Homology of three-dimensional lens space

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is three-dimensional lens space
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Statement

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).