  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.
