Whitehead torsion (Ex)

Show that $1 - t - t^{-1} \in \Zz[\Zz/5]$$<wikitex>; Show that 1 - t - t^{-1} \in \Zz[\Zz/5]$ for $t\in \Zz/5$$t\in \Zz/5$ the generator is a unit and hence defines an element $\eta$$\eta$ in $\textup{Wh}(\Zz/5)$$\textup{Wh}(\Zz/5)$. Prove that we obtain a well-defined map

$\displaystyle \textup{Wh}(\Zz/5) \to \Rr$ $$\displaystyle \textup{Wh}(\Zz/5) \to \Rr$$

by sending the class represented by the $\Zz[\Zz/5]$$\Zz[\Zz/5]$-automorphism $f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$$f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$ to $\ln(|\det(\overline{f})|)$$\ln(|\det(\overline{f})|)$, where $\overline{f} \colon \Cc^n \to \Cc^n$$\overline{f} \colon \Cc^n \to \Cc^n$ is the $\Cc$$\Cc$-linear map

$\displaystyle f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$ $$\displaystyle f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$$

with respect to the $\Zz/5$$\Zz/5$-action on $\Cc$$\Cc$ given by multiplication with $\exp(2 \pi i/5)$$\exp(2 \pi i/5)$. Finally show that $\eta$$\eta$ generates an infinite cyclic subgroup in $\textup{Wh}(\Zz/5)$$\textup{Wh}(\Zz/5)$.

This is a detailed version of [Milnor1966, Example 6.6] and [Kreck&Lück2005, Ex 5.4].

- t - t^{-1} \in \Zz[\Zz/5]$ for $t\in \Zz/5$ the generator is a unit and hence defines an element $\eta$ in $\textup{Wh}(\Zz/5)$. Prove that we obtain a well-defined map $$\textup{Wh}(\Zz/5) \to \Rr$$ by sending the class represented by the $\Zz[\Zz/5]$-automorphism $f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$ to $\ln(|\det(\overline{f})|)$, where $\overline{f} \colon \Cc^n \to \Cc^n$ is the $\Cc$-linear map $$f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$$ with respect to the $\Zz/5$-action on $\Cc$ given by multiplication with $\exp(2 \pi i/5)$. Finally show that $\eta$ generates an infinite cyclic subgroup in $\textup{Wh}(\Zz/5)$. This is a detailed version of {{citeD|Milnor1966|Example 6.6}} and {{citeD|Kreck&Lück2005|Ex 5.4}}. [[Category:Exercises]] [[Category:Exercises with solution]]1 - t - t^{-1} \in \Zz[\Zz/5] for $t\in \Zz/5$$t\in \Zz/5$ the generator is a unit and hence defines an element $\eta$$\eta$ in $\textup{Wh}(\Zz/5)$$\textup{Wh}(\Zz/5)$. Prove that we obtain a well-defined map

$\displaystyle \textup{Wh}(\Zz/5) \to \Rr$ $$\displaystyle \textup{Wh}(\Zz/5) \to \Rr$$

by sending the class represented by the $\Zz[\Zz/5]$$\Zz[\Zz/5]$-automorphism $f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$$f \colon \Zz[\Zz/5]^n \to \Zz[\Zz/5]^n$ to $\ln(|\det(\overline{f})|)$$\ln(|\det(\overline{f})|)$, where $\overline{f} \colon \Cc^n \to \Cc^n$$\overline{f} \colon \Cc^n \to \Cc^n$ is the $\Cc$$\Cc$-linear map

$\displaystyle f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$ $$\displaystyle f \otimes_{\Zz[\Zz/5]} \id_{\Cc} \colon \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc \to \Zz[\Zz/5]^n \otimes_{\Zz[\Zz/5]} \Cc$$

with respect to the $\Zz/5$$\Zz/5$-action on $\Cc$$\Cc$ given by multiplication with $\exp(2 \pi i/5)$$\exp(2 \pi i/5)$. Finally show that $\eta$$\eta$ generates an infinite cyclic subgroup in $\textup{Wh}(\Zz/5)$$\textup{Wh}(\Zz/5)$.

This is a detailed version of [Milnor1966, Example 6.6] and [Kreck&Lück2005, Ex 5.4].