A group-ring handle matrix and its torsion class

By [F1] $u=1-t^2-t^3$ is a unit of $R=\mathbb{Z}[C_5]$ and its class $[u]$ is nonzero in $\mathrm{Wh}(C_5)$; by [F2] the based two-term complex $0\to R\stackrel{u}{\to }R\to 0$ with one basis vector in each of degrees $1$ and $0$ is contractible with contraction $s_0=u^{-1}$, and its contraction torsion is $(-1{)}^{1+1}[u]=[u]\mathrm{\ne }0$.

Take $S^5\subset {\mathbb{C}}^3$ and let a generator of $C_5$ multiply every coordinate by $\zeta =e^{2\pi i\mathrm{/}5}$. This action is free: ${\zeta }^{k}z=z$ for $z\mathrm{\ne }0$ implies ${\zeta }^k=1$. Each point has a small ball in $S^5$ disjoint from its other four translates; these balls give covering charts for $S^5\to M=S^5\mathrm{/}{C}_5$, and the quotient charts have smooth transition maps given by restrictions of the linear action. Distinct finite orbits have disjoint invariant neighborhoods, so the quotient is Hausdorff; images of a countable sphere basis give a countable quotient basis. Thus $M$ is a compact connected smooth boundaryless $5$-manifold. The sphere is simply connected by $S^n$ is simply connected for every $n\ge 2$. Any deck map agrees at one point with one of the five action maps, hence agrees everywhere by uniqueness on a connected cover. Therefore For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group gives ${\pi }_1(M)=C_5$.

Orient $S^5$ by the boundary orientation induced from the standard orientation of ${\mathbb{C}}^3$ and orient $M$ by pushing this orientation forward along the local diffeomorphism $S^5\to M$; this is well defined because scalar multiplication by a unit complex number is complex linear and hence orientation-preserving, so the deck translations preserve the chosen orientation of $S^5$. Thus $M$ is a closed connected oriented $n$-manifold with $n=5$ and ${\pi }_1(M)=C_5$, and the oriented hypothesis of [F3] is met.

Apply [F3] with the matrix $A=(u)$ of size one: there is an h-cobordism $(W;M,M^{\mathrm{\prime }})$ of dimension $n+1\ge 6$ and a handle presentation $H$ relative to $M$ with one $2$-handle and one $3$-handle whose intersection matrix is $A=(u)$, invertible with class $[A]=[u]$, and whose presentation-indexed torsion is ${\tau }_H(W,M)=(-1{)}^2[u]=[u]\mathrm{\ne }0$.

The exhibited presentation therefore has ${\tau }_H=u$-class nonzero in $\mathrm{Wh}(C_5)$, while the same $u$ appears as the contraction torsion of the abstract two-term complex of step 1.1; this is the announced class computation, and it makes no claim about presentations of the same h-cobordism other than $H$, consistent with the presentation-relative scope of the criterion.