# Plus construction

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In [mathematics](/source/mathematics), the '''plus construction''' is a method for simplifying the [fundamental group](/source/fundamental_group) of a space without changing its [homology](/source/homology_(mathematics)) and [cohomology](/source/cohomology) [group](/source/group_(mathematics))s. 

Explicitly, if <math>X</math> is a based connected [CW complex](/source/CW_complex) and <math>P</math> is a [perfect](/source/Perfect_group) [normal subgroup](/source/normal_subgroup) of <math>\pi_1(X)</math> then a map <math>f\colon X \to Y</math> is called a +-construction relative to <math>P</math> if <math>f</math> induces an isomorphism on homology, and <math>P</math> is the kernel of <math>\pi_1(X) \to \pi_1(Y)</math>.<ref>[Charles Weibel](/source/Charles_Weibel), ''An introduction to algebraic K-theory'' IV, Definition 1.4.1</ref>

The plus construction was introduced by {{harvs|txt|author-link=Michel Kervaire|last=Kervaire|first=Michel|year=1969}}, and was used by [Daniel Quillen](/source/Daniel_Quillen) to define [algebraic K-theory](/source/algebraic_K-theory). Given a [perfect](/source/perfect_group) [normal subgroup](/source/normal_subgroup) of the fundamental group of a [connected](/source/Connected_space) [CW complex](/source/CW_complex) <math>X</math>, attach two-cells along loops in <math>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>R</math> is a [unital](/source/unital_algebra) [ring](/source/ring_(mathematics)), we denote by <math>\operatorname{GL}_n(R)</math> the group of [invertible](/source/invertible_matrix) <math>n</math>-by-<math>n</math> [matrices](/source/matrix_(mathematics)) with elements in <math>R</math>. <math>\operatorname{GL}_n(R)</math> embeds in <math>\operatorname{GL}_{n+1}(R)</math> by attaching a <math>1</math> along the diagonal and <math>0</math>s elsewhere. The [direct limit](/source/direct_limit) of these groups via these maps is denoted <math>\operatorname{GL}(R)</math> and its [classifying space](/source/classifying_space) is denoted <math>B\operatorname{GL}(R)</math>. The plus construction may then be applied to the perfect normal subgroup <math>E(R)</math> of <math>\operatorname{GL}(R) = \pi_1(B\operatorname{GL}(R))</math>, generated by matrices which only differ from the [identity matrix](/source/identity_matrix) in one off-diagonal entry. For <math>n>0</math>, the <math>n</math>-th [homotopy group](/source/homotopy_group) of the resulting space, <math>B\operatorname{GL}(R)^+</math>, is isomorphic to the <math>n</math>-th <math>K</math>-group of <math>R</math>, that is, 
: <math>\pi_n\left( B\operatorname{GL}(R)^+\right) \cong K_n(R).</math>

==See also==
* [Semi-s-cobordism](/source/Semi-s-cobordism)

==References==
{{Reflist}}

*{{citation|last=Adams|first=J. Frank| authorlink=Frank Adams| year=1978|title=Infinite loop spaces|pages=82–95|isbn=0-691-08206-5|publisher=[Princeton University Press](/source/Princeton_University_Press)|location=Princeton, N.J.}}
*{{Citation | last1=Kervaire | first1=Michel A. | authorlink=Michel Kervaire| title=Smooth homology spheres and their fundamental groups |mr=0253347 | year=1969 | journal=[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society) | issn=0002-9947 | volume=144 | pages=67–72 | doi=10.2307/1995269| doi-access=free | jstor=1995269 }}
*{{citation|first=Daniel|last= Quillen|authorlink=Daniel Quillen| title=The Spectrum of an Equivariant Cohomology Ring: I|journal= [Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series|volume= 94|issue=3 |year=1971| pages= 549–572|doi=10.2307/1970770 |jstor= 1970770}}.
*{{citation|first=Daniel|last= Quillen|authorlink=Daniel Quillen|title=The Spectrum of an Equivariant Cohomology Ring: II|journal= [Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series|volume= 94|issue=3 |year=1971| pages= 573–602|doi=10.2307/1970771 |jstor= 1970771}}.
*{{citation|first=Daniel|last= Quillen|authorlink=Daniel Quillen|title=On the cohomology and K-theory of the general linear groups over a finite field|journal= [Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series|volume= 96|issue=3 |year=1972| pages= 552–586|doi=10.2307/1970825|jstor= 1970825}}.

==External links==
*{{eom|id=Plus-construction}}

Category:Algebraic topology
Category:Homotopy theory

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Adapted from the Wikipedia article [Plus construction](https://en.wikipedia.org/wiki/Plus_construction) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Plus_construction?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.