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I'm considering the following question - if M is an m-dimensional subspace of $\mathbb{R}^n$, then how to compute the homology of $\mathbb{R}^n - M$. Thanks!

Riley Zhang
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    Can you do the cases $\mathbb R^3 - { \rm point }$, $\mathbb R^3 - \mathbb R$ and $\mathbb R^3 - \mathbb R^2$? [Hint: Convince yourself that these spaces deformation-retract onto $S^2$, $S^1$ and $S^0$ respectively.] Can you see how this idea generalises? – Kenny Wong Oct 15 '17 at 19:37
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    What is $\mathbb R^n -M $ deformation retract onto? – Anubhav Mukherjee Oct 15 '17 at 20:07
  • Thank you both @KennyWong. But I still don't quite get this... Is $M$ always homeomorphic to $\mathbb{R}^k$? And is $\mathbb{R}^n - M$ deformation retract onto $S^{n - k - 1}$? – Riley Zhang Oct 25 '17 at 01:58

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