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 $π_{1} (X)$ then a map $f : X \to Y$ is called a +-construction relative to $P$ if $f$ induces an isomorphism on homology, and $P$ is the kernel of $π_{1} (X) \to π_{1} (Y)$ .^[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 $X$ , 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 ${GL}_{n} (R)$ the group of invertible $n$ -by- $n$ matrices with elements in $R$ . ${GL}_{n} (R)$ embeds in ${GL}_{n + 1} (R)$ by attaching a $1$ along the diagonal and $0$ s elsewhere. The direct limit of these groups via these maps is denoted $GL (R)$ and its classifying space is denoted $B GL (R)$ . The plus construction may then be applied to the perfect normal subgroup $E (R)$ of $GL (R) = π_{1} (B GL (R))$ , 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, $B GL (R)^{+}$ , is isomorphic to the $n$ -th $K$ -group of $R$ , that is,

π_{n} (B GL (R)^{+}) ≅ K_{n} (R) .

References

↑ Charles Weibel, An introduction to algebraic K-theory IV, Definition 1.4.1

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]

[1] Charles Weibel, An introduction to algebraic K-theory IV, Definition 1.4.1

[1]

See also

References

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