# What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

What are some interesting and easy-to-understand (for non-differential geometers) facts about subobjects of $\mathbb{R}^4$ that are not only false in $\mathbb{R}^3$, but also specific to the structure of $\mathbb{R}^4$ and maybe do not easily or naturally generalize to higher dimensions?

• Do knotted 2-spheres count? – N. Owad Jan 11 '16 at 18:23
• Sounds interesting! – Damian Reding Jan 11 '16 at 20:12
• Does it have to be true and uninteresting in $\mathbb{R}^5$, or can it be also false in $\mathbb{R}^5$? – Brian Tung Jan 13 '16 at 21:44
• Sure, I was just trying to weaken the requirement on what happens in $\mathbb{R}^5$. – Damian Reding Jan 14 '16 at 0:30

Only for $n=4$ does there exist an open set $U\subseteq\mathbb{R}^n$ that is homeomorphic to $\mathbb{R}^n$ but not diffeomorphic to $\mathbb{R}^n$ (a small exotic $\mathbb{R}^4$). What this means is not too difficult to explain (no need to explain what a manifold is, only what a homeomorphism and a diffeomorphism are between open subsets of $\mathbb{R}^n$). I don't think it qualifies as "uninteresting for $\mathbb{R}^5$", though (it's definitely not a triviality in any dimension other than $1$), but you seemed to say "false" was also OK.

Due to the theory of quaternions, due to Hamilton, $\bf R^4$ has a structure of a of non commutative field. The only dimension for which $\bf R^n$ is a field are $n=1,2, 4$ . As an application, the special orthogonal group in dimension 4 is not simple : it is the quotient of $U\times U$ by it center $Z/2Z$ where $U$ is the unitary group in complex dimension 2, or the set of quaternions of norm 1. In other dimension (other than 2) the special orthogonal group modulo its center is simple.

• But then this is not true in $\mathbb{R}^5$, is it? – Brian Tung Jan 13 '16 at 21:48

Lets bump knot theory up a dimension. In general, an $n$-sphere can be non-trivially knotted in $\mathbb{R}^{n+2}$. Obviously, an $n$-sphere can be embedded in $\mathbb{R}^{n+1}$, where it is usually defined, but no knotting can occur. In $\mathbb{R}^{n+3}$, we can use the extra dimension and unknot every $n$-sphere.

So, for $n=2$, we have that a $2$-sphere can be knotted in $\mathbb{R}^{4}$, cannot be knotted in $\mathbb{R}^{3}$, and is uninteresting in $\mathbb{R}^{5}$, since there is only one knot type.

If you want to look into this more, you should look at any number of great books: Rolfsen, Adams, and more that I am just not thinking of at the moment.

Edit: As per Mike's comment, we should be assuming PL or locally flat here.

• You have not said precisely what kind of knots you're considering. Your statement that the only knotted spheres in $\Bbb R^{n+2}$ are $n$-spheres; this is true if you're working with locally flat or PL maps, but not smoothly: Haefliger discovered many examples of higher codimension, smoothly knotted spheres; the first is an $S^3$ in $S^6$. It is true that 2-spheres are smoothly unknotted in $S^5$ but I do not think it is uninteresting until $S^7$. – user98602 Jan 15 '16 at 15:58
• You make a good point. I do very little above dimension 4, so I will defer to you in this. So, I suppose I should assume locally flat or PL to make my answer complete. Thank you for the correction. – N. Owad Jan 15 '16 at 17:39