The relative handle chain complex computes $H_{\ast }(W,M_0)$ and has the intersection matrix as its differential

Product collars let us enlarge the lower stage slightly across each attaching seam and retract that enlargement onto the lower stage. Excision then identifies the relative homology of the next index stage with that of the disjoint union of its handle pairs; the retraction and excision preserve the relative core classes. By [F1]–[F2], $H_i(W_k,W_{k-1};\mathbb{Z})$ is zero unless $i=k$, and in that degree is free on these cores. At $k=0$, compressing the initial collar onto $M_0$ gives the same calculation, including disjoint $0$-handles.

Write ${\delta }_k:C_k\to H_{k-1}(W_{k-1})$ for the pair connector and $p_{k-1}:H_{k-1}(W_{k-1})\to C_{k-1}$ for the quotient map. By [F3], ${\mathrm{\partial }}_k=p_{k-1}{\delta }_k$. Exactness of the pair $(W_{k-1},W_{k-2})$ gives ${\delta }_{k-1}{p}_{k-1}=0$, so ${\mathrm{\partial }}_{k-1}{\mathrm{\partial }}_k=p_{k-2}{\delta }_{k-1}{p}_{k-1}{\delta }_k=0$ for $k\ge 2$. For $k=1$ this follows from ${\mathrm{\partial }}_0=0$.

The triple sequences and step 1.1 show inductively that $H_i(W_j,M_0)=0$ for $i>j$. Fix $k\ge 0$. The triple $(W_k,W_{k-1},M_0)$ therefore injects $H_k(W_k,M_0)$ into $C_k$. Its image is the kernel of the connector into $H_{k-1}(W_{k-1},M_0)$, which itself injects into $C_{k-1}$ by the triple $(W_{k-1},W_{k-2},M_0)$ when $k\ge 1$. Naturality and the connector factorization identify that composite with ${\mathrm{\partial }}_k$. For $k=0$ the target is zero. Hence $H_k(W_k,M_0)$ identifies with $\ker {\mathrm{\partial }}_k$.

The triple $(W_{k+1},W_k,M_0)$ identifies $H_k(W_{k+1},M_0)$ with the cokernel of $C_{k+1}\to H_k(W_k,M_0)$, because $H_k(W_{k+1},W_k)=0$. Under step 3.1 this map is ${\mathrm{\partial }}_{k+1}$. Adding stages of index $j\ge k+2$ changes neither $H_k$ nor its map, since both $H_{k+1}(W_j,W_{j-1})$ and $H_k(W_j,W_{j-1})$ vanish. The finite filtration therefore gives $H_k(W,M_0)\cong \ker {\mathrm{\partial }}_k\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{k+1}$.

The pair connector sends an upper core class to its oriented attaching sphere. To read its coefficient at $e_j$, collapse the preceding stage and all other $k$-handle summands. In the outgoing piece $D^k\times S^{d-k-1}$ of $e_j$, $d=\dim W$, the resulting map to $D^k\mathrm{/}{S}^{k-1}$ is projection onto the $D^k$ factor. A regular value at its center has preimages exactly $A_i\cap B_j$; the local degrees are $I(A_i,B_j)$'s local signs by the normal-orientation convention. Summing gives the displayed formula. For $k=0$ the connector records the two signed endpoints, and for $k=d-1$ the same calculation is dual. This local argument applies to a relative triad as well as a closed manifold. The coordinate convention and the two-index assertion now follow from steps 1.1–4.1.