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

We prove that for all n = 4k − 2 and k ≥ 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T<0(M)). 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