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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?

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  • $\begingroup$ Fair point, but the $n=2$ case is trivial (and the $n=3$ case only slightly less so). In any case, I've restricted the question to $n\geq 3$ for clarity. $\endgroup$
    – foliations
    Commented Jun 20, 2015 at 14:28

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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

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  • $\begingroup$ Thanks! Do you know what can happen for $d=3$ (i.e. $n=4$ in the original question)? $\endgroup$
    – foliations
    Commented Jun 20, 2015 at 15:04
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    $\begingroup$ The N that you construct is $S^{d+1}$ in general. For N is really $\Delta \cup -\Delta$, which is the boundary of $\Delta \times I$. Puncturing this gives an h-cobordism between N and a sphere, which is a smooth product by your dimension hypothesis. This argument breaks down if d=3, but (starting with Mazur's original construction) there are lots of homology 3-spheres that bound contractible manifolds, and for all that I know, the double N is $S^4$. $\endgroup$
    – Danny Ruberman
    Commented Jun 20, 2015 at 15:05

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