Quillen's Plus Construction

Quillen's plus construction is one of the two main approaches to defining higher algebraic K-groups. It modifies the classifying space $BGL(R)$ by killing its fundamental group (which is $GL(R)$ ) down to the abelianization $K_1(R)$ , while preserving homology. The resulting space $BGL(R)^+$ has homotopy groups that define the higher K-groups.

The plus construction in general

Definition2.1Quillen's plus construction

Let $X$ be a connected CW complex and $N \trianglelefteq \pi_1(X)$ a perfect normal subgroup (i.e., $N = [N, N]$ ). The plus construction $X^+_N$ (or simply $X^+$ when $N$ is understood) is a CW complex together with a map $\phi: X \to X^+$ satisfying:

$\phi_*: \pi_1(X) \to \pi_1(X^+)$ is the quotient map $\pi_1(X) \twoheadrightarrow \pi_1(X)/N$ .
$\phi_*: H_*(X; M) \xrightarrow{\sim} H_*(X^+; M)$ is an isomorphism for every $\pi_1(X)/N$ -module $M$ (equivalently, for every local coefficient system on $X^+$ ).

The plus construction is unique up to homotopy equivalence (under $X$ ) when it exists. It exists whenever $N$ is perfect.

RemarkConstruction via attaching cells

The plus construction is built by:

Killing $\pi_1$ : For each generator $g$ of $N$ , attach a 2-cell along a loop representing $g$ . This kills $N$ in $\pi_1$ , giving a space $X'$ with $\pi_1(X') = \pi_1(X)/N$ .
Fixing homology: The 2-cells introduce classes in $H_2$ that were not present. Since $N$ is perfect, these can be killed by attaching 3-cells without affecting $\pi_1$ or lower homology.

Formally, one shows that the attaching maps for the 3-cells can be chosen so that the resulting space $X^+$ satisfies $H_*(X; \mathbb{Z}) \cong H_*(X^+; \mathbb{Z})$ as abelian groups (and more generally for local coefficients).

Application to K-theory

Definition2.2Higher algebraic K-groups via plus construction

For a ring $R$ , the group $E(R) = [GL(R), GL(R)]$ is perfect (this follows from the Steinberg relations). Apply the plus construction to the classifying space $BGL(R)$ with $N = E(R)$ :

$K_n(R) = \pi_n(BGL(R)^+) \quad \text{for } n \geq 1.$

By construction:

$\pi_1(BGL(R)^+) = GL(R)/E(R) = K_1(R)$
$H_*(BGL(R)^+; \mathbb{Z}) \cong H_*(BGL(R); \mathbb{Z}) = H_*(GL(R); \mathbb{Z})$

For $n = 0$ , we retain the original definition $K_0(R) =$ Grothendieck group of projective modules. The K-theory space is

$K(R) = K_0(R) \times BGL(R)^+$

so that $\pi_n(K(R)) = K_n(R)$ for all $n \geq 0$ (where $\pi_0 = K_0$ ).

ExampleK-groups of finite fields

Quillen computed the K-groups of finite fields completely. For $\mathbb{F}_q$ with $q = p^f$ :

$K_n(\mathbb{F}_q) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(q^i - 1)\mathbb{Z} & n = 2i - 1 > 0 \\ 0 & n \text{ even}, n > 0 \end{cases}$

The computation uses the fact that $BGL(\mathbb{F}_q)^+ \simeq F\psi^q$ , the homotopy fiber of $\psi^q - 1: BU \to BU$ , where $\psi^q$ is the Adams operation. This connects algebraic K-theory to topological K-theory.

For example: $K_1(\mathbb{F}_q) = \mathbb{F}_q^{\times} \cong \mathbb{Z}/(q-1)$ , $K_3(\mathbb{F}_q) \cong \mathbb{Z}/(q^2 - 1)$ .

Properties of the plus construction

ExampleGroup completion interpretation

The plus construction can be understood via group completion of the monoid

$\coprod_{n \geq 0} BGL_n(R)$

under the $H$ -space structure induced by direct sum $\oplus: GL_m(R) \times GL_n(R) \to GL_{m+n}(R)$ .

The group completion theorem (McDuff--Segal) states that

$\mathbb{Z} \times BGL(R)^+ \simeq \Omega B\left(\coprod_n BGL_n(R)\right)$

where the right side is the loop space of the classifying space of the topological monoid. This provides the connection between the plus construction and the more homotopy-theoretic approaches to K-theory.

RemarkComparison with K₂

For $n = 2$ , the plus construction gives $K_2(R) = \pi_2(BGL(R)^+)$ . Since $BGL(R)$ is a $K(GL(R), 1)$ , we have $\pi_2(BGL(R)) = 0$ . The plus construction introduces new $\pi_2$ from the attached 2-cells. One can show

$K_2(R) = \pi_2(BGL(R)^+) \cong H_2(E(R); \mathbb{Z}) \cong \ker(\operatorname{St}(R) \to E(R))$

recovering Milnor's definition. The isomorphism $K_2(R) \cong H_2(E(R); \mathbb{Z})$ follows from the Hopf formula: for a perfect group $G$ with universal central extension $\tilde{G}$ , $\pi_2(BG^+) \cong H_2(G; \mathbb{Z}) \cong \ker(\tilde{G} \to G)$ .