# 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. It was introduced by 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 ${\displaystyle X}$, attach two-cells along loops in ${\displaystyle X}$ 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 ${\displaystyle R}$ is a unital ring, we denote by ${\displaystyle GL_{n}(R)}$ the group of invertible ${\displaystyle n}$-by-${\displaystyle n}$ matrices with elements in ${\displaystyle R}$. ${\displaystyle GL_{n}(R)}$ embeds in ${\displaystyle GL_{n+1}(R)}$ by attaching a ${\displaystyle 1}$ along the diagonal and ${\displaystyle 0}$s elsewhere. The direct limit of these groups via these maps is denoted ${\displaystyle GL(R)}$ and its classifying space is denoted ${\displaystyle BGL(R)}$. The plus construction may then be applied to the perfect normal subgroup ${\displaystyle E(R)}$ of ${\displaystyle GL(R)=\pi _{1}(BGL(R))}$, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For ${\displaystyle n>0}$, the ${\displaystyle n}$th homotopy group of the resulting space, ${\displaystyle BGL(R)^{+}}$ is the ${\displaystyle n}$th ${\displaystyle K}$-group of ${\displaystyle R}$, ${\displaystyle K_{n}(R)}$.