Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T05:19:22.736Z Has data issue: false hasContentIssue false
  • English
  • Français

Open manifolds with non-homeomorphic positively curved souls

Part of: Differential topology Topological $K$-theory 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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).

References

REFERENCES

Belegradek, I. Vector bundles with infinitely many souls. Proc. Amer. Math. Soc. 131 (2003), no. 7, 2217–2221.10.1090/S0002-9939-03-06863-1CrossRefGoogle Scholar
Belegradek, I., Kwasik, S. and Schultz, R.. Moduli spaces of nonnegative sectional curvature and non-unique souls. J. Differential Geom. 89 (2011), no. 1, 49–85.10.4310/jdg/1324476751CrossRefGoogle Scholar
Belegradek, I., Kwasik, S. and Schultz, R.. Codimension two souls and cancellation phenomena. Adv. Math. 275 (2015), 1–46.10.1016/j.aim.2015.02.006CrossRefGoogle Scholar
Cheeger, J. and Gromoll, D.. On the structure of complete manifolds of nonnegative curvature. Ann. of Math. (2) 96 (1972), 413–443.CrossRefGoogle Scholar
Chinburg, T., Escher, C and Ziller, W.. Topological properties of Eschenburg spaces and 3-Sasakian manifolds. Math. Ann. 339 (2007), no. 1, 3–20.CrossRefGoogle Scholar
Crowley, D. and NordströM, J.. The classification of 2-connected 7-manifolds. Preprint (2014), arXiv:1406.2226.Google Scholar
Crowley, D.. The classification of highly connected manifolds in dimensions 7 and 15. PhD Thesis, Indiana University (2002), arXiv:math/0203253.Google Scholar
Devito, J.. The classification of compact simply connected biquotients in dimensions 4 and 5. Differential Geom. Appl. 34 (2014), 128–138.CrossRefGoogle Scholar
Eschenburg, J.–H.. New examples of manifolds with strictly positive curvature. Invent. Math. 66 (1982), no. 3, 469–480.10.1007/BF01389224CrossRefGoogle Scholar
Eschenburg, J.–H.. Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen. Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie, Band 32 (1984).Google Scholar
Escher, C.. A diffeomorphism classification of generalized Witten manifolds. Geom. Dedicata 115 (2005), 79–120.CrossRefGoogle Scholar
Goette, S., Kerin, M. and Shankar, K.. Highly connected 7-manifolds and non-negative sectional curvature. Preprint (2017), arXiv:1705.05895.Google Scholar
GonzáLez–áLvaro, D.. Nonnegative curvature on stable bundles over compact rank one symmetric spaces. Adv. Math. 307 (2017), 53–71.CrossRefGoogle Scholar
GonzáLez–áLvaro, D. and Zibrowius, M.. The stable converse soul question for positively curved homogeneous spaces. Preprint (2017), arXiv:1707.04711.Google Scholar
Grove, K. and Ziller, W.. Curvature and symmetry of Milnor spheres. Ann. of Math. (2) 152 (2000), no. 1, 331–367.CrossRefGoogle Scholar
Grove, K. and Ziller, W.. Lifting group actions and nonnegative curvature. Trans. Amer. Math. Soc. 363 (2011), no. 6, 2865–2890.CrossRefGoogle Scholar
Hirsch, M.W.. Differential topology. Graduate Texts in Mathematics, 33 (Springer–Verlag, New York, 1976).10.1007/978-1-4684-9449-5CrossRefGoogle Scholar
Husemoller, D.. Fibre bundles. Third edition. Graduate Texts in Mathematics, 20 (Springer–Verlag New York, 1994).CrossRefGoogle Scholar
Kapovitch, V., Petrunin, A. and Tuschmann, W.. Non-negative pinching, moduli spaces and bundles with infinitely many souls. J. Differential Geom. 71 (2005), no. 3, 365–383.CrossRefGoogle Scholar
Kreck, M. and Stolz, S.. A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3)×SU(2)×U(1)-symmetry. Ann. of Math. (2) 127 (1988), no. 2, 373–388.CrossRefGoogle Scholar
Kreck, M. and Stolz, S.. Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature. J. Differential Geom. 33 (1991), no. 2, 465–486.10.4310/jdg/1214446327CrossRefGoogle Scholar
Kreck, M. and Stolz, S.. Nonconnected moduli spaces of positive sectional curvature metrics. J. Amer. Math. Soc. 6 (1993), no. 4, 825–850.10.1090/S0894-0347-1993-1205446-4CrossRefGoogle Scholar
Kruggel, B.. A homotopy classification of certain 7-manifolds. Trans. Amer. Math. Soc. 349 (1997), no. 7, 2827–2843.CrossRefGoogle Scholar
Kruggel, B.. Kreck–Stolz invariants, normal invariants and the homotopy classification of generalized Wallach spaces. Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 4, 469–485.CrossRefGoogle Scholar
Kruggel, B.. Homeomorphism and diffeomorphism classification of Eschenburg spaces. Quart. J. Math. Oxford Ser. (2) 56 (2005), no. 4, 553–577.10.1093/qmath/hah031CrossRefGoogle Scholar
Li, B. and Duan, H.. Spin characteristic classes and reduced KSpin group of a low dimensional complex. Proc. Amer. Math. Soc. 113 (1991), no. 2, 479–491.Google Scholar
Milgram, R.J.. The classification of Aloff–Wallach manifolds and their generalizations. Surveys on surgery theory, Vol. 1, 379–407, Ann. of Math. Stud., 145 (Princeton Univ. Press, Princeton, NJ, 2000).Google Scholar
Milnor, J.. Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. (2) 74 (1961), 575–590.CrossRefGoogle Scholar
Milnor, J. and Stasheff, J.. Characteristic classes. Annals of Mathematics Studies, no. 76. (Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974).10.1515/9781400881826CrossRefGoogle Scholar
Petersen, P. and Wilhelm, F.. An exotic sphere with positive sectional curvature. Preprint (2008), arXiv:0805.0812.Google Scholar
Rigas, A.. Geodesic spheres as generators of the homotopy groups of O, BO. J. Differential Geom. 13 (1978), no. 4, 527–545 (1979).CrossRefGoogle Scholar
Shankar, K.. Strong inhomogeneity of Eschenburg spaces. Appendix A by Mark Dickinson and the author. Michigan Math. J. 50 (2002), no. 1, 125–141.Google Scholar
Shankar, K., Tapp, K. and Tuschmann, W.. Nonnegatively and positively curved invariant metrics on circle bundles. Proc. Amer. Math. Soc. 133 (2005), no. 8, 2449–2459.CrossRefGoogle Scholar
Siebenmann, L.C.. On detecting open collars. Trans. Amer. Math. Soc. 142 (1969), 201–227.CrossRefGoogle Scholar
Thomas, E.. On the cohomology groups of the classifying space for the stable spinor groups. Bol. Soc. Mat. Mexicana (2) 7 (1962), 57–69.Google Scholar
Tuschmann, W.. Spaces and moduli spaces of Riemannian metrics. Front. Math. China 11 (2016), no. 5, 1335–1343.10.1007/s11464-016-0576-1CrossRefGoogle Scholar
Tuschmann, W. and Wraith, D.. Moduli spaces of Riemannian metrics. Second corrected printing. Oberwolfach Seminars, 46 (Birkhäuser Verlag, Basel, 2015).CrossRefGoogle Scholar
Wang, M. and Ziller, W.. Einstein metrics on principal torus bundles. J. Differential Geom. 31 (1990), no. 1, 215–248.CrossRefGoogle Scholar
Wilking, B.. Nonnegatively and positively curved manifolds. Surveys in differential geometry. Vol. XI, 25–62, Surv. Differ. Geom., 11 (Int. Press, Somerville, MA, 2007).Google Scholar
González-Álvaro, D. and Zibrowius, M.. Eschenburg Calculator. Zenodo (2018). doi:10.5281/zenodo.1173635.CrossRefGoogle Scholar