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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?

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

I have the following situation, let $\mathbb{S}^n$ 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?

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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foliations
foliations
  • 1.1k
  • 1
  • 10
  • 9

Topology of hypersurface of sphere fixed by homeomorphic involution

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

I have the following situation, let $\mathbb{S}^n$ 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?