Talk:Whitehead torsion IV (Ex)

Revision as of 13:36, 29 March 2012

Let $<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 for $f|_M=(i')^{-1}\circ i:M\rightarrow M'$ that $\tau(f|_{M})=0$. 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 $M'=\varphi^{-1}(0),M=\varphi^{-1}(1)$. 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,

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

So on the boundary $M$ of $W$ we have for $f|_M=(i')^{-1}\circ i:M\rightarrow M'$ that $\tau(f|_{M})=0$. 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.