Presentation-indexed Whitehead torsion of an h-cobordism

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $(W;M_0,M_1)$ be a nonempty connected smooth h-cobordism (h-Cobordism), put $\pi ={\pi }_1(M_0)={\pi }_1(W)$, the identification being along the homotopy equivalence $M_0\hookrightarrow W$, and fix a finite handle presentation $H$ of $(W,M_0)$ (The based handle chain complex over the fundamental group ring). Choose one oriented lift of each handle as in the based handle complex and any chain contraction $s$ of $C_{\ast }^{\mathrm{h}}(W,M_0;H)$, which exists by The handle complex of an h-cobordism is contractible over the group ring, with an explicit contraction. The presentation-indexed Whitehead torsion is ${\tau }_H(W,M_0):={\tau }_s(C_{\ast }^{\mathrm{h}}(W,M_0;H))\in \mathrm{Wh}(\pi ),$ the contraction torsion of the bounded contractible based free right $\mathbb{Z}[\pi ]$-complex in the sense of AT-22 (Finite based free complexes and contraction torsion, Contraction torsion does not depend on the contraction), followed by the quotient map ${\tilde{K}}_1(\mathbb{Z}[\pi ])\to \mathrm{Wh}(\pi )$ (K₁ of a ring and the Whitehead group of a discrete group). In a presentation with handles only in two adjacent degrees $q,q+1$ and differential matrix $A:C_{q+1}\to C_q$, the convention gives ${\tau }_H(W,M_0)=(-1{)}^q[A]$, which by The torsion of the handle complex is the torsion of the inclusion is the Whitehead torsion of the inclusion $M_0\hookrightarrow W$ for the associated CW structures. For fixed $H$ the class is independent of the contraction and the other auxiliary choices specified in the well-definedness theorem. This notation retains the presentation $H$; it does not assert equality for arbitrary handle presentations.