Talk:Reidemeister torsion (Ex)
Markus Land (Talk | contribs) (Created page with "<wikitex>; Since the chain complex in question is finite and levelwise projective we know that contractibility is implied by being acyclic. And we see directly that if $r \ne...") |
m (A twist in the definition of the contraction is corrected. An easy calculation in K_1 is added.) |
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To compute the Reidemeister torsion we need to construct a chain homotopy $s: C_\bullet \to C_{\bullet+1}$ such that | To compute the Reidemeister torsion we need to construct a chain homotopy $s: C_\bullet \to C_{\bullet+1}$ such that | ||
$$ ds + sd = \id $$ | $$ ds + sd = \id $$ | ||
− | Here we can take $s_0 : C_0 = \Q \to \Q\oplus \Q = C_1$ to be the map $1 \mapsto ( | + | Here we can take $s_0 : C_0 = \Q \to \Q\oplus \Q = C_1$ to be the map $1 \mapsto (1,0)$ as then |
− | $$ s_{-1}(d_0(x)) + d_1(s_0(x)) = d_1( | + | $$ s_{-1}(d_0(x)) + d_1(s_0(x)) = d_1(x,0) = x $$ |
− | and then we need to choose $s_1: C_1 = \Q\oplus \Q \to \Q = C_2$ to be the map $(x,y) \mapsto \frac{ | + | and then we need to choose $s_1: C_1 = \Q\oplus \Q \to \Q = C_2$ to be the map $(x,y) \mapsto \frac{y}{r}$. |
− | To compute the torsion of this complex we have to | + | To compute the torsion of this complex we have to consider the class |
− | $$ \lbrack d+s : C_{\mathrm{ev}} \to | + | $$ \lbrack d+s : C_{\mathrm{ev}} \to C_{\mathrm{odd}} \rbrack \in K_1(\Q) $$ |
which is given by | which is given by | ||
− | $$ (d+s)(x,y) = d_2(x) + s_0(y) = ( | + | $$ (d+s)(x,y) = d_2(x) + s_0(y) = (y,rx) $$ |
− | and hence the matrix is given by | + | and hence the matrix of $d+s$ in the standard basis is given by |
− | $$\begin{pmatrix} r & 0 \\ 0 & 1 \end{pmatrix}$$ | + | $$A:=\begin{pmatrix} 0 & 1 \\ r & 0 \end{pmatrix},$$ |
− | + | whose class in $K_1(\Q)$ is the same as the class of $(-r)$ since | |
+ | $$ A\simeq E_{12}^{-1}E_{21}^{1}AE_{21}^{-r}=\begin{pmatrix} -r & 0 \\ 0 & 1 \end{pmatrix} \simeq (-r) $$ where $\simeq$ stands for equality in $K_1(\Q)$ and $E_{ij}^q$ is the elementary matrix with $q\in\Q$ at the $(i,j)$ entry. | ||
</wikitex> | </wikitex> |
Latest revision as of 18:22, 3 September 2013
Since the chain complex in question is finite and levelwise projective we know that contractibility is implied by being acyclic. And we see directly that if then the complex is exact, hence acyclic. To compute the Reidemeister torsion we need to construct a chain homotopy such that
Here we can take to be the map as then
and then we need to choose to be the map .
To compute the torsion of this complex we have to consider the class
which is given by
and hence the matrix of in the standard basis is given by
whose class in is the same as the class of since
Here we can take to be the map as then
and then we need to choose to be the map .
To compute the torsion of this complex we have to consider the class
which is given by
and hence the matrix of in the standard basis is given by
whose class in is the same as the class of since