Homology of three-dimensional lens space

Homology groups with coefficients in integers
Let $$L(p,q)$$ be the three-dimensional lens space with parameters $$p$$ and $$q$$. The homology groups with coefficients in the integers are:

$$H_k(L(p,q),\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad k = 0,3 \\ \mathbb{Z}/p\mathbb{Z}, & \qquad k = 1 \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

Note that the homology groups do not depend on $$q$$, even though the homeomorphism class, and the homotopy type, of the space depend on $$q$$. (There are some equivalence conditions on $$q$$ under which we get homeomorphic spaces, and weaker equivalence conditions under which we get homotopy-equivalent spaces, but the homology groups coincide even when these conditions are violated).