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October 2021 The stable converse soul question for positively curved homogeneous spaces
David González-Álvaro, Marcus Zibrowius
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J. Differential Geom. 119(2): 261-307 (October 2021). DOI: 10.4310/jdg/1632506394

Abstract

The stable converse soul question (SCSQ) asks whether, given a real vector bundle $E$ over a compact manifold, some stabilization $E \times \mathbb{R}^k$ 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.

Funding Statement

D. González-Álvaro received support from: the MINECO grants MTM2014-57769-3-P, MTM2014-57309-REDT and MTM2017-85934-C3-2-P, the SNSF-Project 200021E-172469 and the DFG-Priority programme Geometry at Infinity (SPP 2026).

Acknowledgment

Part of the research for this publication was conducted in the framework of the DFG Research Training Group 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology.

Citation

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David González-Álvaro. Marcus Zibrowius. "The stable converse soul question for positively curved homogeneous spaces." J. Differential Geom. 119 (2) 261 - 307, October 2021. https://doi.org/10.4310/jdg/1632506394

Information

Received: 29 April 2018; Accepted: 6 December 2019; Published: October 2021
First available in Project Euclid: 27 September 2021

Digital Object Identifier: 10.4310/jdg/1632506394

Subjects:
Primary: 53C21
Secondary: 19L64, 57R22

Rights: Copyright © 2021 Lehigh University

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Vol.119 • No. 2 • October 2021
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