Bundles with Non-multiplicative Â-Genus and Spaces of Metrics with Lower Curvature Bounds

Abstract

Abstract We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing $\hat{A}$-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all ${{\operatorname{Spin}}}$ manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and $S^n \times S^n$ or $S^n \times S^{n+1}$, respectively. We also construct manifolds $M$ whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.