Question: I see this notation a lot, but I can't seem to find an answer in my google searches. In particular I am looking at the question 

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> Let $\alpha:S^n\rightarrow S^n$ be the antipodal map. Prove that $n$ is odd, then $\alpha\simeq \operatorname{id}.$ 
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If it is an arbitrary symmetric group, then I understand, but I have seen $S^1$ be used for the circle.

Answer: 
The notation $S^{n}$ denotes the unit $n$-sphere in $\mathbb{R}^{n+1}$. That is, $$S^{n}=\left\lbrace(x\_{1},x\_{2},\ldots,x\_{n+1})\in\mathbb{R}^{n+1}\::\:\sum\_{j=1}^{n+1}x\_{j}^{2}=1\right\rbrace.$$
$$S^n=\{(x\_1,x\_2, \dots,x\_{n+1}): x\_1^2+x\_2^2+\cdots+x\_{n+1}^2=1\}$$ is the n-dimensional unit sphere... in $n+1$-dimensional Euclidean space $\mathbb E^{n+1}$...

It can also be considered independent of the surrounding space as an abstract *manifold* ...