Bundles with non-multiplicative $\hat{A}$-genus and spaces of metrics with lower curvature bounds

TL;DR

This paper constructs specific smooth bundles with non-zero $$-genus to explore the topology of metric spaces with curvature bounds, revealing non-trivial homotopy groups in these spaces for certain Spin-manifolds.

Contribution

It introduces new bundles with non-multiplicative $$-genus and uses them to identify non-trivial rational homotopy groups in spaces of Riemannian metrics with curvature bounds, expanding understanding of geometric topology.

Findings

Non-vanishing -genus in constructed bundles.

Identification of non-trivial rational homotopy groups in metric spaces.

Existence of elements of infinite order in homotopy groups of positive scalar curvature metrics.

Abstract

We construct smooth bundles with base and fiber products of two spheres whose total spaces have non-vanishing $\hat{A}$-genus. We then use these bundles to locate non-trivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin-manifolds of dimension six or at least ten 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 which lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.