Plus construction

Article summary

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) .…

Key claims

No claims recorded.

Discussion questions

1. Summarise the article in your own words. Which single claim would you verify first, and why?
2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

No sources recorded.