K₁ of a ring and the Whitehead group of a discrete group

Let $R$ be an associative unital ring and let $\mathrm{G}\mathrm{L}(R)\supseteq \mathrm{E}(R)$ be its stable groups (Stable general linear and elementary groups for right modules). By Stable elementary matrices equal the commutator subgroup, $\mathrm{E}(R)$ is normal in $\mathrm{G}\mathrm{L}(R)$ and equals the commutator subgroup, so the quotient $K_1(R):=\mathrm{G}\mathrm{L}(R)\mathrm{/}\mathrm{E}(R)$ is an abelian group. This group is written additively: for $A\in {\mathrm{G}\mathrm{L}}_n(R)$ the symbol $[A]$ denotes the class of the stabilized matrix $A$ in $K_1(R)$, and the group law is the one induced by matrix multiplication, so $[AB]=[A]+[B],[I_n]=0,[A^{-1}]=-[A]$ for compatible $A,B$; the class $[A]$ is unchanged by stabilization and equals the class of $\mathrm{diag}(A,I_m)$ for every $m$. The reduced group ${\tilde{K}}_1(R):=K_1(R)\mathrm{/}⟨[-1]⟩$ is the quotient of $K_1(R)$ by the subgroup generated by the class of the $1\times 1$ matrix $-1$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups); in ${\tilde{K}}_1(R)$ one has $[-A]=[A]$ for every invertible $A$.

Now let $\pi$ be a discrete group, with integral group ring $\mathbb{Z}[\pi ]$, a unital ring with basis the classes $[g]$ of the group elements (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$). The Whitehead group of $\pi$ is $\mathrm{W}\mathrm{h}(\pi ):=K_1(\mathbb{Z}[\pi ])/⟨[\pm g]:g\in \pi ⟩,$ the quotient by the subgroup generated by the classes of the $1\times 1$ unit matrices $(\pm g)$. Reading the generators in the order $[-1]$, then $[g]=[-1]+[-g]$, exhibits the same subgroup as $⟨[-1]⟩+⟨[g]:g\in \pi ⟩$, so there is a natural identification $\mathrm{W}\mathrm{h}(\pi )={\tilde{K}}_1(\mathbb{Z}[\pi ])/⟨[g]:g\in \pi ⟩\mathrm{.}$

Functoriality and well-definedness. A unital ring homomorphism $R\to R^{\mathrm{\prime }}$ carries invertible matrices to invertible matrices and elementary matrices to elementary matrices, hence induces $K_1(R)\to K_1(R^{\mathrm{\prime }})$ and ${\tilde{K}}_1(R)\to {\tilde{K}}_1(R^{\mathrm{\prime }})$; a group homomorphism $\phi :\pi \to {\pi }^{\mathrm{\prime }}$ induces the unital ring homomorphism $\mathbb{Z}[\pi ]\to \mathbb{Z}[{\pi }^{\mathrm{\prime }}]$, $[g]\mapsto [\phi (g)]$, which sends $[\pm g]$ to $[\pm \phi (g)]$ and therefore descends to a homomorphism $\mathrm{W}\mathrm{h}(\pi )\to \mathrm{W}\mathrm{h}({\pi }^{\mathrm{\prime }})$. These assignments are compatible with composition and preserve identities. If $\phi$ is an inner automorphism of $\pi$, given by $c_g(h)=gh{g}^{-1}$, then the induced ring automorphism of $\mathbb{Z}[\pi ]$ is conjugation by the unit $[g]$, so on ${\mathrm{G}\mathrm{L}}_n(\mathbb{Z}[\pi ])$ it acts as $A\mapsto D_{g}A{D}_g^{-1}$ with $D_g$ the scalar matrix $g{I}_n$; conjugation by a fixed invertible matrix is the identity on the quotient $K_1$, because $[D_{g}A{D}_g^{-1}]=[D_g]+[A]-[D_g]=[A]$. Hence inner automorphisms of $\pi$ induce the identity on $\mathrm{W}\mathrm{h}(\pi )$.