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? 
 A: 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.
A: 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. 
A: 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.
