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RBega2
RBega2
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Suppose $\Omega\subset \mathbb{S}^n$ is an open set with $\Sigma=\partial \Omega$ a smooth hypersurface. If $\Omega$ is a homology ball, in the sense that $H_i(\Omega)=H_i(\mathbb{D}^n)$ for $0\leq i \leq n$, then is it possible for $\Omega$ to be non-contractible?

Some (possibly incorrect) observations:

  • When $n=1,2$ it is clear any such $\Omega$ is contractible.
  • For all $n>1$, I believe $\Sigma$ is a homology sphere (via Poincare duality) so when $n=3$ there is no such non-contractible $\Omega$ (using the classification of surfaces and Alexander's theorem -- is there a simpler argument?)
  • By removing a small ball from a homology sphere one can produce a manifold with boundary, $X$, that is a homology ball that is not contractible. However, by the generalized Schoenflies conjecture of Brown and Mazur, such an $X$ can't embed in $\mathbb{S}^n$.

EDIT: For what it is worth, this was essentially asked (and answered) previously in this question

Suppose $\Omega\subset \mathbb{S}^n$ is an open set with $\Sigma=\partial \Omega$ a smooth hypersurface. If $\Omega$ is a homology ball, in the sense that $H_i(\Omega)=H_i(\mathbb{D}^n)$ for $0\leq i \leq n$, then is it possible for $\Omega$ to be non-contractible?

Some (possibly incorrect) observations:

  • When $n=1,2$ it is clear any such $\Omega$ is contractible.
  • For all $n>1$, I believe $\Sigma$ is a homology sphere (via Poincare duality) so when $n=3$ there is no such non-contractible $\Omega$ (using the classification of surfaces and Alexander's theorem -- is there a simpler argument?)
  • By removing a small ball from a homology sphere one can produce a manifold with boundary, $X$, that is a homology ball that is not contractible. However, by the generalized Schoenflies conjecture of Brown and Mazur, such an $X$ can't embed in $\mathbb{S}^n$.

Suppose $\Omega\subset \mathbb{S}^n$ is an open set with $\Sigma=\partial \Omega$ a smooth hypersurface. If $\Omega$ is a homology ball, in the sense that $H_i(\Omega)=H_i(\mathbb{D}^n)$ for $0\leq i \leq n$, then is it possible for $\Omega$ to be non-contractible?

Some (possibly incorrect) observations:

  • When $n=1,2$ it is clear any such $\Omega$ is contractible.
  • For all $n>1$, I believe $\Sigma$ is a homology sphere (via Poincare duality) so when $n=3$ there is no such non-contractible $\Omega$ (using the classification of surfaces and Alexander's theorem -- is there a simpler argument?)
  • By removing a small ball from a homology sphere one can produce a manifold with boundary, $X$, that is a homology ball that is not contractible. However, by the generalized Schoenflies conjecture of Brown and Mazur, such an $X$ can't embed in $\mathbb{S}^n$.

EDIT: For what it is worth, this was essentially asked (and answered) previously in this question

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RBega2
RBega2
  • 2.5k
  • 1
  • 9
  • 18

Can a smooth domain in a sphere be a homology ball without being contractible?

Suppose $\Omega\subset \mathbb{S}^n$ is an open set with $\Sigma=\partial \Omega$ a smooth hypersurface. If $\Omega$ is a homology ball, in the sense that $H_i(\Omega)=H_i(\mathbb{D}^n)$ for $0\leq i \leq n$, then is it possible for $\Omega$ to be non-contractible?

Some (possibly incorrect) observations:

  • When $n=1,2$ it is clear any such $\Omega$ is contractible.
  • For all $n>1$, I believe $\Sigma$ is a homology sphere (via Poincare duality) so when $n=3$ there is no such non-contractible $\Omega$ (using the classification of surfaces and Alexander's theorem -- is there a simpler argument?)
  • By removing a small ball from a homology sphere one can produce a manifold with boundary, $X$, that is a homology ball that is not contractible. However, by the generalized Schoenflies conjecture of Brown and Mazur, such an $X$ can't embed in $\mathbb{S}^n$.