2 fixed grammar

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......)

1

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 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......)