In what capacity is $S^{3}$ locally the same as $S^{2}\times S^{1}$?

Question

Clearly they're not the same topologically as the former is simply connected and the latter not so.

One can consider the three-sphere as diffeomorphic to $SU(2)$

$$S^{3}\simeq SU(2)$$

Now, we also know that there is a relationship for the two-sphere:

$$S^{2}\simeq\frac{SU(2)}{U(1)}$$

And of course there is the isomorphism:

$$S^{1}=U(1)$$

So we can write:

$$S^{2}\times S^{1}\simeq\frac{SU(2)}{U(1)}\times U(1)$$

I'm guessing I can't just cancel out these factors to say that $S^{2}\times S^{1}\simeq SU(2)$, (As these $U(1)$ groups may be different ones in fact) but I'm thinking that maybe locally they're somehow equivalent, perhaps a lie bracket of vector fields on both adhere to the $su(2)$ lie algebra??? Since both manifolds are parallelizable this appears feasible. Could someone please enlighten me?

Answer 1

There are several levels to this questions. On one hand, they are both three-dimensional manifolds, so technically they are trivially locally the same: they are both locally the same as $\mathbb R^3$. But I guess this is not what you are really looking for.

What you really noticed is the following: for both $S^3$ and $S^2\times S^1$ there is a surjective map to $S^2$ such that the fibers (i.e. the pre-images of each point of $S^2$ through this map) are some copies of $S^1$.

Algebraically, for $S^3$ this map is the group projection to the quotient. For $S^2\times S^1$ this is just the projection to the first factor.

This also means the following: both $S^3$ and $S^2\times S^1$ look like a union of a family of $S^1$'s, and this family is parameterized by a $S^2$. For $S^2\times S^1$ this is obvious: the circles all have the form $\{x\}\times S^1$, where the parameter $x$ ranges throught the points of $S^2$. The projection map we were talking about before is the map that "identifies" each thiat circle to a point. You can visualize this, like trying to "shrink" these $S^1$ till they become points.

For $S^3$, yes, this is the Hopf fibration. This describes the 3-sphere as a union of 1-circles parameterized by a 2-sphere. And the corresponding projection map $S^3\to S^2$ can be seen exactly as the projection map to the quotient $SU(2)/U(1)$.

All that can be summarized as follows: both are $S^1$-bundles on $S^2$, as user10354138 observed.

Topologically, this is similar to what happens with the Moebius strip and with the cylinder: both are "unions of lines, that move around a circle", or $\mathbb R^1$-bundles on $S^1$. The fibers just move around in a different way: in the Moebius strip and in the Hopf fibration, they rotate in a notrivial way. In the cylinder or in $S^2\times S^1$ they just move around in the simplest straightforward way.

I think it would be interesting to rephrase all this from the point of view of Lie groups and Lie algebras, or from vector fields, but I warn against some of the most naive attempts, by reminding that there is no Lie group structure on $S^2$, by the hairy ball theorem. In particular we cannot just write a short exact sequence

$$ 1\to U(1) \to G \to SU(2)/U(1) \to 1 $$

of groups, because the last quotient is not a group quotient, is it? I have not looked properly into this, but maybe we can simply look at which Lie groups are $S^2\times S^1$ and see how much do they look like or unlike $SU(2)$