Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?

share|cite|improve this question
It seems like you should be able to prove this more or less by induction. – Aaron Mazel-Gee Nov 30 '10 at 6:41

Let $e_0, \ldots, e_n$ be the standard basis of $\mathbb {R}^{n+1}$. Take $\epsilon$ small. Consider the vector $v_1$ of length 1 in the span of $e_0, e_1$. Then the vector $v_2$ of length $\epsilon$ in the span of $v_1, e_2$, and in general the vector $v_i$ of length $\epsilon^{i-1}$ in the span of $v_{i-1}, e_i$. Now consider the vector $w=v_1+\ldots v_n$ For small $\epsilon$ the set of $w$'s is a torus embedded $\mathbb {R}^{n+1}$ (any $\epsilon < 1$ will do, actually).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.