Show there is a line through the origin in $\mathbb{R}^4$ disjoint from a set.

Question

I found this question on an old qualifying exam, but I don't know what ideas and/or theorems I should use to approach it. I tried to use $\mathbb{RP}^4$, but that is the only idea I have for the following:

Let $f:S^2 \to \mathbb{R}^4$ be a smooth map whose image $f(S^2)$ does not contain the origin. Show that there is a line through the origin in $\mathbb{R}^4$ disjoint from the image $f(S^2)$.

Thanks!

score 3 · Accepted Answer · answered Jul 09 '13 at 21:59

3

Your idea is fine. Assuming the contrary, one obtains a surjective smooth map from the two-dimensional $S^2$ to the three-dimensional $\mathbb R\mathbb P^4$ (use invariance of domain)

answered Jul 09 '13 at 21:59

Hagen von Eitzen

374,180

I had just missed the connection between my original idea and the contradiction that you describe. Some things are so clear in retrospect! – phenomenalwoman4 Jul 11 '13 at 03:05

Show there is a line through the origin in $\mathbb{R}^4$ disjoint from a set.

1 Answers1