On negatively curved bundles with hyperbolic fibers outside the Igusa stable range

TL;DR

This paper demonstrates that the space of negatively curved metrics on hyperbolic manifolds has complex topological structure, with nontrivial rational homotopy groups, and constructs bundles with negatively curved fibers representing elements of infinite order.

Contribution

It establishes the nontriviality of rational homotopy groups of negatively curved metric spaces and constructs bundles with negatively curved fibers representing infinite order elements.

Findings

Nontrivial rational homotopy groups in the space of negatively curved metrics.

Existence of bundles over spheres with negatively curved fiber metrics.

Representation of infinite order elements in the homotopy groups of the classifying space.

Abstract

We prove that the Teichm\"{u}ller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in $\pi_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.