Find strong deformation retract to corona/crown

Question

I'm looking to find the retraction ($r:X \rightarrow A$) and the deformation $H:X \times [0,1] \rightarrow X$) but I can't think of how, I'm bad thinking about functions, any help is appreciated!

Answer 1

First, parametrize the space $X$ using a vector function $\vec{p}(u,v,w)$ defined as $$ \vec{p}(u,v,w)=((|v|+1+w)\cos(u),(|v|+1+w)\sin(u),v) $$ where $v\in\Bbb R$, $w\in[0,1]$, and $u\in[0,2\pi]$.

For the retraction $r:X\to A$, just send $v$ to $0$.

Now, the deformation retraction $H:X\times I\to X$ that forms a homotopy $\operatorname{id}_X\simeq r:X\to A$ can be easily defined via straightline homotopy $$ H(\vec{p}(u,v,w),t)=(((1-t)|v|+1+w)\cos(u),((1-t)|v|+1+w)\sin(u),(1-t)v) $$ It's easy to check that $H(x,0)=\operatorname{id}_X$ and $H(x,1)=r$.

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I could add the derivation of the parametric equation for $X$ if you want.....