Explicitly, if is a based connected CW complex and is a perfectnormal subgroup of then a map is called a +-construction relative to if induces an isomorphism on homology, and is the kernel of .[1]
The plus construction was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfectnormal subgroup of the fundamental group of a connectedCW complex, attach two-cells along loops in 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 is a unitalring, we denote by the group of invertible-by-matrices with elements in . embeds in by attaching a along the diagonal and s elsewhere. The direct limit of these groups via these maps is denoted and its classifying space is denoted . The plus construction may then be applied to the perfect normal subgroup of , generated by matrices which only differ from the identity matrix in one off-diagonal entry. For , the -th homotopy group of the resulting space, , is isomorphic to the -th -group of , that is,