Function from a sphere This is a very basic question but i have difficulties understanding the following:
If I have a function which has as its domain a 1-Sphere $S^1$ then how do I have to imagine such a function? 
Will it have only one parameter? 
Could you make a concrete example of such a function?
Thanks
 A: You can think of a function $f:\mathbb{S}^1\to X$, where $X$ is some topological space, as being a function $F:\mathbb{R}\to X$ that has the property that $F(x)=F(x+1)$ for all $x\in\mathbb{R}$. These two notions are equivalent because there is a quotient map $q:\mathbb{R}\to\mathbb{S}^1$ defined by $q(x)=(\cos(2\pi x),\sin(2\pi x))$, and so we can associate a periodic function $F:\mathbb{R}\to X$ with the function $f:\mathbb{S}^1\to X$ defined by 
$$f(q(x))=F(x)$$
Note that because $\cos(2\pi x)=\cos(2\pi y)$ and $\sin(2\pi x)=\sin(2\pi y)$ implies $x=y+k$ where $k\in\mathbb{Z}$, the output of $f\circ q$ doesn't depend on whether we chose to look at $x$ or $y$, because $F(x)=F(y)$. So, this function $f$ is well-defined.
You can think of a function $g:\mathbb{S}^1\to X$ has having the single parameter $x\in[0,1]$, as long as it satisfies $g(0)=g(1)$. This is just another way of saying what I wrote above.
Of course, there are the examples of the identity function $\text{id}:\mathbb{S}^1\to\mathbb{S}^1$, and of the various constant functions $f:\mathbb{S}^1\to X$ where $X$ is any topological space, but those are probably not very enlightening. Here is one nice example: the function $g:\mathbb{S}^1\to \mathbb{R}$ defined by $g(a,b)=a$. It sends a point on the circle to its $x$-coordinate.
