Talk:Whitehead torsion IV (Ex)

Let $\varphi:W\rightarrow [0,1]$$<wikitex> Let $\varphi:W\rightarrow [0,1]$ a Morse function with $M'=\varphi^{-1}(0),M=\varphi^{-1}(1)$. We define $f:W\rightarrow M'\times I$ by $w\mapsto\left((i')^{-1}(w),\varphi(w)\right)$. For the Whitehead torsion of $(i')^{-1}\circ i:M\rightarrow M'$ we have, using the quoted results of Milnor, $$\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$$ So on the boundary $M$ of $W$ we have $\tau(f|_{M})=0$. Since the odd-dimensional L-groups $L_{2k+1}^s(\Zz\pi)$ vanish for finite groups $\pi$ of odd order there is no obstruction to do surgeries in the interior of $W$ to turn it into an s-cobordism without changing the boundary. Thus $M$ and $M'$ are s-cobordant. </wikitex>\varphi:W\rightarrow [0,1]$ a Morse function with $M'=\varphi^{-1}(0),M=\varphi^{-1}(1)$$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 $\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.