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 Daniel Quillen. Given a perfect normal subgroup of the fundamental group of a connected CW complex <math>X<math>, attach two-cells along loops in <math>X<math> 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. If the aforementioned subgroup is the entire fundamental group, this second step is unnecessary.
The most common application of the plus construction is in algebraic K-theory. If <math>R<math> is a unital ring, we denote by <math>GL_n(R)<math> the group of invertible <math>n<math>-by-<math>n<math> matrices with elements in <math>R<math>. <math>GL_n(R)<math> embeds in <math>GL_{n+1}(R)<math> by attaching a <math>1<math> along the diagonal and <math>0<math>s elsewhere. The direct limit of these groups via these maps is denoted <math>GL(R)<math> and its classifying space is denoted <math>BGL(R)<math>. The plus construction may then be applied to the perfect normal subgroup <math>E(R)<math> of <math>GL(R) = \pi_1(BGL(R))<math>, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For <math>i>0<math>, the <math>n<math>th homotopy group of the resulting space, <math>BGL(R)^+<math> is the <math>n<math>th <math>K<math>-group of <math>R<math>, <math>K_n(R)<math>.