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 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 GL_n(R) the group of invertible n-by-n matrices with elements in R. GL_n(R) embeds in GL_{n+1}(R) by attaching a 1 along the diagonal and 0s elsewhere. The direct limit of these groups via these maps is denoted GL(R) and its classifying space is denoted BGL(R). The plus construction may then be applied to the perfect normal subgroup E(R) of GL(R) = \pi_1(BGL(R)), generated by matrices which only differ from the identity matrix in one off-diagonal entry. For i>0, the nth homotopy group of the resulting space, BGL(R)^+ is the nth K-group of R, K_n(R).

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