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Let $K$ be a knot in $S^3$ and $M^3$ be a 3-manifold obtained by 0-surgery from $S^3$ along $K$. By using Mayer-Vietoris sequence, we can see that $H_i(M^3)=H_i(S^1\times S^2)$. Therefore, we have a surjection from $\pi_1(M^3)\to H_1(S^1\times S^2)=\mathbb{Z}$ and for $n>0$, we have n-fold cyclic covering of $M$, say $M_n$.

On the other hand, by using collar neighborhood of Seifert surface in $S^3$, we can make a n-fold branched cyclic covering of $S^3$ over $K$, say $L_n$.

To extract more information by using local coefficient system, we have given a character $\phi\colon \pi_1(L_n)\to \mathbb{Z}_m$. (For convenience, $m$ and $n$ are prime-power order.)

These two 3-manifolds, $M_n$ and $L_n$ can be used to study knot $K$. Since $\Omega(K(\mathbb{Z}_m,1))=\mathbb{Z}_m$ is torsion group, $rL_n=\partial W_n$ and over $\mathbb{Z}_m$ for some $r>0$ and some 4-manifold $W_n$.

I feel that concrete understanding on intersection form of $W_n$ is needed and important.

  1. Is it true that can we obtain a $V_n$ from $W_n$ by attaching $r$ 2-handle, where $V_n$ satisfies $\partial V_n= rM_n$ over $\mathbb{Z}_m$ ?

  2. What is the difference of intersection form on $H_2(V_n;\mathbb{Q})$ and $H_2(L_n;\mathbb{Q})$? ($V_n$ is a 4-manifold satisfying condition in Question 1. i.e.)$\partial V_n=rM_n$.

  3. How about $H_2(V_n;\mathbb{Q}(\mathbb{Z}_m))$ and $H_2(L_n;\mathbb{Q}(\mathbb{Z}_m))$? Here I'm using homology with local coefficients.

  4. What is the influence of different choice of $W_n$ on intersection form ? (i.e. Both $W_n$ and $W_n'$ satisfies $\partial W_n= rL_n$ and $\partial W_n' = rL_n$.) How much different that intersection form on $H_2(W_n;\mathbb{Q})$ and $H_2(W_n';\mathbb{Q})$ ?

Please give me any detailed references, if any.

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Does $W_n=M_n$? –  HJRW Aug 17 '10 at 16:59
    
NoNo $M_n$ is a 3-manifold which is n-fold covering of $M$. $W_n$ is a 4-manifold satisfying $\partial W_n=rL_n$. –  Topologieee Aug 17 '10 at 17:04
    
Oh right, I see. The question would have be easier to read for me if you inserted the phrase 'for some 4-manifold $W_n$' at the end of your third paragraph. –  HJRW Aug 17 '10 at 19:14
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