Modification lemma: prescribed class changes by isotopy of an embedded boundary sphere

Fix a $(q+1)$-handle ${\phi }_j$. Choose a parallel copy $P_j$ of its attaching sphere just outside its closed attaching tube, using the extension over a neighborhood of the normal disk factor. It lies in $N_q^{\circ }$, the open complement of all closed upper attaching tubes. In $N_{q+1}$ it bounds the outgoing disk $D^{q+1}\times \{z\}$, $z\in S^{n-q-1}$, together with the short annular collar from its boundary to $P_j$. This is an embedded disk with interior in that handle's outgoing piece, disjoint from any sphere in the common part. Its normal disk coordinates give the standard bounding-disk framing. In $C_q$, $[P_j]={\mathrm{\partial }}_{q+1}[{\phi }_j]$ with compatible orientations, by [F1].

$N_q^{\circ }$ is connected: the upper attaching cores have codimension $n-q\ge 2$, so paths can be rerouted in finitely many product charts off the cores, and radial retraction in each punctured tube pushes them outside the closed smaller tubes. Apply [F2] within this common part to band-sum $f$ with $P_j$. The band interior can avoid both spheres and all other attaching tubes. The resulting $q$-sphere $g$ is embedded and remains in $N_q^{\circ }$. An oriented pair-of-pants bordism in a thin neighborhood of the band has boundary $g-f-P_j$, so in $C_q$ its fundamental chain gives $[g]=[f]+[P_j]$. Choose the orientation of the added parallel copy, rather than reverse $f$, to obtain either sign.

In $N_{q+1}$, slide the added lobe back along the band and across the embedded disk of step 1.1. The disk interior is in the new handle's outgoing region, whereas $f$ and the band are in the common part. A thin product neighborhood of their union therefore gives the usual embedded isotopy from the band-sum to $f$, fixed outside that neighborhood. If $f$ has a normal frame, use the bounding-disk frame for $P_j$ and the matching band frame; this isotopy transports the full frame and gives a framed isotopy as well. Thus no implication from a vanishing homology class to embedding triviality is used.

Repeat with fresh disjoint parallel copies $\mathrm{\mid }{x}_j\mathrm{\mid }$ times, using the sign of $x_j$, and then over the finite upper-handle list. At every iteration the new sphere is in the common part and is isotopic one level higher to the preceding sphere. The additive computation gives $[g]=[f]+{\sum }_j{x}_j{\mathrm{\partial }}_{q+1}[{\phi }_j]$. In particular, a standard trivial framed starting sphere yields a sphere framed-isotopic to it in $N_{q+1}$. This proves the assertion, including zero coefficients and an empty upper-handle list.