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 $w\mapsto\left((i')^{-1}(w),\varphi(w)\right)$. | 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, | For the Whitehead torsion of $(i')^{-1}\circ i:M\rightarrow M'$ we have, using the quoted results of Milnor, |
Latest revision as of 15:04, 1 April 2012
Let a Morse function with . We define by . For the Whitehead torsion of we have, using the quoted results of Milnor,
So on the boundary of we have for that . Since the odd-dimensional L-groups vanish for finite groups of odd order there is no obstruction to do surgeries in the interior of to turn it into an s-cobordism without changing the boundary. Thus and are s-cobordant.