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On pg. 47 of Hathcer's Algebraic Topology, the author discusses the fundamental group of $\mathbf R^n-(A\cup B)$, where $A$ and $B$ are circles in $\mathbf R^3$ which are linked.

The author writes that $\mathbf R^3-A\cup B$ deformation retracts to the wedge sum of a torus and a $2$-sphere.

enter image description here

I was unable to see how this is so. Can somebody please help me visualize a deformation retract.

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  • $\begingroup$ Can you see how 1) everything outside the sphere deforms onto the sphere, and 2) everything inside the torus deforms onto the torus? $\endgroup$
    – Arthur
    Jan 21 '16 at 23:41
  • $\begingroup$ Yes. That is clear. The points inside the sphere and outside the torus are difficult to deform. $\endgroup$ Jan 21 '16 at 23:45
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    $\begingroup$ Blow up the other circle until it touches the torus. Continue to blow up until everything is pressed to the torus or the sphere. $\endgroup$ Jan 21 '16 at 23:47
  • $\begingroup$ @DanielFischer Never discuss this at an airport. $\endgroup$ Jan 22 '16 at 2:27
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    $\begingroup$ It's easy to see how you can deformation-retract the complement of the other circle to the complement of a solid torus whose soul is the other circle without moving any point at a distance greater than $\varepsilon$ from the circle. Now imagine that the boundary of the solid torus is made of balloon rubber, and that the space inside the sphere with the two tori removed is filled with a really soft and very compressible substance. Something like this, only more so. Now inflate the small torus, considering the sphere and the first torus rigid. $\endgroup$ Jan 22 '16 at 13:20
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This is a partial answer.

To see the deformation retraction, see the flow lines in the diagram below. It shows the flow lines in the vertical and horizontal slices. For the intermediary slices, we gradually move from horizontal slice to the vertical slice.

(In the diagram, dashed circles are the slices of the torus; bold circles and points are the slices of the removed circles).

enter image description here

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  • $\begingroup$ The picture here "i.stack.imgur.com/yt7WE.png" is relevant to the discussion. What is the appropriate place to post it? Should I post it in a comment? $\endgroup$ Apr 13 '17 at 17:15
  • $\begingroup$ I added your image to your post. IIRC you need something like 25 rep points to add the image yourself. I also handled the flags. It seems to me that the votes to close came because the first version of your answer sounded a bit like "I, too, have this same question" to the voters. That would be a valid reason to vote to delete, because in this slot you should only post answers. A partial answer, marked as such (like you did), is fine. Welcome to the site. We are a bit strict about a few things. You will learn the do's and do-not's quickly enough :-) $\endgroup$ Apr 13 '17 at 20:16

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