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

Abstract

We prove that the Teichmüller 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 $π_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.

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