Let $M$ be path connected closed 3-manifold, $\pi_{1}(M)=Z/5$,how to compute $H_n(M)$?

Question

Let $M$ be path connected closed 3-manifold, $\pi_{1}(M)=Z/5$,how to compute $H_n(M)$?

I guess to use poincare duality,could any one give me some details?Thanks

Answer 1

$H_1= Z/5$ (it is the abelianziation of $\pi_1$). $H^1=Hom(\pi_1, \Bbb Z)=0$. Since $H^1=0$ (in $\Bbb F_2$ coefficients) every vector bundle over $M$ is orientable hence $M$ is orientable (alternatively $M$ admits no nontrivial double cover as $\Bbb Z/5$ admits no index two subgroup). Hence $H_3=\Bbb Z$ and we can use PD to get the rest.