# Deformation Retract of Complement of Two Linked Circles in $\mathbf R^3$

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.

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

• Can you see how 1) everything outside the sphere deforms onto the sphere, and 2) everything inside the torus deforms onto the torus? Jan 21 '16 at 23:41
• Yes. That is clear. The points inside the sphere and outside the torus are difficult to deform. Jan 21 '16 at 23:45
• Blow up the other circle until it touches the torus. Continue to blow up until everything is pressed to the torus or the sphere. Jan 21 '16 at 23:47
• @DanielFischer Never discuss this at an airport. Jan 22 '16 at 2:27
• 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. Jan 22 '16 at 13:20