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  1. First question, which maps we have from $\mathbb{S}^n$ to $\mathbb{S}^m$(if $m$ is not equal to $n$ )? The rotation or another mapping? Ok, how to express this mapping.

  2. Second, let $z$ be any element of $\mathbb{S}^n$, then why $\mathbb{S}^n-\{z\}$ is homeomophic to the Euclid space $\mathbb{R}^n$, especially for $n \geq 3$? Yes, I knew the "normal" sphere projection, but for any point in $\mathbb{S}^n$? I want to express it explicitly. Or if you can give a formulation of the rotation between $\mathbb{S}^n$?

  3. Third, if $S^n-\{x\}$ is homeomophic to $S^m-\{y\}$, as here $x$ is an element of $\mathbb{S}^n$ and $y$ is an element of $\mathbb{S}^m$, then $m=n$.

How to prove this? I knew if we use the sphere projection, the answer is obvious, but I hope a (pure) set topological way other than a algebra topological way(if $S^n-\{x\}$ is homeomophic to $S^m-\{y\}$, then $R^n$ is homeomophic to $R^m$, and so $m=n$).

In other words, I'd like this way: if $S^n-\{x\}$ is homeomophic to $S^m-\{y\}$, then $S^n$ is homeomophic to $S^m$. If we know the homeomorphism $f$ from $S^n-\{x\}$ to $S^m-\{y\}$, can we know the homeomorphism expression $g$ from $S^n$ to $S^m$?

Thanks a lot.

David Chan
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  • (1) Of course there are infinitely many maps $\Bbb S^n \to \Bbb S^m$ for $m > 0$ (even infinitely many constant maps). Are you just looking for maps up to, e.g., homotopy? If so, then this is basically the classification of homotopy groups of spheres, which is well-studied (but still open for large enough $m, n$). (2) This is just conjugation of the "standard" stereographic projection (which is what I assume you mean by "normal" here). (3) This follows immediately from Invariance of Domain. – Travis Willse Oct 08 '15 at 13:48
  • @travis (1) I'd like some non-trival continuous maps from $S^n →S^m$(2) Yes, standard stereographic projection.(3)I want to escape Invariance of Domain to some extent. At least, If we know the homeomorphism $f$ from $S^n −{x}$ to $S^m −{y} $, can we know the homeomorphism expression $ g$ from $S^n$ to $ S^m$ ? – David Chan Oct 08 '15 at 13:55
  • (1) The Hopf Fibration $\pi: \Bbb S^3 \to \Bbb S^2$ is a standard but beautiful example, and it is nontrivial in the sense that it is not null-homotopic. (In fact, its fibers are all copies of $\Bbb S^1$, so there are three spheres involved.) In a sense that can be made precise, it actually generates all continuous maps $\Bbb S^3 \to \Bbb S^2$ up to homotopy. (3) See http://math.stackexchange.com/questions/24873/elementary-proof-that-mathbbrn-is-not-homeomorphic-to-mathbbrm (my favorite method is to look at cohomology groups of punctured Euclidean spaces, but this is not "elementary"). – Travis Willse Oct 08 '15 at 14:07

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