2017/07/15 by David González-Álvaro, González-Álvaro, David, Marcus Zibrowius +1 · 1 citation
Mathematics · Medicine · #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.1707.04711
openalex publication_date 2017/07/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The stable converse soul question (SCSQ) asks whether, given a real vector\nbundle (E ) over a compact manifold, some stabilization (E\× Rk ) admits\na metric with non-negative (sectional) curvature. We extend previous results to\nshow that the SCSQ has an affirmative answer for all real vector bundles over\nany simply connected homogeneous manifold with positive curvature, except\npossibly for the Berger space (B13 ). Along the way, we show that the same\nis true for all simply connected homogeneous spaces of dimension at most seven,\nfor arbitrary products of simply connected compact rank one symmetric spaces of\ndimensions multiples of four, and for certain products of spheres. Moreover, we\nobserve that the SCSQ is "stable under tangential homotopy equivalence": if it\nhas an affirmative answer for all vector bundles over a certain manifold (M ),\nthen the same is true for any manifold tangentially homotopy equivalent\nto~ (M ). Our main tool is topological K-theory. Over (B13 ), there is\nessentially one stable class of real vector bundles for which our method fails.\n