Let $S^n \times \mathbb{T}^n$ be the trivial Torus bundle over $S^n$. Assume that we have a continuous fiber preserving map $\phi :TS^n \to S^n \times \mathbb{T}^n$ which restriction to each fiber gives us a covering space structure.

Does this imply that $n=1,3,7$?

**Example:** Existence of such structure gives us a covering space structure $\alpha: TS^n \to S^n \times \mathbb{T}^n$. Then a lifting process gives us a non vanishing section for the tangent bundle of sphere. Hence for $n=2k$ such structure can not exist.

More generally assume that $E$ and $X$ are $n$ dimensional vector and (not necessarily trivial) torus bundles over a topological space $Y$, respectively. What type of obstructions would appear for existence of a fiber preserving covering map $\alpha :E\to X$?

**Added:** According to comment of Mark Grant we ask :"Is there a nontrivial vector bundle $E$ and torus bundle $X$ of the same dimension over a space $Y$ which admit a fiber preserving map $\alpha: E \to X$ whose restriction to each fiber is a covering map?