I am trying to compute the homotopy groups of the weighted projective space $\mathbb{P}(w) = S^{2n+1}/S^1$ with weights $w=(w_0,...,w_n)$ which is the orbit space of the $S^1$-action $$ s\cdot (x_0,...,x_n) = (s^{w_0}x_0,...,s^{w_n}x_n). $$ I want to use the long exact homotopy sequence for Serre-fibrations in order to relate the homotopy of $\mathbb{P}(w)$ to the homotopy of spheres, so I need the projection $p: S^{2n+1} \to S^{2n+1}/S^1$ to have the homotopy-lifting property for cubes $I^k$. Is there a way to show this?
Alternatively, since $\mathbb{P}(w) = \mathbb{P}^n\mathbb{C}/G_w$ with $G_{w_i}$ being the group of $w_i$-roots of unity and $G_w = G_{w_0}\times ... \times G_{w_n}$ acting coordinate-wise on $\mathbb{P}^n\mathbb{C}$, it would also work if the projection $\tilde{p}: \mathbb{P}^n\mathbb{C} \to \mathbb{P}^n\mathbb{C}/G_w$ were a Serre-fibration. Then, I could identify the homotopy of $\mathbb{P}(w)$ with the homotopy of $\mathbb{P}^n\mathbb{C}$. Is this possible?