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.
Explicitly, if <math>X</math> is a based connected CW complex and <math>P</math> is a perfect normal subgroup of <math>\pi_1(X)</math> then a map <math>f\colon X \to Y</math> is called a +-construction relative to <math>P</math> if <math>f</math> induces an isomorphism on homology, and <math>P</math> is the kernel of <math>\pi_1(X) \to \pi_1(Y)</math>.<ref>Charles Weibel, An introduction to algebraic K-theory IV, Definition 1.4.1</ref>
The plus construction was introduced by Template:Harvs, 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 <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.
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>\operatorname{GL}_n(R)</math> the group of invertible <math>n</math>-by-<math>n</math> matrices with elements in <math>R</math>. <math>\operatorname{GL}_n(R)</math> embeds in <math>\operatorname{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>\operatorname{GL}(R)</math> and its classifying space is denoted <math>B\operatorname{GL}(R)</math>. The plus construction may then be applied to the perfect normal subgroup <math>E(R)</math> of <math>\operatorname{GL}(R) = \pi_1(B\operatorname{GL}(R))</math>, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For <math>n>0</math>, the <math>n</math>-th homotopy group of the resulting space, <math>B\operatorname{GL}(R)^+</math>, is isomorphic to the <math>n</math>-th <math>K</math>-group of <math>R</math>, that is,
- <math>\pi_n\left( B\operatorname{GL}(R)^+\right) \cong K_n(R).</math>