Realization of prescribed Whitehead torsion by h-cobordisms

Choose an invertible matrix $A=(a_{ij})\in {\mathrm{G}\mathrm{L}}_c(\mathbb{Z}[\pi ])$ with class $u$ by [F1]. Attach $c$ pairwise disjoint standard framed $2$-handles along circles bounding disks in $M$, giving $W_2$ and its outgoing level $N$. The circles are nullhomotopic, so [F5] identifies ${\pi }_1(N)\cong {\pi }_1(W_2)\cong \pi$.

For each handle take its standard framed cancelling $2$-sphere from [F2], meeting its belt once and the others not at all. For every monomial $\pm \gamma$ in column $i$ of $A$, take a disjoint parallel copy of the corresponding sphere, reversing its orientation for a negative sign. Join those finitely many copies by framed bands whose core paths represent the prescribed labels. Such paths exist in the full belt complement by [F5] and step 1.1 and can be chosen embedded and away from the copied spheres; $1+2<n$ permits the needed relative general-position avoidance. Their transverse $2$-disk thickenings give the bands of [F2]. The connected sum is a framed embedded $2$-sphere with intersection vector ${\sum }_j[{\phi }_j^2]\cdot a_{ji}$. The framing is the one explicitly glued from the copies and framed bands; no generic unframed sphere is declared to have trivial normal bundle.

Construct the finite columns successively. Perturb and route the new copies and bands relative to their fixed end disks so that the column spheres remain pairwise disjoint: two $2$-sphere images have expected intersection dimension $4-n<0$, and band cores avoid previously constructed $2$-spheres since $3-n<0$. Normal parallel copies and their glued frames are retained. Attach the $c$ $3$-handles along these framed column spheres. By construction their algebraic attaching-belt matrix is exactly $A$, so [F3] gives the contractible relative complex with contraction torsion $(-1{)}^2[A]=u$. This construction realizes arbitrary columns directly; it does not use a homology lemma that only handles unit rows.

The incoming inclusion induces a fundamental-group isomorphism because the $2$-handle attaching loops are null and $3$-handles change no fundamental group. In the reverse presentation the handle indices are $n-2$ and $n-1$, both at least three for $n\ge 5$, so the outgoing inclusion also induces a fundamental-group isomorphism. Its relative complex is the dual of the two-term complex, with adjoint-transpose differential of $A$, hence invertible as well. Apply the homotopy-equivalence criterion [F4] at both ends. Thus the result is the required oriented h-cobordism with its prescribed presentation and torsion.