Teichm\"uller space of negatively curved metrics on Complex Hyperbolic Manifolds is not contractible

TL;DR

This paper demonstrates that for certain complex hyperbolic manifolds, the Teichmüller space of negatively curved metrics is not contractible, revealing complex topological properties in high-dimensional geometric structures.

Contribution

It proves the existence of non-contractible Teichmüller spaces for negatively curved metrics on specific complex hyperbolic manifolds in dimensions 4k-2.

Findings

Existence of manifolds with non-trivial fundamental group of the Teichmüller space

Teichmüller space of negatively curved metrics is not contractible in these cases

Results hold for all dimensions n=4k-2, k≥2

Abstract

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $\pi_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichm\"uller 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.