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 {\displaystyle X} is a based connected CW complex and P {\displaystyle P} is a perfect normal subgroup of π 1 ( X ) {\displaystyle \pi _{1}(X)} then a map f : X → Y {\displaystyle f\colon X\to Y} is called a +-construction relative to P {\displaystyle P} if f {\displaystyle f} induces an isomorphism on homology, and P {\displaystyle P} is the kernel of π 1 ( X ) → π 1 ( Y ) {\displaystyle \pi _{1}(X)\to \pi _{1}(Y)} . The plus construction was introduced by Michel 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 X {\displaystyle X} , attach two-cells along loops in X {\displaystyle 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 {\displaystyle R} is a unital ring, we denote by GL n ( R ) {\displaystyle \operatorname {GL} _{n}(R)} the group of invertible n {\displaystyle n} -by- n {\displaystyle n} matrices with elements in R {\displaystyle R} . GL n ( R ) {\displaystyle \operatorname {GL} _{n}(R)} embeds in GL n + 1 ( R ) {\displaystyle \operatorname {GL} _{n+1}(R)} by attaching a 1 {\displaystyle 1} along the diagonal and 0 {\displaystyle 0} s elsewhere. The direct limit of these groups via these maps is denoted GL ( R ) {\displaystyle \operatorname {GL} (R)} and its classifying space is denoted B GL ( R ) {\displaystyle B\operatorname {GL} (R)} . The plus construction may then be applied to the perfect normal subgroup E ( R ) {\displaystyle E(R)} of GL ( R ) = π 1 ( B GL ( R ) ) {\displaystyle \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 {\displaystyle n>0} , the n {\displaystyle n} -th homotopy group of the resulting space, B GL ( R ) + {\displaystyle B\operatorname {GL} (R)^{+}} , is isomorphic to the n {\displaystyle n} -th K {\displaystyle K} -group of R {\displaystyle R} , that is,
π n ( B GL ( R ) + ) ≅ K n ( R ) . {\displaystyle \pi _{n}\left(B\operatorname {GL} (R)^{+}\right)\cong K_{n}(R).}
See also Semi-s-cobordism
References
Adams, J. Frank (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, JSTOR 1995269, 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, JSTOR 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, JSTOR 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, JSTOR 1970825.
External links "Plus-construction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
