I need to find:

the fundamental group of the space obtained by cutting out the three-dimensional $S^3$ sphere of two circles, once linked with each other.

Can you help me? I have no idea about it, i just find http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf but this doesn't help me.

  • 2
    $\begingroup$ Consider $S^3$ as the unit sphere of $\Bbb C^2$ and the linked circles as the set $\{|z_0|=0 \text{ or } |z_1|=0\}$. Can you write down an explicit deformation retraction onto a nicer subspace? $\endgroup$ – user98602 Nov 19 '14 at 23:08
  • $\begingroup$ Hint: Look at Example 1.23 in Hatcher. $\endgroup$ – Brandon Carter Nov 19 '14 at 23:11
  • 1
    $\begingroup$ Retraction not so explicit, but this is the same as $\mathbb R^3$ without the $z$-axis and without the standard unit circle $x^2 + y^2 = 1$ in the $xy$-plane. $\endgroup$ – Will Jagy Nov 19 '14 at 23:51

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