Let $M$ be a smooth (or complex) manifold, $N$ — topological manifold and $p: M \to N$ covering map. Consider the smooth (complex) structure on $N$ obtained by the well-known procedure making $p$ into smooth (holomorphic) map.
Is that true that from triviality of the (holomorphic) tangent bundle $TM$ follows triviality of $TN$? Is there any relation between those two whatsoever?