What does the notation $S^n$ mean?

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

Let $\alpha:S^n\rightarrow S^n$ be the antipodal map. Prove that $n$ is odd, then $\alpha\simeq \operatorname{id}.$

If it is an arbitrary symmetric group, then I understand, but I have seen $S^1$ be used for the circle.

Answer 1

The notation $S^n$ denotes the locally $n$-dimensional unit sphere. For example $S^2$ is your classic sphere cut out in $(x,y,z)$-space by the equation $x^2+y^2+z^2 = 1$.

Answer 2

The notation $S^{n}$ denotes the unit $n$-sphere in $\mathbb{R}^{n+1}$. That is, $$S^{n}=\left\lbrace(x_{1},x_{2},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\::\:\sum_{j=1}^{n+1}x_{j}^{2}=1\right\rbrace.$$

Answer 3

$$S^n=\{(x_1,x_2, \dots,x_{n+1}): x_1^2+x_2^2+\cdots+x_{n+1}^2=1\}$$ 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 ...