# Thursday, October 15

:::{.exercise title="Lens Spaces"}

Given two coprime $p, q$, define an action of $\ZZ/p\ZZ = \gens{\tau}$ on $S^3 \subset \CC^2$ as 
\[  
(w, z) &\mapsto (e^{2\pi i / p}w, e^{2\pi i q/p}z )
.\]
Note that $\tau^p = \id$.
Then define $L(p, q) = S^3 / \ZZ/p\ZZ = S^3 /\sim$ where $(w, z) \sim \tau^j(w, z)$ for all $j$.

1. Show that $L(p, q)$ is a smooth manifold, so that the quotient map $\pi: S^3 \to L(p, q)$ is smooth.
  Show that $L(p, q) = A\union B$ where $A, B\cong S^1 \cross \DD^2$ and $A\intersect B \cong S^1 \cross S^1$.

> Hint: think in polar coordinates, replacing $w = r_1 e^{i\theta_1}$.
  Write a subset of $S^3$ as $\ts{r_1^2 = {1\over 2}, r_2^2 = {1\over 2}}$, this set is fixed by the action.
:::