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I was trying to understand homology of a $3$-manifold $M$ obtained by rational surgery on an $m$-component oriented link $L$. I have a few questions regarding the following paragraph in the book 4-Manifolds and Kirby Calculus by Gompf and Stipsicz:

1) I don't see why $H_1(S^3 - L; \mathbb{Z}) \cong \mathbb{Z}^m$. (I tried it twice by using long exact sequence as suggested but I guess I am making a mistake.)

2) How can I see that $\lambda_i$ is the boundary of a Seifert surface? Also, how can I prove that $F_i$ determines that relation?

4-Manifolds and Kirby Calculus

The image is taken from the book 4-Manifolds and Kirby Calculus by Gompf and Stipsicz.

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Geometrically, at least, each component of $L$ contributes a meridian as a generator of $H_1(S^3-L)$. Try looking at the Wirtinger representation of a link diagram and seeing which generators are homologous (but not homotopic). – Neal Nov 18 '12 at 3:15
up vote 1 down vote accepted

1) LES of pair $0=H_2(S^3)\to H_2(S^3,S^3-L)=H_2(\nu L, \partial (\nu L))=\mathbb{Z}^m \to H_1(S^3-L)\to H_1(S^3)=0$. The middle bit is excision and homotopy invariance, and then the fact that $H_2(\nu L, \partial (\nu L))=H_2(\nu L/ \partial (\nu L))=H_2( \vee_m S^2)$.

2) Any knot in $S^3$ bounds a Seifert surface (

Your question about $F_i$ is a bit unclear. The core of a 2-handle from the surgery kills $p_i \mu_i+ q_i \lambda_i$, but we want to express $\lambda_i$ in terms of the generators of the $H_1(S^3-L)$ that we understand (those being $\mu_i$). $F_i$ is just there to provide cobordism between $\lambda_i$ and some linear combination of $\mu_j$'s, so that we have $\lambda_i=\sum lk(L_i, K_j)\mu_j$ which we then substitute in.

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