Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with 
$H\_{\*}(M;Q)=H\_{\*}(S^3;Q)$


As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's Lecture notes in Algebraic toplogy book, we have a $Q/Z$ valued linking form, $\lambda\colon H\_{1}(M;Z)\times H\_{1}(M;Z)\to Q/Z$ defined by the adjoint to following isomorphism. 


(Actually we can define linking form in more general setting, e.g.) odd dimensional manifold without the restriction such as rational homology sphere condition. Because I want to just intuitive idea about linking form, I restricted the case)


$H\_1(M;Z)\cong H^2(M;Z)\cong H^1(M;Q/Z)=Hom(H\_1(M;Z),Q/Z)$ 


First isomorphism : Poincare duality, 


Second isomorphism : Inverse of Bockstein homomorphism $\delta$ induced form $0\rightarrow Z \rightarrow Q\rightarrow Q/Z\rightarrow 0$. More precisely, induced long exact sequence is that $\ldots\rightarrow H^1(M;Q)\rightarrow H^1(M;Q/Z)\rightarrow H^2(M;Z)\rightarrow H^2(M;Q)$ and here $H^1(M;Q)$ and $H^2(M;Q)$ vanishes. Therefore, the long exact sequence shows that $H^1(M;Q/Z)$ and $H^2(M;Z)$ are naturally isomorphic (if $M$ is rational homology 3-sphere) and we call that homomorphism as Bockstein homomorphism.


Third isomorphism : Universal Coefficient theorem


In short, $\lambda\colon H\_{1}(M;Z)\times H\_{1}(M;Z)\to Q/Z$ is defined by $\lambda(x,y)=<\tilde{x}\cup\delta^{-1}(\tilde{y}),[M]>$, where $\tilde{x},\tilde{y}$ are Poincare dual to $x,y$ and $\delta\colon H^1(M;Q/Z)\to H^2(M;Z)$ is a Bockstein homomorphism as defined above. Cup products are defined in $H^1(M;Z)\times H^2(M;Q/Z) \to H^3(M;Q/Z)$ induced from the Bilinear map $Z \times Q/Z \to Q/Z$.


I understand this linking form only algebraic viewpoint. Therefore, I can play with this form only by using dilluminating algebraic topology. 


I'm struggle to find geometric interpretation but I have no idea to express the Q/Z valued in terems of geometric language which seems to be highly algebraic. (Feeling like Injective, divisible, Ext or something linke that )


Are there any intuitve and geometric (clean) interpretation about this linking form? (e.g.)such as Alexander duality, like the argument that .........we can find a dual basis which represents meridian......) 



Intutive interpretation about Linking forms
This may be a good place to explain the well-known principle 
$$\text{intersection in the interior = linking in the boundary}$$
in an oriented $m$-dimensional manifold with boundary $(M,\partial M)$. Let
$$f~:~(K,\partial K)\subset (M,\partial M)~,~g~:~(L,\partial L) \subset (M,\partial M)$$
be embeddings of oriented manifolds with boundary, such that
$${\rm dim}(K)~=~k~,~{\rm dim}(L)~=~\ell~,~k+\ell~=~m~,~
f(\partial K) \cap g(\partial L)~=~\emptyset \subset \partial M~.$$
Assume there exists an isotopy (= homotopy through embeddings) rel $\partial K$
$$f\_t~:~K \to M~~(0 \leqslant t \leqslant 1)$$
such that $f\_0=f$ and $f\_1(K)\subset \partial M$, with each $f\_t(K), g(L)$ intersecting transversely in $M$, so that $f\_t(K) \cap g(L)\subset M$ is a finite set with an
intersection index $I(x)\in \{\pm 1\}$ at each point $x \in f\_t(K) \cap g(L)$
according to the orientations. A continuity argument shows that the function
$$\lambda~:~[0,1] \to {\mathbb Z}~:~t\mapsto \lambda(t)=
\sum\limits\_{x \in f\_t(K) \cap g(L)}I(x)$$
is constant, so that 
$${\rm intersection}(f(K),g(L)\subset M)~=~\lambda(0)~=~\lambda(1)$$
$$=~{\rm linking}(f(\partial K),g(\partial L) \subset \partial M) \in {\mathbb Z}~.$$
This is best seen by drawing pictures for $(M,\partial M)=(D^2,S^1)$.


The localization exact sequence in algebraic $L$-theory is based on an abstract homological version of this principle.


