Let $L\_n(p)$ be the $2n+1$ dimensional Lens space
$$
S^{2n+1}/\mathbb{Z}\_p
$$
where the action is given as $z\_i\rightarrow e^{\frac{2\pi}{p}}z\_i$, $i=1,...,n+1$, with $z\_i$ the coordinates of $\mathbb{C}^{n+1}$ such that $S^{2n+1}$ is $|z\_1|^2+...+|z\_{n+1}|^2=1$. For $k\neq 0,2n+1$ the homology groups with coefficients in a commutative ring $R$ are
$$
H\_k(L\_n(p),R)=\left\{\begin{array}{cc}
R/pR & \text{if $k$ is odd}\\
T\_p(R) & \text{if $k$ is even }
\end{array}\right.
$$
where $T\_p(R)\subset R$ is the $p-$torsion subgroup:
$$
T\_p(R)=\left\{x\in R \ | \ px=0\right\} \ .
$$
Since $R$ is an abelian group, we can consider $R^{\vee}$, its Pontryagin. Fix an extension $\Gamma$ of $R$ by $R^{\vee}$:
$$
1\rightarrow R^{\vee}\rightarrow \Gamma \rightarrow R\rightarrow 1 \ .
$$
This induces a long exact sequence of homology and cohomology groups, with connecting homomorphisms given by the Bockstein maps. For instance in cohomology $\beta :H^k(L\_n(p),R)\rightarrow H^{k+1}(L\_n(p),R^{\vee})$.


Given $\Sigma \in H\_{2n}(L\_n(p),R)$ we consider its Poincare' dual cocycle $A=PD(\Sigma)\in H^1(L\_n(p),R)$, and its image under Bockstein $\beta(A)\in H^2(L\_n(p),R^{\vee})$. Since $R^{\vee}$ is also a ring, the cup product associated with the product $R^{\vee}\times R^{\vee}\rightarrow R^{\vee}$ allows to construct a class $\beta(A)^n\in H^{2n}(L\_n(p),R^{\vee})$, and the natural pairing $R\times R^{\vee}\rightarrow \mathbb{R}/\mathbb{Z}$ allows the construction of the self-linking invariant:
$$
lk(\Sigma)=\int \_{L\_n(p)}A\cup \beta(A)^n \in \mathbb{R}/\mathbb{Z}
$$


1. Preliminary question: how this self-linking invariant is related with the more standard linking form which only uses the sequence $\mathbb{Z}\rightarrow \mathbb{Q}\rightarrow \mathbb{Q}/\mathbb{Z}$?
2. Let us first fix $n=1$, $p=2$ so that $L\_1(p)=\mathbb{RP}^3$, and $R=\mathbb{Z}\_2$. Since $T\_2(\mathbb{Z}\_2)=\mathbb{Z}\_2$ there is precisely one 2-cycle $\Sigma \in H\_2(L\_1(2),\mathbb{Z}\_2)$, and one 1-cycle $\gamma\in H\_1(L\_1(2),\mathbb{Z}\_2)$. How can I compute
$$
lk(\Sigma)=\int \_{L\_1(2)}A\cup \beta(A)
$$
explicitely? Moreover I have the intuition that $\beta(A)$ should be the Poincare' dual of $\gamma$, but I don't know how to prove this.
3. For general $n$ and $p$ fix $R=\mathbb{Z}\_p$, so that $R/pR=T\_p(R)=\mathbb{Z}\_p$, and again choose one generator $\Sigma $ of $H\_{2n}(L\_n(p),\mathbb{Z}\_p)$. How can I compute $lk(\Sigma)$? By extending the intuition before I also guess that, fixing a generator $\gamma$ of $H\_1(L\_n(p),\mathbb{Z}\_p)$, $\beta(A)$ is the Poincare' dual of $\beta(A)^n$. Is this correct?



Computation of the linking invariant on Lens spaces
In my thesis, I gave the calculation of the linking form on homology, which is equivalent to the question you asked. I credited the calculation to de Rham ([Sur L'analysis situs des varietés a n dimensions](http://www.numdam.org/item/THESE_1931__129__1_0.pdf), J. Math. Pures et Appl., 10 (1931), 115-200.) See Proposition 4 in [Imbedding punctured lens spaces and connected sums](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwj6vZHruPX-AhUjF1kFHcolCOwQFnoECBwQAQ&url=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fpacific-journal-of-mathematics%2Fvolume-113%2Fissue-2%2FImbedding-punctured-lens-spaces-and-connected-sums%2Fpjm%2F1102709207.pdf&usg=AOvVaw30d8npdEhbIfukeNw-zYcx). Pacific J. Math. 113 (1984), no. 2, 481–491.


The answer is given for a lens space $L(m;q\_1,\ldots,q\_{2k})$ of dimension $4k-1$; I imagine that it's similar for dimension $4k+1$. With respect to a certain generator $e$ of $H\_{2k-1}(L)$, de Rham showed $\lambda(e,e) = \frac1m q\_1\cdots q\_k \cdot r\_{k+1}\cdots r\_{2k}$ where $q\_jr\_j \equiv 1 \pmod{m}$. You'd have to look in de Rham's paper to find the details.


