Consider the $2n+1$ dimensional sphere $S^{2n+1}$ as the submanifold of $\mathbb{C}^{n+1}$ defined by $$ |z_1|^2+...+|z_{n+1}|^2=1 \ . $$ Given an integer $p$ we consider the $\mathbb{Z}_p$ action on $S^{2n+1}$ whose generator is $$ z_i\rightarrow e^{\frac{2\pi i }{p}}z_i \ , \ \ \ \ i=1,...,n+1 \ . $$ The quotient defines the Lens space $$ L_n(p)=S^{2n+1}/\mathbb{Z}_p \ . $$ For instance $L_1(2)=\mathbb{RP}^3$. Let $A$ be an abelian group. The homology groups with coefficients in $A$ are
$$ H_i(L_n(p),A)=\left\{\begin{array}{cc} A & \text{if $i=0,2n+1$}\\ A/pA & \text{if $i$ is odd}\\ T_p(A) & \text{if $i$ is even} \end{array} \right. $$ where $T_p(A)\subset A$ is the $p-$torsion subgroup of $A$: $$ T_p(A)=\left\{a\in A \ | \ pa=0 \ \right\} $$ I would like to have the explicit representatives of the generators, in some simple cases. Specifically I would need the generators of: $$ H_i(L_1(2),\mathbb{Z}_2)=\mathbb{Z}_2 \ , \ i=1,2 $$ and $$ H_i(L_2(4),\mathbb{Z}_4)=\mathbb{Z}_4 \ , \ \ i=2,4 \ . $$
In the first case, since $L_1(2)=\mathbb{RP}^3$, I think a very explicit description (to visualize them) of the these cycles should be possible. For the second case I would appreciate any kind of description.