Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

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.