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The Whitehead group of the trivial group is zero

Statement

For the trivial group $1$ , one has $K_{1} (Z [1]) ≅ {\pm 1}$ by the determinant and hence $Wh (1) = 0$ . Consequently a homotopy equivalence between finite simply connected CW complexes is simple.

Facts & Assumptions

Given: The trivial group and finite simply connected CW complexes for the consequence.

[F1]

$K_{1} (R) = G L (R) / E (R)$ and $Wh (1) = K_{1} (Z) / ⟨ [- 1] ⟩$ (K₁ of a ring and the Whitehead group of a discrete group).

[F2]

Integer division and Bézout operations reduce a finite list of integers of gcd $1$ to a list with a single $\pm 1$ by elementary additions and swaps (Division with remainder in $Z$ : for $a \in Z$ and $b > 0$ there are unique $q, r \in Z$ with $a = q b + r$ and $0 \leq r < b$ , Bézout's identity: for integers $a, b$ not both zero, $\gcd (a, b)$ is the least positive element of ${a x + b y : x, y \in Z}$ ; in particular $a x + b y = \gcd (a, b)$ has an integer solution).

[F3]

A finite CW homotopy equivalence is simple precisely when its Whitehead torsion vanishes (Whitehead torsion is the complete obstruction to finite CW simple homotopy).

Proof

technique · direct

1.1

Since $Z [1] = Z$ , integer determinant sends $G L_{m} (Z)$ to ${\pm 1}$ and sends each elementary matrix to $1$ . It therefore induces a homomorphism $K_{1} (Z) \to {\pm 1}$ , which is onto because the one-by-one matrices $(1)$ and $(- 1)$ occur.

2.1

Let $A \in G L_{m} (Z)$ . Its first column is primitive: if an integer $d > 1$ divided every entry, it would divide the determinant $\pm 1$ , a contradiction. By [F2], elementary integer row additions reduce this column to $(\pm 1, 0, \dots, 0)^{T}$ ; a row swap is a product of elementary matrices and a diagonal $- 1$ , so its Whitehead class is accounted for by $[- 1]$ . Clear the remainder of the first row by elementary column additions and repeat on the invertible $(m - 1) \times (m - 1)$ minor. Induction gives an elementary-equivalent diagonal matrix with entries $\pm 1$ . Stabilized diagonal $- 1$ entries add to a single $[- 1]$ class, since $[- 1] + [- 1] = [1] = 0$ . Hence the determinant homomorphism is injective, and $K_{1} (Z) ≅ {\pm 1}$ .

F1F2step 1.1

3.1

The quotient defining $Wh (1)$ kills this entire two-element group, so $Wh (1) = 0$ . A finite simply connected homotopy equivalence has torsion in $Wh (1)$ on each connected component and thus has zero torsion; [F3] makes it simple. ∎

F1F3step 2.1

Depends on

Used by

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Sources

Lück, §2.1, printed p.25 (standard reference, not scraped)
Lurie, Example 9, printed p.3 (standard reference, not scraped)