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.

score 3 · Answer 1 · answered Aug 09 '13 at 05:15

3

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?

answered Aug 09 '13 at 05:15

Tim kinsella

5,903

Dear Tim, this is a very nice answer:+1. It is not obvious however that $M\to T^n$ is a covering. But it is true: a proper local diffeomorphism is indeed a covering. In the absence of properness this is completely false, as any open embedding shows. – Georges Elencwajg Aug 09 '13 at 06:22
@GeorgesElencwajg : Thanks! Yes so I guess compactness of M is used in each of the three lines of the proof. – Tim kinsella Aug 09 '13 at 06:42

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

1 Answers1