Whitehead torsion V (Ex)

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Exercise 0.1.

  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.
  2. Show that addition is well-defined in the Whitehead group via
\displaystyle  [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.

  1. Show that Wh(e) is trivial.
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