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 $\backslash pi\_1(X)$ then a map $f\backslash colon\; X\; \backslash to\; Y$ is called a +-construction relative to $P$ if $f$ induces an isomorphism on homology, and $P$ is the kernel of $\backslash pi\_1(X)\; \backslash to\; \backslash pi\_1(Y)$.Charles Weibel, An introduction to algebraic K-theory IV, Definition 1.4.1

The plus construction was introduced by {{harvs|txt|author-link=Michel Kervaire|last=Kervaire|first=Michel|year=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 $\backslash operatorname\{GL\}\_n(R)$ the group of invertible $n$-by-$n$ matrices with elements in $R$. $\backslash operatorname\{GL\}\_n(R)$ embeds in $\backslash operatorname\{GL\}\_\{n+1\}(R)$ by attaching a $1$ along the diagonal and $0$s elsewhere. The direct limit of these groups via these maps is denoted $\backslash operatorname\{GL\}(R)$ and its classifying space is denoted $B\backslash operatorname\{GL\}(R)$. The plus construction may then be applied to the perfect normal subgroup $E(R)$ of $\backslash operatorname\{GL\}(R)\; =\; \backslash pi\_1(B\backslash operatorname\{GL\}(R))$, 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, $B\backslash operatorname\{GL\}(R)^+$, is isomorphic to the $n$-th $K$-group of $R$, that is,

: $\backslash pi\_n\backslash left(\; B\backslash operatorname\{GL\}(R)^+\backslash right)\; \backslash cong\; K\_n(R).$