Simply connected h-cobordisms have zero Whitehead obstruction

Given: A connected smooth h-cobordism $(W;M_0,M_1)$ of dimension $n+1\ge 6$ with $M_0$ closed, connected and simply connected.

The trivial group has vanishing Whitehead group: $\mathbb{Z}[1]=\mathbb{Z}$; the determinant is a well-defined surjection $K_1(\mathbb{Z})\to \{\pm 1\}$ because every elementary matrix has determinant $1$; and it is injective, since a common divisor of the entries of a column of an invertible integer matrix divides the determinant $\pm 1$, the division algorithm with the Bézout identity reduces such a primitive column to $(\pm 1,0,\dots ,0{)}^{\mathsf{T}}$ by elementary row additions and swaps, and induction on the size then presents every invertible integer matrix, up to permutation, as elementarily equivalent to a diagonal matrix with entries $\pm 1$, whose class is a sum of classes $[\pm 1]$; hence $K_1(\mathbb{Z})\cong \{\pm 1\}$ and $\mathrm{Wh}(1)=K_1(\mathbb{Z})\mathrm{/}⟨[\pm 1]⟩=0$ (K₁ of a ring and the Whitehead group of a discrete group, Division with remainder in $\mathbb{Z}$: for $a\in \mathbb{Z}$ and $b>0$ there are unique $q,r\in \mathbb{Z}$ with $a=qb+r$ and $0\le r<b$, Bézout's identity: for integers $a,b$ not both zero, $\gcd (a,b)$ is the least positive element of $\{ax+by:x,y\in \mathbb{Z}\}$; in particular $ax+by=\gcd (a,b)$ has an integer solution).

For a simply connected closed $M_0$ the fundamental group ${\pi }_1(M_0)$ is trivial, and the presentation-indexed torsion of every finite handle presentation of $(W,M_0)$ is an element of $\mathrm{Wh}({\pi }_1(M_0))=\mathrm{Wh}(1)$ (Simply connected topological spaces, h-Cobordism).

The presentation-relative s-cobordism criterion: $W$ is diffeomorphic to $M_0\times [0,1]$ relative to $M_0$ if and only if some finite handle presentation $H$ of $(W,M_0)$ has ${\tau }_H(W,M_0)=0$ (The smooth s-cobordism theorem: a vanishing presentation implies a product).

A product cobordism $M_0\times [0,1]$ has the critical-point-free height-function presentation relative to $M_0\times \{0\}$ (Product cobordisms have critical-point-free presentations).

A finite handle presentation of $(W,M_0)$ exists: the two-index normal-form lemma, applied at any allowed index to the oriented data below, produces a handle decomposition of $W$ relative to $M_0$ under the countable-choice hypothesis ${\mathrm{A}\mathrm{C}}_{\omega }$ assumed here (h-cobordisms admit two-index normal form presentations, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The oriented hypotheses of the criterion hold automatically. If $M_0$ were nonorientable, its orientation double cover would be a connected two-sheeted cover of the simply connected space $M_0$ (The orientation double cover is canonically oriented and preserves closedness), contradicting the triviality of connected covers of a simply connected space (A connected covering of a locally path-connected simply connected space is one-sheeted and trivial); hence $M_0$ is orientable, and the homotopy equivalence $M_0\hookrightarrow W$ of the h-cobordism carries ${\pi }_1(M_0)=1$ to ${\pi }_1(W)=1$, so the boundaryless interior of $W$ is simply connected: pushing boundary-collar coordinates a small positive distance inward gives a homotopy equivalence $\mathrm{int}W\hookrightarrow W$, with its homotopies keeping positive coordinates positive. Apply the same cover argument to $\mathrm{int}W$. Its orientation extends over the product boundary collars to $W$, with $M_1$ inheriting a compatible boundary orientation (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type, h-Cobordism, Simply connected topological spaces).