Plus construction

In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.

Explicitly, if $X$ is a based connected CW complex and $P$ is a perfect normal subgroup of $\pi _{1}(X)$ then a map $f\colon X\to Y$ is called a +-construction relative to $P$ if $f$ induces an isomorphism on homology, and $P$ is the kernel of $\pi _{1}(X)\to \pi _{1}(Y)$ .^[1]

The plus construction was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex $X$ , attach two-cells along loops in $X$ whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.

The most common application of the plus construction is in algebraic K-theory. If $R$ is a unital ring, we denote by $\operatorname {GL} _{n}(R)$ the group of invertible $n$ -by- $n$ matrices with elements in $R$ . $\operatorname {GL} _{n}(R)$ embeds in $\operatorname {GL} _{n+1}(R)$ by attaching a $1$ along the diagonal and $0$ s elsewhere. The direct limit of these groups via these maps is denoted $\operatorname {GL} (R)$ and its classifying space is denoted $B\operatorname {GL} (R)$ . The plus construction may then be applied to the perfect normal subgroup $E(R)$ of $\operatorname {GL} (R)=\pi _{1}(B\operatorname {GL} (R))$ , generated by matrices which only differ from the identity matrix in one off-diagonal entry. For $n>0$ , the $n$ -th homotopy group of the resulting space, $B\operatorname {GL} (R)^{+}$ , is isomorphic to the $n$ -th $K$ -group of $R$ , that is,