a generalization of punctured cylinder

Question

Let $S^1\times \mathbb{R}$ be the infinite cylinder.

Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$.

How about the puctured space $S^3\times \mathbb{R}-*$?

Do we have $S^3\times \mathbb{R}-*\simeq Skeleton^3(S^3\times \mathbb{R})\simeq S^3\vee S^3$?

Answer 1

Since $X=S^n\times\mathbb{R}\cong S^n\times (0,1)$, you can think of $X$ as an n-dimensional annulus or shell, without boundary. Usually, $X$ retracts onto $S^n$. But after removing a point, $X\setminus\{*\}$ retracts onto $S^n$, with a bump. This "bump" is caused by the removed point and looks like a copy of $S^n$ (you can easily visualize this for $n=2,3$). But this space is homotopy equivalent to $S^n\vee S^n$.

So, $S^n\times\mathbb{R}\setminus\{*\}\simeq S^n\vee S^n$.