Define a retraction

Question

I read in the book that there is a retraction $r_n:I^n\times I\rightarrow (\partial I^n\times I)\cup(I^n \times \{0\})$

But I do not know how to define this retraction.Some instructions would be very helpful to me. Thank you!

Answer 1

This is a pretty commonly-used retraction - though I'm surprised that a book would not bother defining what it is. The basic idea is the set $\partial I^n \times I \cup I^n \times \{0\}$ is sort of like the lower "shell" of the cube $I^{n+1}$ - it consists of the bottom face along with the faces about the boundary of that face, and we can simply push the interior out onto that shell. More precisely, let us use the coordinates $I=[-1,1]$ and then fix a point $P=\{0,0,\ldots,0,2\}$ which lies in $\mathbb R^{n+1}$ above the top face of $I^{n+1}$. For any point $X$ in $I^{n+1}$, you can draw the ray $\overline{PX}$, and it will intersect the shell at exactly one location $Y$. The function associating $X$ to $Y$ is a retract - and, in fact, can extend to a strong deformation retract. Here's a rough picture that shows the construction: