I'm not an topologist, so I apologize in advance if this is a silly question.


I have the following situation, let $\mathbb{S}^n$ (for $n\geq 3$) be the standard (smooth if it matters) $n$-sphere and $\Sigma\subset \mathbb{S}^n$ a connected hypersurface. Let $\Omega\_\pm$ be the two components of of $\mathbb{S}^n\backslash \Sigma$ (there are exactly two by Jordan-Brouwer). Suppose moreover there is a non-trivial homeomorphic involution $\phi:\mathbb{S}^n\to \mathbb{S}^n$ which fixes $\Sigma$ in the strong sense that it restricts to the identity map and swaps $\Omega\_\pm$.


My question is how much can one say about $\Sigma$?


So far what I can say (I only sketched this in my head so may have made a mistake):


* By Mayer-Vietoris, the homology groups of the $\Omega\_\pm$ vanish and $\Sigma$ is a homology sphere.
* By Seifert-Van Kampen, $\pi\_1(\Omega\_\pm)=0$ and so by Hurewicz, all the homotopy groups of $\Omega\_\pm$ vanish.


However, I'm now stuck as I don't see a reason for $\pi\_1(\Sigma)$ to vanish and don't know enough examples to know if this can even be true. 


Can one go further? Is $\Sigma$ a homotopy sphere?



Topology of hypersurface of sphere fixed by homeomorphic involution
In "Smooth Homology Spheres and their Fundamental Groups" Kervaire proves that i) every 4-dimensional homology sphere bounds a contractible smooth manifold, and ii) for $d \geq 5$, every $d$-dimensional homology sphere bounds a contractible smooth manifold, after perhaps changing it by connect-sum with an exotic sphere.


Hence, let $\Sigma^d$ be any homology sphere of dimension $d \geq 4$, and modify it if necessary by connected-sum with a homotopy sphere so that it bounds a contractible manifold $\Delta^{d+1}$. Then
$$N := \Delta \cup\_\Sigma \Delta$$
has an obvious (smooth) involution with fixed set $\Sigma$. Furthermore, this is easily seen to be a homotopy sphere (by Mayer--Vietoris and Seifert--van Kampen). It may not be diffeomorphic to $S^{d+1}$, but as it has an orientation-reversing involution it will have order $2$ in the group $\Theta\_{d+1}$ of exotic spheres. When $d+1=5,6, 12, 13$ this group is trivial or $\mathbb{Z}/3$, and so $N$ must in fact be diffeomorphic to $S^{d+1}$ in these cases.


Thus $\Sigma$ need not be a homotopy sphere in general


