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 $$X$$ is a based connected CW complex and $$P$$ is a perfect normal subgroup of $$\pi_1(X)$$ then a map $$f\colon X \to Y$$ is called a +-construction relative to $$P$$ if $$f$$ induces an isomorphism on homology, and $$P$$ is the kernel of $$\pi_1(X) \to \pi_1(Y)$$.

The plus construction 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 $$X$$, 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 $$\operatorname{GL}_n(R)$$ the group of invertible $$n$$-by-$$n$$ matrices with elements in $$R$$. $$\operatorname{GL}_n(R)$$ embeds in $$\operatorname{GL}_{n+1}(R)$$ by attaching a $$1$$ along the diagonal and $$0$$s elsewhere. The direct limit of these groups via these maps is denoted $$\operatorname{GL}(R)$$ and its classifying space is denoted $$B\operatorname{GL}(R)$$. The plus construction may then be applied to the perfect normal subgroup $$E(R)$$ of $$\operatorname{GL}(R) = \pi_1(B\operatorname{GL}(R))$$, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For $$n>0$$, the $$n$$-th homotopy group of the resulting space, $$B\operatorname{GL}(R)^+$$, is isomorphic to the $$n$$-th $$K$$-group of $$R$$, that is,
 * $$\pi_n\left( B\operatorname{GL}(R)^+\right) \cong K_n(R).$$