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!



Is the space of immersions of $S^n$ into $\mathbb R^{n+1}$ simply connected?
I'm not a professional topologist by any means, but let me give this a shot. There's some discussion in the first few lectures of John Francis from [this course](http://math.northwestern.edu/~jnkf/classes/hprin/). Please correct me if I've made mistakes below...


By [Smale](http://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-0104227-9/) and [Hirsch](http://www.ams.org/journals/tran/1959-093-02/S0002-9947-1959-0119214-4/home.html), the space of immersions of $S^n$ into $R^{n+1}$ is homotopy equivalent to the space of **unbased** maps from $S^n$ into $V\_{n}(\mathbb{R}^{n+1})$, $Map(S^n,V\_n(\mathbb{R}^{n+1})$. Here $V\_n(\mathbb{R}^{n+1})$ is the [Stiefel manifold](http://en.wikipedia.org/wiki/Stiefel_manifold) of $n$ frames in $n+1$-space. $V\_{n}(\mathbb{R}^{n+1})$ is homeomorphic to $SO(n+1)$. 


See Ryan's answer for some more details.


Your question is then basically equivalent to characterizing $\pi\_1(Map(S^{n},SO(n+1)))$, the fundamental group of that mapping space. 


As B.S. points out in his answer, since $SO(n+1)$ is a group, $Map(S^n,SO(n+1))$ is homotopy equivalent to $SO(n+1)\times\Omega^n(SO(n+1))$, therefore the above fundamental group is $\pi\_1(SO(n+1))\times\pi\_1(\Omega^n(SO(n+1)))\cong\pi\_1(SO(n+1))\times\pi\_{n+1}(SO(n+1))$.


Though B.S. has pointed out that we can see that it's always nontrivial just from the first factor, we can in fact compute the group from known results on homotopy groups of $SO(n+1)$ (see e.g. [this table](http://www.math.uni-frankfurt.de/~johannso/SkriptAll/SkriptTopAlg/SkriptTopCW/homotop14.pdf) compiled from the literature by [Klaus Johannson](http://www.math.uni-frankfurt.de/~johannso/SkriptAll/SkriptTopAlg/SkriptTopCW/)).


The result for all $n$ is (scroll right in the grey box below):



```
n| 1   | 2      | 3              | 4         | 5    | 6      | 8s-1                | 8s             | 8s+1        | 8s+2      | 8s+3           | 8s+4      | 8s+5      | 8s+6   |
-| --- | ------ | -------------- | --------- | ---- | ------ | ------------------- | -------------- | ----------- | --------- | -------------- | --------- | --------- | ------ |
 | Z   | Z_2+Z  | Z_2+Z_2+Z_2    | Z_2+Z_2   | Z_2  | Z_2+Z  | Z_2+Z_2+Z_2+Z_2     | Z_2+Z_2+Z_2    | Z_2+Z+Z_2   | Z_2+Z_2   | Z_2+Z_2+Z_2    | Z_2+Z_2   | Z_2+Z_4   | Z_2+Z  |

```

In the rightmost columns, $s$ is any integer greater than or equal to 1, and the plus signs denote direct sum. Apologies for the ugly formatting of the table. 


