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 ${\displaystyle X}$ is a based connected CW complex and ${\displaystyle P}$ is a perfect normal subgroup of ${\displaystyle \pi _{1}(X)}$ then a map ${\displaystyle f\colon X\to Y}$ is called a +-construction relative to ${\displaystyle P}$ if ${\displaystyle f}$ induces an isomorphism on homology, and ${\displaystyle P}$ is the kernel of ${\displaystyle \pi _{1}(X)\to \pi _{1}(Y)}$.[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 ${\displaystyle X}$, 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 ${\displaystyle \operatorname {GL} _{n}(R)}$ the group of invertible ${\displaystyle n}$-by-${\displaystyle n}$ matrices with elements in ${\displaystyle R}$. ${\displaystyle \operatorname {GL} _{n}(R)}$ embeds in ${\displaystyle \operatorname {GL} _{n+1}(R)}$ by attaching a ${\displaystyle 1}$ along the diagonal and ${\displaystyle 0}$s elsewhere. The direct limit of these groups via these maps is denoted ${\displaystyle \operatorname {GL} (R)}$ and its classifying space is denoted ${\displaystyle B\operatorname {GL} (R)}$. The plus construction may then be applied to the perfect normal subgroup ${\displaystyle E(R)}$ of ${\displaystyle \operatorname {GL} (R)=\pi _{1}(B\operatorname {GL} (R))}$, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For ${\displaystyle n>0}$, the ${\displaystyle n}$-th homotopy group of the resulting space, ${\displaystyle B\operatorname {GL} (R)^{+}}$, is isomorphic to the ${\displaystyle n}$-th ${\displaystyle K}$-group of ${\displaystyle R}$, that is,

${\displaystyle \pi _{n}\left(B\operatorname {GL} (R)^{+}\right)\cong K_{n}(R).}$