I'm having a lot of trouble describing the cell structure of the following space $X$:
Consider the action of $S^1$ on the odd sphere $ S^{3} \subseteq \mathbb{C}^{2} $ as follows: $$ g . (z_{1}, z_{2}) = (g . z_1, g ^{3} . z_2) $$
Define $ X = S^{3} / S^{1} $, the quotient of the 3-sphere under this particular action.
How can I visualize this space? Ideally I would like to describe it's cell structure completely.
In general, how do I describe cell-structure of spaces under action by continuous groups? For discrete groups, it's relatively simple, because you can usually get a cell structure which get permutated, (if the action is relatively nice). What do I do in these situation?
I wish I could provide evidence of my work, but I got nothing.
There is a homeomorphism $S^2\cong X$. View $D^2$ as a subset of $\mathbb{C}$ and consider the map $f\colon D^2\to S^3, f(z) = (\sqrt{1-\vert z\vert^2},z)$. Then $f$ maps $S^1$ to a single orbit, hence induces a map $D^2/S^1\to X$. It is easy to check this is a bijection, hence a homeomorphism.
You can also see this by giving $S^3$ an equivariant cell structure. The $0$-cell is $\{0\}\times S^1$ (this has isotropy group the third roots of unity) and one attaches an equivariant $2$-cell via the map $$D^2\times S^1\to S^3, (w,z)\mapsto (z\sqrt{1-\vert w\vert^2}, z^3 w).$$ Thus, after passing to orbits you obtain a CW-complex with one vertex and one $2$-cell, which can only be $S^2$.