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 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 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 n > 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 ) .
