Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the $n$-torus $T^n = S^1 × . . . × S^1?$

Question

I do not know how to do the following qualifying exam problem. Any helped is nice.

Let $M$ be a smooth simply-connected compact manifold of dimension $n$. Is there an immersion of $M$ into the $n$-torus $T^n = S^1 × . . . × S^1?$

Answer 1

No. It must be surjective since its image is open and the torus is connected. But then it's the universal cover of the torus (I.e. $\mathbb{R}^n$) which is not compact. Does that check out?