Can we embedd $S^2$ into $S^3$?

Question

More generally, for what values of $m<n$ can we have a topological embedding $f:S^m\longrightarrow S^n$? Is this problem resolved.

In particular, I want to know two cases:

a)$ m=2, \; n=3$

b) $m=2,\; n=5$ or $7$ or any odd

Answer 1

Short answer: any values of $m,n$ as long as $n\geq m$. These are simply restrictions of embeddings of $\mathbb{R}^m \to \mathbb{R}^n$:

$$ F(x_1, \ldots, x_m) \;\; =\;\; (x_1, \ldots, x_m, \underbrace{0,0,\ldots, 0}_{n-m \; \text{times}}). $$

Answer 2

As $\dim(S^n) = n$ (covering dimension e.g.) and if $X$ embeds into $Y$ implies $\dim(X) \le \dim(Y)$ we have the immediate restriction $m \le n$. And to achieve that we just add $0'$ to the point of $S^m$ (a generalisation of the circle ($S^1$) embedding as the equator in the $2$-sphere etc.) The same holds for $\Bbb R^m$ and $\Bbb R^n$ of course, same argument.