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 formula_1 is a based connected CW complex and formula_2 is a perfect normal subgroup of formula_3 then a map formula_4 is called a +-construction relative to formula_2 if formula_6 induces an isomorphism on homology, and formula_2 is the kernel of formula_8.
The plus construction was introduced by Michel 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 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_11 is a unital ring, we denote by formula_12 the group of invertible formula_13-by-formula_13 matrices with elements in formula_11. formula_12 embeds in formula_17 by attaching a formula_18 along the diagonal and formula_19s elsewhere. The direct limit of these groups via these maps is denoted formula_20 and its classifying space is denoted formula_21. The plus construction may then be applied to the perfect normal subgroup formula_22 of formula_23, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For formula_24, the formula_13-th homotopy group of the resulting space, formula_26, is isomorphic to the formula_13-th formula_28-group of formula_11, that is, 
 formula_30
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