4
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ Does $W_n=M_n$? $\endgroup$
    – HJRW
    Aug 17, 2010 at 16:59
  • $\begingroup$ NoNo $M_n$ is a 3-manifold which is n-fold covering of $M$. $W_n$ is a 4-manifold satisfying $\partial W_n=rL_n$. $\endgroup$ Aug 17, 2010 at 17:04
  • $\begingroup$ 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. $\endgroup$
    – HJRW
    Aug 17, 2010 at 19:14

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.