I am intending to learn low dimensional topology from Saveliev's Book "Lectures on the Topology of 3-manifolds" by myself.
At the very beginning, he gives a Heegaard splitting of $S^3$ stating that
"the sphere $S^3$ is represented as the result of revolving the 2-sphere $S^2=R^2\cup\{\infty\}$ about the circle $l\cup\{\infty\}$ where $l$ is a straight line in $R^2$."
I do not understand why the revolution of $S^2$ about $S^1$ results in $S^3$. Is it a fact that can be generalized to higher dimensions, i.e. does the revolution of $S^{n+1}$ about $S^{n}$ result in $S^{n+2}$?. Thanks for any help and suggestions.
Here the technical mechanism of ‘’revolving’’ means that for each element of the 2-sphere $S^2$ (or any other surface $F$), there is a circle $S^1$ attached. This kind of space is called ‘’circle bundle’’ over the 2-sphere (or over $F$, respectively).
For the hypersphere $S^3$ it happens that can be fibered as $S^1\hookrightarrow S^3\stackrel{h}\to S^2$, and explicitly the map $h$ is constructed employing complex coordinates.
Any tetrad $(x,y,s,t)$ of real numbers which satisfy $x^2+y^2+s^2+t^2=1$ is an element in $S^3$, but with the complex numbers $z=x+iy$ and $w=s+it$ the pair $(z,w)$ parametrize points in the hypersphere $S^3$ if $z\overline{z}+w\overline{w}=1$. Here $\overline{z}=x-iy$ and $z\overline{z}=|z|^2$ is the squared norm of the complex number $z$.
Now $h(z,w)=(2z\overline{w},z\overline{z}-w\overline{w})\in\Bbb R^2\times\Bbb R$ is really a map onto the 2-sphere because $$\|h(z,w)\|=4z\overline{z}w\overline{w}+(z\overline{z}-w\overline{w})^2 =(z\overline{z}+w\overline{w})^2=1.$$
In the hypersphere taking another point of the form $(\lambda z,\lambda w)$ with any complex number $\lambda$ satisfiying $|\lambda|=1$ are mapped onto the same point $h(z,w)$, because $\|h(\lambda z,\lambda w)\|=\|h(z, w)\|=1$
So the set $S=\{(\lambda z,\lambda w):|\lambda|=1\}$ is the fiber of the map $h$ and is a $S^1$, determined by $\lambda$, which is a ‘’circle’’, and that $(\lambda z,\lambda w)$ goes thru the location $(z,w)$ in $S^3$.
We emphasize that the circle bundle $S^3$ is different from the trivial $S^1\hookrightarrow S^1\times S^2\stackrel{\pi}\to S^2$, where $\pi$ is the projection $\pi(\theta,\xi)=\xi$.
Objects (or spaces) $E$ fibered as $S^1\hookrightarrow E\to F$ are called circle bundles over $F$.