Whitehead torsion V (Ex)
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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 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 $A$ 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. | # Show that $Wh(e)$ is trivial. | ||
{{endthm}} | {{endthm}} |
Revision as of 12:13, 26 March 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 is an matrix, is an matrix with , denotes the zero and matricies and denotes matrix multiplication.
- Show that is trivial.
The exercises and hints on this page were sent by Alex Koenen and Arkadi Schelling.