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 ${\displaystyle X}$ is a based connected CW complex and ${\displaystyle P}$ is a perfect normal subgroup of ${\displaystyle \pi _{1}(X)}$ then a map ${\displaystyle f\colon X\to Y}$ is called a +-construction relative to ${\displaystyle P}$ if ${\displaystyle f}$ induces an isomorphism on homology, and ${\displaystyle P}$ is the kernel of ${\displaystyle \pi _{1}(X)\to \pi _{1}(Y)}$.^[1]

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

${\displaystyle \pi _{n}\left(B\operatorname {GL} (R)^{+}\right)\cong K_{n}(R).}$

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, 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.

External links

• "Plus-construction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]