I have a question during reading the paper $O(2)$-actions on the 5-sphere (https://link.springer.com/article/10.1007%2FBF01389237).
A pseudofree $S^1$-action on a sphere $S^{2k-1}$ is a smooth $S^1$-action which is free except for finitely many exceptional orbits whose isotropy types $\Bbb Z_{a_1},\dots,\Bbb Z_{a_n}$ have pairwise relatively prime orders. Suppose we have given a pseudofree $S^1$-action on $S^5$. In the second page of the paper, it is written that it is easy to see that the quotient space $X:=S^5/S^1$ is a 4-manifold with isolated singularities whose neighborhoods are cones on lens spaces corresponding to the exceptional orbits of the action, and that $X$ is simply connected and $H_2(X;\Bbb Q)=\Bbb Q$. I can see that $X$ is a 4-manifold with isolated singularities corresponding to exceptional orbits, and that $X$ is simply connected. But I can't see why the followings are true:
How do we know that neighborhoods of singularities of $X$ are cones on lens spaces?
How do we know that $H_2(X;\Bbb Q)=\Bbb Q$?
This is due t the classification of representations of finite cyclic groups and the equivariant slice theorem. I have written a kind of "sketch" of the argument, please ask me if you want more details.
Suppose that $p$ with isotropy group $\mathbb{Z}_a$ where $a>0$ is an integer. Then, in other words, $p$ is a fixed point of the action of $\mathbb{Z}_a \subset S^1$. Hence, $T_pS^5$ inherits the structure of $\mathbb{Z}_a$-representation. Due to the classification of representations of these finite groups , if $\xi = e^{\frac{2 \pi i}{a}}$, then under an identification $T_{p}(S^5) = \mathbb{C}^2 \oplus \mathbb{R}$, the action looks like the action $\xi. (z_1,z_2,x) = (\xi^{w_1}z_1, \xi^{w_2} z_2,x)$. Note that the $\mathbb{R}$ factor will be exactly the tangent space of the exceptional orbit. When taking the orbit we quotient "exactly in the direction of the orbit", hence the orbit space is locally homeomorphic to $\mathbb{C}^2/\mathbb{Z}_a$, one may verify that it is homeomorphic to a cone on the lens space.
Then, by the equivariant slice theorem (see Audins book "Torus actions on symplectic manifolds, Birkhauser, 2004" Theorem I.2.1 ), this is a local model of the point in the quotient space and the claim follows.