  Consider the space of analytic, quasi-periodic cocycles on the higher
dimensional torus. We provide examples of cocycles with nontrivial Lyapunov
spectrum, whose homotopy classes do not contain any cocycles satisfying the
dominated splitting property. This shows that the main result in the recent
work "Complex one-frequency cocycles" by A. \'Avila, S. Jitomirskaya and C.
Sadel does not hold in the higher dimensional torus setting.
