Whitehead torsion V (Ex)
From Manifold Atlas
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{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
− | + | # Show that the operations (1)-(5) from {{citeD|Lück2001|Section 1.4}} used in the definition of the Whitehead group in fact yield equivalence classes of invertible matrices of arbitrary size. | |
− | $$ | + | # Show that addition is well-defined in the Whitehead group via |
− | + | $$ [A] + [B] = \left [\begin{array}{cc} A & 0 \\ 0 & B \end{array} \right] = [A \cdot (B \oplus \text{Id}_{n-m})].$$ | |
− | + | Here $A$ and $B$ are invertible matricies over the group ring $\Zz[\pi]$, $A$ is an $n \times n$ matrix, $B$ is an $m \times m$ matrix with $n \geq m$, $O$ denotes the zero $n \times m$ and $m \times n$ matricies and $\cdot$ denotes matrix multiplication. | |
− | + | # Show that $Wh(e)$ is trivial. | |
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{{endthm}} | {{endthm}} | ||
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</wikitex> | </wikitex> | ||
[[Category:Exercises]] | [[Category:Exercises]] | ||
+ | [[Category:Exercises without solution]] |
Latest revision as of 15:05, 1 April 2012
Exercise 0.1.
- Show that the operations (1)-(5) from [Lück2001, Section 1.4] used in the definition of the Whitehead group in fact yield equivalence classes of invertible matrices of arbitrary size.
- Show that addition is well-defined in the Whitehead group via
Here and are invertible matricies over the group ring , is an matrix, is an matrix with , denotes the zero and matricies and denotes matrix multiplication.
- Show that is trivial.