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Every Whitehead class is represented by an invertible matrix and conversely

Statement

Let $π$ be a group and $R = Z [π]$ . For every $u \in Wh (π)$ there are $c \geq 1$ and an invertible matrix $A \in {G L}_{c} (R)$ whose class in $K_{1} (R)$ maps to $u$ under the quotient $K_{1} (R) \to Wh (π)$ . Conversely every invertible matrix over $R$ defines an element of $Wh (π)$ by this quotient. Stabilizing a representative by $A \mapsto A \oplus I_{b}$ does not change the class.

Facts & Assumptions

Given: A group $π$ , its integral group ring $R = Z [π]$ , and an element $u \in Wh (π)$ .

[F1]

The stable general linear group is the union $G L (R) = ⋃_{n \geq 0} {G L}_{n} (R)$ along the stabilizations $A \mapsto diag (A, 1)$ , so every element of $G L (R)$ is represented by an invertible $n \times n$ matrix for some finite $n$ , and a matrix is invertible when it has a two-sided inverse; the class $[A] \in K_{1} (R)$ of an invertible matrix is unchanged by stabilization, that is $[diag (A, I_{b})] = [A]$ , and the group law satisfies $[A B] = [A] + [B]$ and $[I_{n}] = 0$ (Stable general linear and elementary groups for right modules, K₁ of a ring and the Whitehead group of a discrete group, Invertible matrices and the general linear group ${GL}_{n} (F)$ ).

[F2]

$K_{1} (R) = G L (R) / E (R)$ is the quotient of the stable general linear group by the stable elementary subgroup, and $Wh (π) = K_{1} (Z [π]) / ⟨ [\pm g] : g \in π ⟩$ is the further quotient by the subgroup generated by the classes of the $1 \times 1$ units $\pm g$ ; the quotient map $K_{1} (Z [π]) \to Wh (π)$ is the canonical projection (K₁ of a ring and the Whitehead group of a discrete group, The group ring $R [G]$ of finitely supported formal $R$ -linear combinations of group elements).

Proof

1.1F1F2

By [F2] the group $K_{1} (R)$ is the quotient of $G L (R)$ by $E (R)$ , so the canonical projection $G L (R) \to K_{1} (R)$ is surjective. By [F1] every element of $G L (R)$ is represented by an invertible $c \times c$ matrix over $R$ for some finite $c \geq 0$ , and $K_{1} (R)$ is the quotient of that group, so every class in $K_{1} (R)$ is the class $[A]$ of such a matrix.

2.1F1F2step 1.1

By [F2] the group $Wh (π)$ is the quotient of $K_{1} (R)$ by the subgroup generated by the classes of the units $\pm g$ , so the projection $q : K_{1} (R) \to Wh (π)$ is surjective: for the given $u \in Wh (π)$ there is a class $[A] \in K_{1} (R)$ with $q ([A]) = u$ . Combining this with step 1.1 exhibits an invertible matrix $A$ of some size $c \geq 1$ whose class in $K_{1} (R)$ maps to $u$ ; if $u = 0$ one may take $A = I_{1}$ .

2.2F1F2step 1.1

Conversely, if $A \in {G L}_{c} (R)$ is invertible, then by [F1] it represents the class $[A] \in K_{1} (R)$ of the stabilized matrix $diag (A, I_{b})$ for every $b \geq 0$ , and [F2] turns this class into the element $q ([A]) \in Wh (π)$ . Stabilization does not change the underlying stable class, because $diag (A, I_{b})$ is by definition the image of $A$ in ${G L}_{c + b} (R)$ under the iterated stabilization, and hence $[diag (A, I_{b})] = [A]$ in $K_{1} (R)$ and $q ([diag (A, I_{b})]) = q ([A])$ in $Wh (π)$ .

3.1step 2.1step 2.2∎

Steps 2.1 and 2.2 give the two asserted directions, and step 2.2 also gives the stabilization statement; the argument is a direct reading of the definitions of $K_{1}$ and of $Wh$ and uses no choice principle and no property of the group $π$ beyond the definition of its group ring.

Depends on

Used by

Realization of prescribed Whitehead torsion by h-cobordisms Proposition

Dependency tree · two levels

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Sources

Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, electronic edition) (standard reference, not scraped)