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