Talk:Whitehead torsion IV (Ex)

Let $\varphi:W\rightarrow [0,1]$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\varphi:W\rightarrow [0,1]$ a Morse function with $M'=\varphi^{-1}(0),M=\varphi^{-1}(1)$$M'=\varphi^{-1}(0),M=\varphi^{-1}(1)$. We define $f:W\rightarrow M'\times I$$f:W\rightarrow M'\times I$ by $w\mapsto\left((i')^{-1}(w),\varphi(w)\right)$$w\mapsto\left((i')^{-1}(w),\varphi(w)\right)$. For the Whitehead torsion of $(i')^{-1}\circ i:M\rightarrow M'$$(i')^{-1}\circ i:M\rightarrow M'$ we have, using the quoted results of Milnor,

$\displaystyle \tau\left((i')^{-1}\circ i\right)=-\tau(W,M')+\tau(W,M)=(-1)^{2k+1}\hat\tau(W,M)+\tau(W,M)=-\tau(W,M)+\tau(W,M)=0$ $$\displaystyle \tau\left((i')^{-1}\circ i\right)=-\tau(W,M')+\tau(W,M)=(-1)^{2k+1}\hat\tau(W,M)+\tau(W,M)=-\tau(W,M)+\tau(W,M)=0$$

So on the boundary $M$$M$ of $W$$W$ we have for $f|_M=(i')^{-1}\circ i:M\rightarrow M'$$f|_M=(i')^{-1}\circ i:M\rightarrow M'$ that $\tau(f|_{M})=0$$\tau(f|_{M})=0$. Since the odd-dimensional L-groups $L_{2k+1}^s(\Zz\pi)$$L_{2k+1}^s(\Zz\pi)$ vanish for finite groups $\pi$$\pi$ of odd order there is no obstruction to do surgeries in the interior of $W$$W$ to turn it into an s-cobordism without changing the boundary. Thus $M$$M$ and $M'$$M'$ are s-cobordant.