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 $X$ 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 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 $X$, 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 $B\operatorname {GL} (R)$. The plus construction may then be applied to the perfect normal subgroup $E(R)$ of $\operatorname {GL} (R)=\pi _{1}(B\operatorname {GL} (R))$, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For $n>0$, the $n$-th homotopy group of the resulting space, $B\operatorname {GL} (R)^{+}$, is isomorphic to the $n$-th $K$-group of $R$, that is,

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

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]