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Open manifolds with non-homeomorphic positively curved souls

Part of: Topological $K$-theory Differential topology Global differential geometry

Published online by Cambridge University Press:  11 July 2019

DAVID GONZÁLEZ-ÁLVARO  and
MARCUS ZIBROWIUS
DAVID GONZÁLEZ-ÁLVARO
Affiliation:
Université de Fribourg, Switzerland. e-mail: david.gonzalezalvaro@unifr.ch
MARCUS ZIBROWIUS
Affiliation:
Heinrich–Heine–Universität Düsseldorf, Germany e-mail: marcus.zibrowius@cantab.net
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Abstract

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

MSC classification

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Secondary: 19L64: Computations, geometric applications 57R22: Topology of vector bundles and fiber bundles
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 357 - 376
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

Supported by SNF grant 200021E-172469, the DFG Priority Programme SPP 2026 (Geometry at Infinity), and MINECO grants MTM2014-57769-3-P and MTM2014-57309-REDT.

‡

Partially supported by the DFG Research Training Group GRK 2240 (Algebro-geometric Methods in Algebra, Arithmetic and Topology).

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