Let $$L$$ be a framed link of $$m$$ components and $$(L_i,f_i)$$ be the component-framing pair. The linking matrix $$A=[a_{i,j}]$$ where $$a_{i,i}=f_i$$ and $$a_{i,j}=lk(L_i,L_j)$$ when $$i\ne j$$. I have read that this matrix is a presentation matrix for the first homology group $$H_1(M; \Bbb Z)$$ and that the nullity of $$A$$ is equal to the first betti number of $$M$$, where $$M$$ is the 3-manifold obtained from $$S^3$$ by surgery on $$L$$. Can someone explain this fact to me or point me to a proof or a reference?
EDIT: I am reading Nicolaescu's Reidemeister Torsion of 3-manifolds p.77 and there I made some progress but I have some questions as well. Suppose the $$c_i$$ are the meridians of the solid tori we glue back on the link complement $$E$$ to obtain $$M$$. This means that we send $$c_i \to f_iμ_i + λ_i$$, where $$μ_i, λ_i$$ are the meridians and longitudes in the regular neighbourhood of $$L$$. It is clear to me that $$μ_i$$ form a basis for $$H_1(E)$$ and taking $$K$$ to be the free abelian generated by the $$c_i$$'s and sending them to $$f_iμ_i + \sum_{k \ne i} a_{i,j}μ_k$$ we have a morphism $$K \to H_1(E)$$ with matrix the linking matrix mentioned above. There Nicolaescu says that by taking the natural map $$a:H_1(E) \to H_1(M)$$ we complete the presentation as this map is onto. Here is when it's unclear to me. By $$a$$ I guess he means the map that sends $$μ_i$$ in $$E$$ to $$μ_i$$ in $$M$$. If this is the case then I don't understand why it's onto and while it's clear to me that the $$f_iμ_i + \sum_{k \ne i} a_{i,j}μ_k$$ will be in the kernel because now they bound a disk in $$M$$ I don't see why they should generate it.
• I have tried looking how the meridians $μ_i$ would behave in $M$ and I tried to determine the kernel of $a$ but since I am unsure if this is indeed the intended morphism in Nicolaescu's book I am unsure how to proceed. Jul 10, 2020 at 12:23
• Does the map $a$ enter in any long exact sequence you can think of? Jul 10, 2020 at 13:43
• The only one that comes to mind is the Mayer-Vietoris derived from the neighbourhood of the link but $a$ doesn't look like it's one of those morphisms, unless of course I'm missing something Jul 10, 2020 at 14:19
• Could it be the map $H_1(E) \oplus H_1(U) \to H_1(M)$ restricted to $H_1(E)$ derived from the inclusions? It would map the meridians to themselves. ($U$ is the glued neighbourhood) Jul 10, 2020 at 14:57