allen hatcher page 46 ex 1.23 I've recently asked a question about Hatcher's explanation of the deformation retraction of $R^3-A$, where $A$ a circle, to the wedge sum of $S^1$ & $S^2$ (page 46, ex 1.23). I didn't get an explanation. I thought of taking the circle to be $S^1$, to visualize it better , but I still don't understand why $R^3$ minus a  circle deformation retracts to a sphere with a diameter, then to the wedge sum of $S^1$ and $S^2$. Where does this diameter come from? Can anyone help me understand it please? Thank you.
 A: Hatcher's explanation is terse but I thought it's perfectly fine. I guess I will give you some details. Draw some pictures as you go.
Think of $A$ as the circle $\{(x,y,0)|x^2+y^2=0.25\}$ in $\mathbb{R}^3$. Note that this is strictly contained in the interior of the unit ball $B^2= \{(x,y,z)|x^2+y^2+z^2\leq 1 \}$


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*Your original space $\mathbb{R}^3 - A$ deformation retracts onto $B^2-A$ by sending every point outside of $B^2$ to $S^2 =  \{(x,y,z)|x^2+y^2+z^2= 1 \}$ via the map: $(x,y,z;t)\mapsto \frac{(x,y,z)}{|(x,y,z)|^t}$. 

*Choose an $\epsilon$ small enough (say $\epsilon = 0.1$) such that an $\epsilon$-neighbourhood $T$ of your circle $A$ is a torus. Then $B^2 - A$ deformation retracts onto $B^2 -T$ where $T$ (do you see why there is a deformation retract?). Observe that $B^2-T$ is a solid $2$-ball with an inner torus $T$ cut out.

*Fatten the torus outwards in all directions so that it either meets itself or touches $S^2$ (this once again describes a deformation retraction). You are then left with $S^2 \cup D$ where $D=\{(0,0,z)|-1\leq z \leq 1\}$ is the line connecting the poles $(0,0,1)$ and $(0,0,-1)$. This line is your diameter. If it helps, you can think of the torus as a torus shaped balloon and you're filling it with air within the confines of the ball.

*To get a homotopy from $S^2 \cup D$ to $S^2 \vee S^1$, simply contract a
meridian between the two poles. To get a deformation retraction, you will need to move the endpoints of your diameter closer and closer together when you're fattening the torus (by bending the torus).
I hope this helps. Let me know if you have questions on any of the steps.
