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) .…

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). ==See also== Semi-s-cobordism ==References== . . . ==External links==

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