Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

47 Views Asked by At

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$.

For example, we can have:

(1) $S^1$ and $S^1$ can be linked in $\mathbb{R}^3$ $\to$ True.

My question is that can we have:

(2) $S^2$ and $S^1$ can be linked in $\mathbb{R}^4$? $\to$ True?

(3) $S^1$ and $S^1$ can be linked in $\mathbb{R}^5$? $\to$ False?

(4) $S^1$ and $S^2$ can be linked in $\mathbb{R}^5$? $\to$ False?

(5) $S^1$ and $S^3$ can be linked in $\mathbb{R}^5$? $\to$ True?

(6) $S^2$ and $S^2$ can be linked in $\mathbb{R}^5$? $\to$ True?

And the answers are the same if we replace the $\mathbb{R}^d$ space by the ${S}^d$ space, correct?