Is $S^1\vee\ldots\vee S^1$ a deformation retract of $\mathbb R^2-\{x^1,\ldots,x^n\}$?

Question

It is easy show that $\mathbb R^2-\{0\}$ can be retracted to $S^1$ via the map $r: x\mapsto x/\|x\|$.

How can I get a retraction from $\mathbb R^2-\{x_0,x_1\}$ to $S^1\vee S^1$? Any idea?

Answer 1

For two points it is very easy. Let $x_0=(0,0), x_1 = (2,0)$. Identify $S^1 \vee S^1$ with $C_0 \cup C_1$, where $C_i$ is the circle with center $x_i$ and radius $1$. Let $D_i$ be the closed disk with center $x_i$ and radius $1$. Then $D_i \setminus \{x_i\}$ strongly deformation retracts to $C_i$ (radial retraction to the boundary). The "exterior" $E = \mathbb R^2 \setminus \operatorname{int}(D_0 \cup D_1)$ strongly deformation retracts to $L = C_0 \cup C_1 \cup (-\infty,-1] \times \{0\} \cup [3,\infty) \times \{0\}$ (shift each point of $E$ vertically in direction of the $x$-axis until it reaches $L$). But $C_0 \cup C_1$ is clearly a strong deformation retract of $L$.

For more than two points it is technically more complicated, but can be done.

Answer 2

Hint. A closed disk minus a point in its interior can be deformation retracted onto its boundary. Now, given two points $x_1,x_2\in\mathbb R^2$, consider the figure-8 shape given by the union of two touching circles of the same radius with centres at $x_1$ and $x_2$ respectively, and call this space $X\cong S^1\vee S^1$.

Now, think about $\mathbb R^2-\{x_1,x_2\}$ and how you might deformation retract it onto $X$. You should see that there isn't an issue in the region enclosed by $X$. It remains to try to deformation retract the outer regions to $X$, and you can try to conceptualise such a map. What are the possible issues? Are these really problems or can they be solved somehow?

If you want, you can then spend time making the above rigorous and spelling out all the details, e.g. writing out the maps explicitly, proving they patch together continuously and so on. Often this is not done if you can explain your intuitive construction of the map clearly enough such that continuity of the relevant maps (and the other hypotheses you need for a deformation retract) becomes obvious, though.