# How to calculate the fundamental group of $S^3$ without two linked cirles

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.

• 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? – user98602 Nov 19 '14 at 23:08
• Hint: Look at Example 1.23 in Hatcher. – Brandon Carter Nov 19 '14 at 23:11
• 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. – Will Jagy Nov 19 '14 at 23:51