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 Kervaire (1969), 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 , attach two-cells along loops in 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 is a unital ring, we denote by the group of invertible -by- matrices with elements in . embeds in by attaching a along the diagonal and s elsewhere. The direct limit of these groups via these maps is denoted and its classifying space is denoted . The plus construction may then be applied to the perfect normal subgroup of , generated by matrices which only differ from the identity matrix in one off-diagonal entry. For , the th homotopy group of the resulting space, is the th -group of , .
- Adams (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 82–95, ISBN 0-691-08206-5
- Kervaire, Michel A. (1969), "Smooth homology spheres and their fundamental groups", Transactions of the American Mathematical Society, 144: 67–72, doi:10.2307/1995269, ISSN 0002-9947, MR 0253347
- Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: I", Annals of Mathematics. Second Series, 94 (3): 549–572, doi:10.2307/1970770.
- Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: II", Annals of Mathematics. Second Series, 94 (3): 573–602, doi:10.2307/1970771.
- Quillen, Daniel (1972), "On the cohomology and K-theory of the general linear groups over a finite field", Annals of Mathematics. Second Series, 96 (3): 552–586, doi:10.2307/1970825.