The space of metrics of positive scalar curvature

TL;DR

This paper explores the topology of the space of positive scalar curvature metrics on high-dimensional spheres and spin manifolds, revealing infinite order elements in their homotopy and homology groups and constructing fiber bundles with non-vanishing A-hat-genera.

Contribution

It introduces new infinite order elements in the homotopy and homology groups of these metric spaces and constructs fiber bundles with non-vanishing A-hat-genera, advancing understanding of their topological structure.

Findings

Infinite order elements in higher homotopy groups of metric spaces.

Construction of fiber bundles with non-zero A-hat-genera.

Demonstration of non-multiplicativity of the A-hat-genus in certain bundles.

Abstract

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics. Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing A-hat-genera, thus establishing the non-multiplicativity of the A-hat-genus in fibre bundles with simply connected base.