Teichmüller space of negatively curved metrics on Gromov–Thurston manifolds is not contractible

Abstract

In this paper we prove that for all n = 4k − 2, k ≥ 2 there exists closed ndimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that π1(T <0 (M )) is non-trivial. T <0 (M ) denotes the Teichmüller space of all negatively curved Riemannian metrics on M , which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity. Gromov Thurston branched cover manifolds provide examples of negatively curved manifolds that do not have the homotopy type of a locally symmetric space. These manifolds will be used in this paper to prove the above stated result.

Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

“…This paper builds on arguments from the paper [18] that proves a similar result for Gromov-Thurston manifolds that support negatively curved Riemannian metrics but are not hyperbolic. Let us recall some terminology from that paper:…”

“…This section also uses most arguments from [18] but since the Pontryagin numbers of a complex manifold need not all vanish, we have to change some arguments for this paper.…”

“…This construction has been discussed in more details with the references and arguments needed to carry it out in [18].…”

“…This will be achieved by assuming that there is such a smooth isotopy and arriving at a contradiction. This section also uses most arguments from [18] but since the Pontryagin numbers of a complex manifold need not all vanish, we have to change some arguments for this paper.…”

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

“…The Teichmüller space T (M ) = BDiff 0 (M ) classifies equivalence classes of smooth bundles with abstract fiber M and equipped with a fiber homotopy trivialization, whereas T <0 (M ) classifies equivalence classes of negatively curved bundles with abstract fiber M and equipped with a fiber homotopy trivialization. It follows from the results obtained in [FO09,Sor14,FS17] that there are negatively curved bundles which are nontrivial as smooth bundles with abstract fiber a hyperbolic manifold, a Gromov-Thurston manifold or a complex hyperbolic manifold. However they all represent elements of finite order in the homotopy groups of T (M ).…”