Elementary matrix operations are realized by handle slides

Identify the handle-chain coefficient with the intersection count, rather than assuming homology bilinearity. Contract the lower stage and the disk-normal directions of the $k$-handles to obtain the relative cell model of [F2]. For the coefficient of an upper attaching sphere $A_i$ on core $c_j$, collapse every other cell, obtaining a continuous map $A_i\cong S^k\to D^k\mathrm{/}{S}^{k-1}\cong S^k$. In the outgoing product $D^k\times S^{n-k-1}$ of the $j$th handle it is projection to the core coordinate modulo its boundary; it is the basepoint off that region. The fibre over the core centre is exactly $A_i\cap B_j$. The product chart in [F6] shows that each local degree is the corresponding intersection sign, with core generators oriented dually to the belt orientations (a common dimension-dependent convention sign has no effect on the transformations below). By the local-degree sum and cellular coefficient formula in [F6], $M_{ij}$ is the coefficient of the upper handle boundary at $c_j$. The same collapse with mod-two coefficients counts the preimages without signs.

Write that boundary as ${\sum }_r{M}_{ir}{c}_r$. Under the lower slide, [F2] gives $c_{j^{\mathrm{\prime }}}^{\mathrm{\prime }}=c_{j^{\mathrm{\prime }}}+\epsilon c_j$ and $c_j^{\mathrm{\prime }}=c_j$, with the other basis vectors fixed; use the comparison of [F3] to transport the upper attaching data. Substitution of $c_{j^{\mathrm{\prime }}}=c_{j^{\mathrm{\prime }}}^{\mathrm{\prime }}-\epsilon c_j^{\mathrm{\prime }}$ gives the new coefficients $M_{ij}^{\mathrm{\prime }}=M_{ij}-\epsilon M_{i{j}^{\mathrm{\prime }}}$ and $M_{i{j}^{\mathrm{\prime }}}^{\mathrm{\prime }}=M_{i{j}^{\mathrm{\prime }}}$, all others unchanged. Thus the operation is $C_j\mapsto C_j-\epsilon C_{j^{\mathrm{\prime }}}$. It is the inverse dual change, rather than the core change applied directly to belt spheres.

For an upper slide the target core basis stays fixed while [F2] replaces the upper core by $d_{i^{\mathrm{\prime }}}^{\mathrm{\prime }}=d_{i^{\mathrm{\prime }}}+\epsilon d_i$. Its boundary is $\mathrm{\partial }{d}_{i^{\mathrm{\prime }}}+\epsilon \mathrm{\partial }{d}_i$, so step 1.1 gives $R_{i^{\mathrm{\prime }}}\mapsto R_{i^{\mathrm{\prime }}}+\epsilon R_i$, with the other rows unchanged. This slide is in the printed range $k+1\le n-2$.

Reversing an upper core orientation reverses its attaching-sphere orientation and hence its row; reversing a lower cocore orientation reverses its belt-sphere orientation and hence its column. These changes are $-1$ multiplications by the local sign convention of [F1]. Over ${\mathbb{Z}}_2$ signs disappear. Together with steps 2.1–2.2 this realizes the elementary additions and sign changes in the stated ranges; it does not turn an algebraic unit into a single geometric intersection.