Does a manifold $S^{3}\times S^{1}$ have the Lie group structure $SU(2)\times U(1)$?

Question

The question is pretty straightforward, I was just wondering since the three-sphere $S^{3}$ has the Lie group structure $SU(2)$ and the circle $S^{1}$ has the structure $U(1)$.

Incidentally this would be the topology of a closed cyclic FLRW (Freidmann-Lemaitre-Robertson-Walker) universe, and and also the Lie group structure of the Electroweak sector of quantum field theory.

Answer 1

Yes. This follows from the following two items:

• A manifold $M$ "has the Lie group structure" of a Lie group $G$ if there is a diffeomorphism $f:M\to G$.

• If $f:M\to G$ and $g:N\to H$ are diffeomorphisms, then $$f\times g:M\times N\to G\times H,\quad (p,q)\mapsto(f(p),g(q))$$ is a diffeomorphism.