Exercise 10. Groups and Covering spaces. Lima Let $X$ be the space obtained from the sphere $S^2$ by gluing the north pole  to the south pole, let $Y=\mathbb{R}^3-S^1$, where $S^1=\left\{(x,y,0)\in\mathbb{R}^3:x^2+y^2=1\right\}$ and let $Z$ be the union of a torus of revolution with a disk whose boundary is the smallest of the parallels of the torus. Prove that $X$,$Y$ and $Z$ have the same homotopy type.
How can I prove that $Z$ have the same homotopy type with to $X$ or $Y$?
 A: Goal
Provide a plan along with some guidance for the reader to complete the details.
Notation: $\sigma=\sqrt{x^2+y^2}$. $Y=\mathbb R^3\setminus\{\sigma=1,\;z=0\}$.
Plan
Step 1. Define $B_2=\{(x,y,z)\mid x^2+y^2+z^2\le4\}$. Replace $Y$ with $\bar Y=B_2\setminus\{\sigma=1,\;z=0\}$.
Step 2. Define $(0,0,z)\sim(0,0,0)$. Replace $\bar Y$ with $\bar Y/{\sim}$.
Step 3. Replace $\bar Y/{\sim}$ with $T^*=\{(\sigma-1)^2+z^2\le1\}\setminus\{\sigma=1,\;z=0\}$.
Step 4. Replace $T^*$ with $T=\{(\sigma-1)^2+z^2=1\}$.
Guidance
Step 1. Put $\rho=\sqrt{\sigma^2+z^2}$.
$$
f(x,y,z)=\begin{cases}
   \tfrac{2}{\rho}(x,y,z)    &\rho\ge 2,\\
   (x,y,z)          &\rm otherwise.
\end{cases}
$$
Step 2. Define $g\colon\bar Y/{\sim}\to\bar Y$ as $g\colon[x,y,z]\mapsto(x,y,\frac{\sigma}{2}z)$.
Step 3. Write $\zeta=\sqrt{1-(\sigma-1)^2}$. Define $h\colon\bar Y/{\sim}\to T^*$ as
$$
h([x,y,z]) = \begin{cases}
  (x,y,\zeta)   &z\ge\zeta,\\
  (x,y,z)       &-\zeta\le z\le\zeta,\\
  (x,y,-\zeta)  &z\le-\zeta.
\end{cases}
$$
Step 4. Write $\xi=\sqrt{(\sigma-1)^2+z^2}$. Define $r\colon T^*\to T$ as
$$
r(x,y,z) = \begin{cases}
   \tfrac{1}{\xi}(x-x/\sigma,y-y/\sigma,z)+(x/\sigma,y/\sigma,0)   &\sigma\ne0,\\
   (0,0,0)  &\sigma=0.
\end{cases}
$$
To do

*

*Draw pictures to understand the idea.

*Prove the continuity of the functions defined above.

*Use the functions to build homotopies between domain and codomain.

*Complete the solution to the problem. (see also How can I prove that the horn torus and $\mathbb{T} \cup D_1$ have the same homotopy type?)

