deformation retraction of the torus with 1 pt. deleted onto 2 circles intersecting at a pt. and that of $\mathbb R^n \setminus \{0\}$ onto $S^{n-1}$.

Question

Here are two questions with their answers:

My question is:

Why in the second term of $H(x,t)$, in the first problem we have the term $t (g^{-1} \circ f)$ while, in the second term of $H(x,t)$, in the second problem we have the term $t (x/|x|)$?

Thanks!

Answer 1

May as well make this an answer. Because you have different spaces you're retracting onto: $g^{-1}\circ f$ maps $I^2 - \{0\}$ to $\partial I^2$, while $\frac{x}{|x|}$ maps $\mathbb{R}^n-\{0\}$ to $S^{n-1}$.

(As a technical note, the argument in 0.1 only shows that $I^2 - \{0\}$ deformation retracts onto $\partial I^2$. To complete the argument, you need to show that $g^{-1}$ and $f$ factor through the quotient map onto $T^2-\{\mbox{a point}\}$.)