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. It was introduced by , 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 formula_1, attach two-cells along loops in formula_1 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 formula_3 is a unital ring, we denote by formula_4 the group of invertible formula_5-by-formula_5 matrices with elements in formula_3. formula_4 embeds in formula_9 by attaching a formula_10 along the diagonal and formula_11s elsewhere. The direct limit of these groups via these maps is denoted formula_12 and its classifying space is denoted formula_13. The plus construction may then be applied to the perfect normal subgroup formula_14 of formula_15, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For formula_16, the formula_5-th homotopy group of the resulting space, formula_18, is isomorphic the formula_5-th formula_20-group of formula_3, that is, 