h-cobordisms admit two-index normal form presentations

Start from any given finite presentation. By the constructive rearrangement and zero-handle procedure of [F1], put it into adapted index order with no $0$-handles, using attaching isotopies, disjoint commutations and geometric $0\mathrm{/}1$ cancellations. The incoming inclusion is a homotopy equivalence. For each $1$-handle form the circle from a parallel half-core and a joining arc in the incoming boundary with its attaching balls removed. Choose the relative path class of this arc using the surjection ${\pi }_1(M_0)\to {\pi }_1(W)$ so that it cancels the loop class of the half-core and a reference joining arc; relative one-dimensional general position gives an embedded representative avoiding the other attaching balls. Shrink the existing $2$-handle attaching tubes and perturb the circle away from their core circles, using $1+1<n$, so it lies in the common part of the levels before and after the $2$-handles. Its loop is therefore null in $W$ and hence in the level after the $2$-handles, since later handles have index at least three and the reverse trace to that level has index at least three. Choose an embedded nullhomotopy disk in that level, possible because $n\ge 5$, and use its normal frame to make the circle a framed embedding trivial one level higher. It meets the selected $1$-handle belt once and avoids the other belts. [F2] trades that handle for a $3$-handle. After finitely many steps all indices are at least two.

For $r=2,\dots ,q-1$, assume all indices are at least $r$ and fix an $r$-handle $e$. Contractibility of the based handle complex and $C_{r-1}=0$ give coefficients $x_j$ with ${\sum }_j{d}_{r+1}[{\phi }_j]\cdot x_j=[e]$. Start with a trivial framed $r$-sphere $\alpha$ in the outgoing level after the $r$-handles; its class is zero. The modification construction gives a sphere $\beta$ with class $[e]$ isotopic to $\alpha$ one level higher. It supplies a framing by transporting the trivial normal frame along that higher-level isotopy; $\beta$ lies in the unchanged open part common to the two levels. Since $r\le q-1\le n-3$, the corrected homology lemma applies. For $r=2$, [F6] supplies its incoming fundamental-group injection, because the presentation still represents the same h-cobordism. It puts $\beta$ into single-point position. Carry the higher attaching data along that ambient isotopy, preserving its one-level-higher triviality and framing. The elimination lemma trades $e$ for an $(r+2)$-handle. Repeating deletes every handle of index below $q$.

Reverse the triad. An original handle of index $k$ becomes a dual handle of index $n+1-k$. Remove dual $0$- and $1$-handles by the same low-index argument, then eliminate dual indices $r=2,\dots ,n-q-1$. This upper bound is at most $n-3$, so only the already proved homology range and its $r=2$ h-cobordism complement condition are used. Trading such a dual $r$-handle creates a dual $(r+2)$-handle, which is an original handle of index $n-r-1\ge q$; the dual $1$-handle trade creates original index $n-2\ge q$. Thus this phase removes all original indices at least $q+2$ without reintroducing any original index below $q$. The remaining indices are exactly $q,q+1$, for every $2\le q\le n-2$. No arbitrary-sphere flipped endpoint is invoked.

The remaining based handle complex is concentrated in degrees $q,q+1$ and contractible. Its only differential, the intersection matrix, is therefore an isomorphism; its two free modules have equal rank and the two handle numbers agree, because tensoring this isomorphism with $\mathbb{Z}$ along the augmentation gives an isomorphism of finite free abelian groups. All changes preserve the relative diffeomorphism type and are the indicated elementary presentation modifications. This proves the full stated normal form without simple connectivity.