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Frank Stones Memorial Colloquium

Ravi Shankar, University of Oklahoma

New Metrics of Non-Negative Sectional Curvature on a Family of 2-Connected 7-Manifolds

3:30 pm, Tuesday, November 14, TUC 352

It is a problem of great interest to determine which simply connected, closed manifolds admit metrics of non-negative sectional curvature. Considering Gromov’s Betti number bound there remains a wide gap between known examples and obstructions. We present a new method of construction, originally suggested by B. Wilking, for non-negative sectional curvature and use it to construct such metrics on a family of 2-connected 7-manifolds. Specifically we show that there are many 7-manifolds homeomorphic to, but not necessarily diffeomorphic to, S^3-bundles over S^4 that admit non-negative sectional curvature. In particular, all 28 smooth structures on the sphere S^7 admit such metrics. This completes the picture for exotic 7-spheres, where such metrics were known only for the Milnor spheres (those that are diffeomorphic to S^3-bundles over S^4) due to Gromoll – Meyer and Grove – Ziller. This is joint work with Sebastian Goette and Martin Kerin.

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