TITLE: What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?
QUESTION [12 upvotes]: 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?

REPLY [6 votes]: 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.