The group-labelled homology lemma realizes group-ring handle bases by isotopy

Given: A nonempty connected compact smooth $(n+1)$-manifold $W$, $n+1\ge 6$, with a finite relative handle decomposition whose handles all have index at least $q$ for a fixed $2\le q\le n-3$, a fixed $q$-handle $\phi$, an element $\gamma \in \pi ={\pi }_1(W)$, and an embedded sphere $f:S^q\hookrightarrow {\mathrm{\partial }}_1{W}_q$ with class $[\phi ]\cdot (\pm \gamma )$ in $C_q=C_q^{\mathrm{h}}(W,M_0)$, with integral chains in the ambient universal cover and their deck-induced right $\mathbb{Z}[\pi ]$-action.

With the ambient-cover, right-module conventions of The based handle chain complex over the fundamental group ring, the relative handle complex of the presentation is the free $\mathbb{Z}[\pi ]$-complex with one basis class per handle in its index, and in the middle level ${\mathrm{\partial }}_1{W}_q$ the coefficient of the class of an embedded sphere in a $q$-handle basis element is the sum of the signed group labels of its transverse intersection points with the belt sphere of that handle, with the orientations induced by the handle framings; the sphere can first be isotoped to be transverse to all belt spheres, and the labels are computed by arcs in the two sheets (The relative handle chain complex computes $H_{\ast }(W,M_0)$ and has the intersection matrix as its differential, Handle boundary coefficients are attaching-belt intersection numbers, Attaching-belt intersection matrix of adjacent-index handles, K handle core cocore attaching region and belt sphere).

For a transverse pair of a $q$-sphere and a belt sphere with two intersection points, arcs joining them in the two sheets and avoiding all other intersection points exist, and the resulting Whitney circle is null-homotopic exactly when the two points carry equal fundamental-group labels; the labels must use paths compatible with the chosen arcs. Only the orientation-free circle criterion is quoted from the label lemma: signs here are the lifted core/normal incidence signs of [F1], and opposite incidence signs with equal labels are treated by [F3] (Arcs joining two points of a connected submanifold avoiding finitely many points, The fundamental-group label controls contractibility of the Whitney circle).

Whitney moves: if the two sheets of a transverse pair have dimensions $a,b\ge 3$ in a manifold of dimension $a+b$, and the two double points have opposite signs, then a null-homotopic Whitney circle admits a clean framed Whitney disk and an isotopy removing the pair and creating no new intersections; and if the isotoped sheet has dimension at most $2$ while the fixed sheet has dimension at least $3$ and the fundamental-group complement condition holds, the same conclusion holds in the two-dimensional borderline (The high-dimensional Whitney trick, The Whitney trick in the codimension-two borderline case).

Coefficient sums: if signed labels of intersection points sum to $0$, or to a single $\pm \gamma$, and there are at least two points, then two of them carry equal labels and opposite signs (A vanishing group-ring coefficient sum pairs off opposite-signed equal labels).

Deleting the actual $q$-handle belts identifies their complement with the incoming boundary minus its attaching cores. For $q=2$ the explicit incoming fundamental-group injection therefore gives complement injection; an h-cobordism satisfies this condition. The helper also constructs a Whitney disk in the full belt complement and avoids additional $2$-spheres. Belt-sphere complements in low handle levels preserve the fundamental group