Plus construction

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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 Daniel Quillen. 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. If the aforementioned subgroup is the entire fundamental group, this second step is unnecessary.

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

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[1] Daniel Quillen, The Spectrum of an Equivariant Cohomology Ring: I, Annals of Mathematics, 2nd Ser., Vol. 94, No. 3 (Nov., 1971), pp. 549-572.
[2] Daniel Quillen, The Spectrum of an Equivariant Cohomology Ring: II, Annals of Mathematics, 2nd Ser., Vol. 94, No. 3 (Nov., 1971), pp. 573-602.
[3] Daniel Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Annals of Mathematics, 2nd Ser., Vol. 96, No. 3 (Nov., 1972), pp. 552-586.