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

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 [math]\displaystyle{ X }[/math], attach two-cells along loops in [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ R }[/math] is a unital ring, we denote by [math]\displaystyle{ \operatorname{GL}_n(R) }[/math] the group of invertible [math]\displaystyle{ n }[/math]-by-[math]\displaystyle{ n }[/math] matrices with elements in [math]\displaystyle{ R }[/math]. [math]\displaystyle{ \operatorname{GL}_n(R) }[/math] embeds in [math]\displaystyle{ \operatorname{GL}_{n+1}(R) }[/math] by attaching a [math]\displaystyle{ 1 }[/math] along the diagonal and [math]\displaystyle{ 0 }[/math]s elsewhere. The direct limit of these groups via these maps is denoted [math]\displaystyle{ \operatorname{GL}(R) }[/math] and its classifying space is denoted [math]\displaystyle{ B\operatorname{GL}(R) }[/math]. The plus construction may then be applied to the perfect normal subgroup [math]\displaystyle{ E(R) }[/math] of [math]\displaystyle{ \operatorname{GL}(R) = \pi_1(B\operatorname{GL}(R)) }[/math], generated by matrices which only differ from the identity matrix in one off-diagonal entry. For [math]\displaystyle{ n\gt 0 }[/math], the [math]\displaystyle{ n }[/math]-th homotopy group of the resulting space, [math]\displaystyle{ B\operatorname{GL}(R)^+ }[/math], is isomorphic to the [math]\displaystyle{ n }[/math]-th [math]\displaystyle{ K }[/math]-group of [math]\displaystyle{ R }[/math], that is,

[math]\displaystyle{ \pi_n\left( B\operatorname{GL}(R)^+\right) \cong K_n(R). }[/math]

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 
• 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 .

External links

• Hazewinkel, Michiel, ed. (2001), "Plus-construction", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Plus-construction