The group-ring modification lemma for embedded spheres

Put $N={\mathrm{\partial }}_1{W}_q$. Since all initial indices are at least two, connectedness of $W$ forces connectedness of its nonempty incoming collar and $W_q$. The surgery description of $N$ replaces tubes of $(q-1)$-spheres, of codimension $n-q+1\ge 3$, by $D^q\times S^{n-q}$, with connected gluing regions; thus $N$ is connected. By [F2], ${\pi }_1(N)\to {\pi }_1(W)$ is an isomorphism. Removing the higher attaching cores, of codimension $n-q\ge 2$, preserves connectedness and surjects on fundamental groups: perturb paths and loops transverse to them using [F3], with expected intersection dimensions $1+q-n<0$. Radial collar retraction replaces core complements by tube complements. Consequently every desired group label is represented by a path in $N^{\circ }$.

For handle ${\phi }_j$, take a parallel copy $t_j=S^q\times \{z\}$ of its attaching sphere, with $z$ on the boundary of the normal disk. Push it slightly into $N^{\circ }$. Its lift represents $d[{\phi }_j]$ by [F1], with orientation chosen accordingly. After the handle is attached it bounds the outgoing disk $D^{q+1}\times \{z\}$, and is therefore a trivial framed sphere in ${\mathrm{\partial }}_1{W}_{q+1}$. The disk is disjoint from $f$, which lies in the old common open region.

Fix a monomial $\epsilon \gamma$ in $x_j$. Choose a joining path from $f$ to $t_j$ in $N^{\circ }$ with the required relative homotopy class by step 1.1. Smooth it, preserve its embedded endpoint germs, approximate it by an embedded arc and perturb its interior off the two spheres by [F3]; the inequalities are $2<n$ and $1+q<n$. Thicken it to a sufficiently thin framed band as in [F3]. Lifting that band fixes the lift of its second sphere; by choosing the path class this is $T_{{\gamma }^{-1}}{\tilde{t}}_j$, namely ${\tilde{t}}_j\cdot \gamma$. The band sum thus has class $[\tilde{f}]+\epsilon d[{\phi }_j]\cdot \gamma$: collapsing the band to its core gives the pinch map whose two oriented sphere classes add, and the negative orientation gives the sign $\epsilon$. This uses right multiplication throughout.

In the higher outgoing level, the disk bounded by $t_j$ together with a thin neighborhood of the joining band lets the added sphere shrink along the band back to its end disk on $f$. This is an isotopy supported in that disk-and-band neighborhood; it preserves a supplied normal framing by transporting it along the same local motion. It is the local band-sum isotopy in Lück’s Modification Lemma, printed pp. 15–16. The resulting sphere remains in $N^{\circ }$, while its comparison isotopy takes place one level higher. When it is attaching data, [F4] transports all subsequent handles.

Expand each $x_j$ as its finite signed sum of group elements. Repeat steps 2.1–3.1 for those finitely many monomials, always choosing fresh sufficiently thin parallel copies and bands. Compatible lifts agree on the unchanged part of the sphere, so the class additions sum to the displayed right-linear formula. Concatenating the higher-level isotopies proves the isotopy and framing assertions. Countable choice is inherited through the geometric suppliers; the coefficient expansion and the number of modifications are finite.