TITLE: Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected?
QUESTION [18 upvotes]: The title is the question.  Sorry, this isn't quite research level.  I imagine the answer is well-known, just not to me.  Thanks for any help!

REPLY [13 votes]: Your space is never simply connected (for $n\geq 1$). As already answered, it is (weakly) homotopy equivalent by Smale-Hirsch h-principle to the space of unbased maps from $S^n$ to $SO(n+1)$, which is itself $SO(n+1)\times \Omega^n SO(n+1)$
($\Omega^n$ = based maps from $S^n$). So $\pi_1$ is at least $\pi_1(SO(n+1))$ (and more in general).
In fact it is not even connected as soon as $\pi_n SO(n+1)$ isn't trivial, which occurs for $n=1,3,4,5$ and many more (maybe all except $n=2$?).
But it is connected for $n=2$, which awarded its celebrity to Smale (sphere eversion), even if his advisor Raoul Bott didn't believed it at first.
EDIT: according to j.c. $\pi_n SO(n+1)$ is trivial only for $n=2,6$. Quite an interesting fact !