Let $ X = \mathbb R^3 \setminus A$, where $A$ represents two linked circles. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on which to use van Kampen.

Any help would be appreciated. Thanks


The complement of two linked circles deformation retracts onto a wedge sum of a sphere and a torus. You should imagine the sphere as a big shell surrounding both circles, and you should imagine the torus as a tube tightly wrapping one circle and threading through the other; the torus lies in the interior of the sphere, and touches the sphere at one point.

This is hard to describe in words, so please see the image in Hatcher, Example 1.23, bottom of page 46.

The problem then reduces to the problem of finding the fundamental group of $S^2 \vee (S^1 \times S^1)$. This can be done using Van Kampen: take the open set $C$ to be an open neighbourhood of the $S^2$ and take $D$ to be an open neighbourhood of the $S^1 \times S^1$...

  • $\begingroup$ :how can i intuitively see that given space deformation retracts onto wedge sum of sphere and torus??i am really finding it difficult to get an idea about this....moreover,is it not possible to apply van kampen theorem directly on the space,ie.without using dformation retract?? $\endgroup$ – Abhishek Shrivastava Oct 15 '17 at 13:41
  • $\begingroup$ @AbhishekShrivastava Regarding visualisation... did you look at the diagram in the Hatcher's book? It may also be helpful to start with some simpler examples: (i) $\mathbb R^3$ minus the origin can be deformation-retracted onto a sphere, (ii) $\mathbb R^3$ minus a circle can be deformation-retracted onto a sphere with the north and south poles joined by a straight line segment, which in turn can be deformation retracted onto a sphere wedge a circle (see Hatcher, same example). In each case, you need to imagine that the space is made of clay, and you're trying to "flatten it out". $\endgroup$ – Kenny Wong Oct 15 '17 at 13:50
  • $\begingroup$ @AbhishekShrivastava As for using Van Kampen directly on the original space... I'm afraid I can't help you. I actually find it much easier to visualise these deformation retractions! $\endgroup$ – Kenny Wong Oct 15 '17 at 13:52
  • $\begingroup$ :thanks for your reply..i do understand the deformation retract in each of the examples you told,but,this one is the only one in which i am not able to visualize the deformation retract.I will be really thankful if u could demonstrate this...(i already looked in hatcher's but could not get intuition about this deformation retract) $\endgroup$ – Abhishek Shrivastava Oct 15 '17 at 14:13
  • $\begingroup$ @AbhishekShrivastava This is so hard to explain in words! I'm so sorry! Take a look here: math.stackexchange.com/questions/1621724/… $\endgroup$ – Kenny Wong Oct 15 '17 at 17:51

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