Three-dimensional lens space as a circle bundle
= Three-dimensional lens space as a circle bundle
{title2=$S^1\to L(p)\to S^2$}
The diagonal quotient $L(p)=S^3/(\mathbb Z/p)$ is the unit <circle bundle> of a complex line bundle over $S^2$ whose <Euler class> is $p$ times a generator. Its <Gysin sequence> gives
$$
H^j(L(p);\mathbb Z)\cong
\begin{cases}
\mathbb Z,&j=0,3,\\
\mathbb Z/p,&j=2,\\
0,&\text{otherwise},
\end{cases}
$$
while $H^j(L(p);\mathbb F_p)\cong\mathbb F_p$ for $0\leq j\leq3$.