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Non-Negative Curvature, Elliptic Genus and Unbounded Pontryagin Numbers

Part of: Differential topology Global differential geometry

Published online by Cambridge University Press:  26 February 2018

Martin Herrmann  and
Nicolas Weisskopf
Martin Herrmann*
Affiliation:
Fakultät für Mathematik, Karlsruher Institut für Technologie, Kaiserstraße 89–93, 76133 Karlsruhe, Germany (martin.herrmann@riemannian-topology.de)
Nicolas Weisskopf
Affiliation:
Département de Mathématiques, Université de Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland (nic.weisskopf@gmail.com)
*
*Corresponding author.
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Abstract

We discuss the cobordism type of spin manifolds with non-negative sectional curvature. We show that in each dimension 4k ⩾ 12, there are infinitely many cobordism types of simply connected and non-negatively curved spin manifolds. Moreover, we raise and analyse a question about possible cobordism obstructions to non-negative curvature.

Keywords

non-negative curvaturespin manifoldselliptic genusPontryagin numbers

MSC classification

Primary: 53C20: Global Riemannian geometry, including pinching
Secondary: 57R75: ${rm O}$- and ${rm SO}$-cobordism
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 449 - 456
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538.Google Scholar
2.Dessai, A., Characteristic numbers of positively curved Spin-manifolds with symmetry, Proc. Amer. Math. Soc. 133(12) (2005), 3657–3661.Google Scholar
3.Dessai, A., Obstructions to positive curvature and symmetry, Adv. Math. 210(2) (2007), 560–577.Google Scholar
4.Dessai, A. and Tuschmann, W., Nonnegative curvature and cobordism type, Math. Z. 257(1) (2007), 7–12.Google Scholar
5.Hirzebruch, F., Berger, T. and Jung, R., Manifolds and modular forms, Aspects of Mathematics, Volume E20 (Friedr. Vieweg & Sohn, Braunschweig, 1992). With appendices by Nils-Peter Skoruppa and by Paul Baum.Google Scholar
6.Hirzebruch, F. and Slodowy, P., Elliptic genera, involutions, and homogeneous spin mani-folds, Geom. Dedicata, 35(1–3) (1990), 309–343.Google Scholar
7.Kotschick, D., Pontryagin numbers and nonnegative curvature. J. Reine Angew. Math. 646 (2010), 135–140.Google Scholar
8.Lott, J., Â-genus and collapsing, J. Geom. Anal. 10(3) (2000), 529–543.Google Scholar
9.Milnor, J. W. and Stasheff, J. D., Characteristic classes, Annals of Mathamatics Studies, Volume 76 (Princeton University Press, Princeton, NJ, 1974).Google Scholar
10.Ochanine, S., Genres elliptiques équivariants, In Elliptic curves and modular forms in algebraic topology, Lecture Notes in Mathematics, Volume 1326, pp. 107–122 (Springer, Berlin, 1988).Google Scholar
11.Singhof, W., On the topology of double coset manifolds, Math. Ann. 297(1) (1993), 133–146.Google Scholar
12.Stewart, T. E., Lifting group actions in fibre bundles. Ann. of Math. (2) 74 (1961), 192–198.Google Scholar
13.Weisskopf, N., Positive curvature and the elliptic genus, 2013, arXiv:1305.5175 [math.DG].Google Scholar