Talk:Whitehead torsion IV (Ex)

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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,

\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.

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