Trying to brush up on some geometric and algebraic topology, I got a little confused about the following:
Suppose we have the standard unit sphere $S^2$, but we remove its north and south poles. Is this topological space homeomorphic or homotopic to $S^1 \times \mathbb{R}$? I would think that they are not homotopic since I don't think both spaces are deformation retracts, are they? Now I do know that the stereographic projection is a map from $S^2$ to the plane, but that just involves the removal of either the north or the south pole, correct?
Best Answer
The two spaces are homeomorphic. $S^2$ minus one point is identified by stereographic projection with $\mathbb{R}^2$, so $S^2$ minus two points is homeomorphic with say $\mathbb{R}^2 \backslash \{0\}$. Identifying $\mathbb{R}^2$ with $\mathbb{C}$ and $S^1$ with $\mathbb{R}/\mathbb{Z}$, the homeomorphism
$\mathbb{R}\times \mathbb{R}/\mathbb{Z}\cong \mathbb{C} \backslash \{0\}$
is given by $(r,\theta)\mapsto e^re^{2\pi i\theta}$.