I am attempting to find an explicit deformation retract from the torus minus some point to a figure of eight. Thus far, I have realised it is sufficient to show that:
If $I=[-1,1]$, then $I^2 – \{0,0\}$ deformation retracts to $\partial I^2$, since if we identify the edges of the square, we have the torus and the boundary of the square represents two circles intersecting at a single point, which is homeomorphic to a figure of 8.
I have also got the deformation retraction:
$f_t(x,y)= \begin{cases}
(x+t(1-x),y) &\quad\text{if $|x| > |y|$} \\
(x,y+t(1-y)) &\quad\text{if $|x| < |y|$} \\
\end{cases}$
I am not sure what to do if $|x|=|y|$. (Sorry if it is obvious!)
Best Answer
Here's an explicit formula, based on my comment above.
Let's suppose $|y|>|x|$. Then we want to linearly slide the $y$-coordinate from $y$ at $t=0$ to $sgn(y)$ at $t=1$. That's easy: $t\frac{y}{|y|}+(1-t)y$. According to my comment, we also want to scale $x$ by the same factor, to preserve the slope of the line from $(0,0)$ to $(x,y)$. So we get $t\frac{x}{|y|}+(1-t)x$. Of course, this works for $|x|>|y|$ very similarly, and so our homotopy is $$ f_t(x,y)=\begin{cases}(t\frac{x}{|y|}+(1-t)x, t\frac{y}{|y|}+(1-t)y)& |y|>|x|\\ (t\frac{x}{|x|}+(1-t)x, t\frac{y}{|x|}+(1-t)y)& |x|>|y| \end{cases} $$ And it's clear these branches agree when $|x|=|y|$.
EDIT: You can write the above deformation without branching:
$$ f_t(x,y) = \left(\frac{t}{\max(|x|,|y|)}+1-t\right)(x,y) $$