Whitehead Torsion and the S Cobordism Theorem — Examples

By [F1] the Whitehead group of the trivial group vanishes, and by [F2] the fundamental group of the simply connected closed manifold $M_0$ is trivial, so every presentation-indexed class ${\tau }_H(W,M_0)$ of a finite handle presentation of $(W,M_0)$ lies in $\mathrm{Wh}(1)=0$ and is therefore the zero class.

Take the finite handle presentation $H$ of $(W,M_0)$ supplied by [F5]; by step 1.1 its class vanishes, and by [F6] the oriented hypotheses of the criterion are met, so applying [F3] with this presentation yields a diffeomorphism $W\cong M_0\times [0,1]$ relative to $M_0$.

The product model is consistent with the conclusion: the height function of $M_0\times [0,1]$ has no critical points and presents it with the empty handle list by [F4], whose torsion class is zero by the criterion; conversely the argument of steps 1.1 and 2.1 shows that every simply connected h-cobordism in dimension $n+1\ge 6$ is a product, which is the simply connected h-cobordism theorem as the case $\pi =1$ of the presentation-relative criterion.

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.

In the old target coordinates the new basis has columns $e_k^{\mathrm{\prime }}$, hence matrix $P=I+E_{ij}r$, with $P^{-1}=I-E_{ij}r$. The source basis is unchanged. Since an old target coordinate column equals $P$ times its new column, the new differential is $A^{\mathrm{\prime }}=P^{-1}A$. Thus row $i$ changes by subtracting $r$ times row $j$. The target core change acts inversely on differential coordinates; it must not be copied directly onto the belt basis.

For the concrete matrix $A=I_2$, take $i=1,j=2,r=1$. Then $P=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ and $A^{\mathrm{\prime }}=\left(\begin{array}{cc}1& -1\\ 0& 1\end{array}\right)\mathrm{\ne }{I}_2$. Both are invertible. More generally $P^{-1}A=A$ for an invertible $A$ would imply $P=I$, impossible for $r=\pm g$.

By [F2], $[P]=0$ and $[P^{-1}A]=-[P]+[A]=[A]$ in the Whitehead group. Multiplying by the parity sign in [F3] gives ${\tau }_{H^{\mathrm{\prime }}}={\tau }_H$. This supplies the promised matrix computation and the unchanged torsion class.

Let $C$ be the based free right $R$-complex $0\to C_1=R\stackrel{d_1}{\to }{C}_0=R\to 0$ with differential $d_1(x)=xu$, generated in degree one and zero by one basis vector each. By [F1] $u$ is a unit, and the map $h_0(y)=y{u}^{-1}$, $h_1=0$ satisfies $d_1{h}_0={\mathrm{i}\mathrm{d}}_{C_0}$ and $h_0{d}_1={\mathrm{i}\mathrm{d}}_{C_1}$, so $C$ is contractible with this contraction; by [F2] with $q=1$ its contraction torsion is $\tau (C)=(-1{)}^2[u]=[u]$, which is nonzero in $\mathrm{W}\mathrm{h}(C_5)$ by [F1].

Forgetting the $R$-module structure, the contraction $h$ of step 1.1 is a homomorphism of abelian groups, so it contracts the underlying complex of abelian groups of $C$; hence that underlying complex is contractible, and in particular acyclic, with differential the isomorphism $x\mapsto xu$ and inverse $y\mapsto y{u}^{-1}$. Separately, by [F3] the augmentation $\epsilon :\mathbb{Z}[C_5]\to \mathbb{Z}$ is a ring homomorphism with $\epsilon (1)=1$ and $\epsilon (t^k)=1$ for every $k$, so $\epsilon (u)=1-1-1=-1$, which is a unit of $\mathbb{Z}$; the base change along $\epsilon$ gives the complex $0\to \mathbb{Z}\stackrel{-1}{\to }\mathbb{Z}\to 0$, contracted by $n\mapsto -n$, whose homology also vanishes.

The complex $C$ is bounded and based free over $R=\mathbb{Z}[C_5]$, its contraction torsion class $[u]$ is nonzero in $\mathrm{W}\mathrm{h}(C_5)$ by step 1.1, and its underlying complex of abelian groups is contractible, in particular acyclic, by step 2.1; therefore the false claim fails. This is exactly why the handle chain complex of a cobordism is carried over $\mathbb{Z}[{\pi }_1]$ with its chosen lifts and basis and not over $\mathbb{Z}$, as recorded in [F4]: an abelian acyclicity check cannot see the class $[u]$.