Negatively curved bundles in the Igusa stable range

Abstract

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of Riemannian metrics without conjugate points on a high‐dimensional manifold with hyperbolic fundamental group. As a consequence, we show that spaces of negatively curved Riemannian metrics have in general nontrivial rational homotopy groups. We also show that smooth M‐bundles over spheres equipped with fiberwise negatively curved metrics represent elements of finite order in the homotopy groups πiBDifffalse(Mfalse) of the classifying space for smooth M‐bundles, provided i≪dimM.

Riemannian foliation with exotic tori as leaves

“…Note that here we do not need to assume is odd as for even in the Igusa stable range (see also [, Theorem 4.1]). Now we apply [, Lemma 3] by taking both fibration as , we have for . Hence we have for .…”

“…Proof All the ingredients of the proof can be found in . We first need Morlet's comparison theorem (see [; , Section 2]).…”

“…Proof All the ingredients of the proof can be found in . We first need Morlet's comparison theorem (see [; , Section 2]). Let and be the th orthogonal group.…”

“…Now Morlet's comparison theorem says, as long as , there exists a map which induces an injective correspondence of connected components and a weak homotopy equivalence on each component. Now we take to be the exotic torus , we have for As indicated at the end of [, Section 4], the fiber is simply connected and for as . Note that here we do not need to assume is odd as for even in the Igusa stable range (see also [, Theorem 4.1]).…”