的答案，那么对于任何等价于 ~\(M\) 的流形切线同伦也是如此。我们的主要工具是拓扑 K 理论。在 \(B^{13}\) 上，本质上存在一类稳定的实向量束，但我们的方法失败了。