Negatively curved bundles in the Igusa stable range

TL;DR

This paper investigates the rational homotopy groups of negatively curved metric spaces on high-dimensional manifolds and shows that certain negatively curved bundles over spheres have finite order in the homotopy groups of the classifying space.

Contribution

It introduces new connections between smoothing theory and the topology of negatively curved bundles, revealing finite order elements in the homotopy groups of classifying spaces.

Findings

Rational homotopy groups of negatively curved metric spaces are characterized.

Negatively curved bundles over spheres represent finite order elements in homotopy groups.

High-dimensional manifolds exhibit specific topological properties related to negative curvature.

Abstract

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.