Showing two things are homotopic to each other

Question

I want to show that $\mathbb{C} - \{0\} \simeq S^1$ and the unit square is $\simeq S^1$ where $\simeq$ is homotopic in this case. In other words I want to find an equation for each sort of speak that defines a unit square with one value and a circle for another value. I have done this before with the annulus and the circle by letting $H(z,t) = (1-t)z + \displaystyle\frac{z}{|z|}$. I want to do something similar for the other two examples I brought up. The help would be greatly appreciated!

Answer 1

You won't be able to show that $I^2$ (the unit square, where $I=[0,1]$) is homotopy equivalent to $S^1$ because that's not true.

To show that $\mathbb C-\{0\}$ is homotopy equivalent to $S^1$, it suffices to find a deformation retraction of $\mathbb C-\{0\}$ onto $S^1$. Here's one way to do that: for each point $z\in\mathbb C-\{0\}$, the point on $S^1$ which is closest to $z$ is $z/|z|$. Make the deformation retraction by sending $z$ to $z/|z|$ at constant speed along the line connecting them. (Draw a picture if you don't understand!) Points that are farther away will have to move more quickly. Can you find the formula for this deformation retraction?

Edit: The border of the unit square (call it $X$) is homotopy equivalent to $S^1$. You can show it by making a straight-line homotopy (along the lines of the other part of this question) from $1_X$ to $1_{S^1}$. That is, connect each point of $X$ with the corresponding point on the unit circle (what do I mean by corresponding point?) and move it along the line connecting them.