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| In [[mathematics]], the '''plus construction''' is a method for simplifying the [[fundamental group]] of a space without changing its [[homology (mathematics)|homology]] and [[cohomology]] [[group (mathematics)|group]]s. It was introduced by {{harvs|txt|authorlink=Michel Kervaire|last=Kervaire|year=1969}}, and was used [[Daniel Quillen]] to define algebraic K-theory. Given a [[perfect group|perfect]] [[normal subgroup]] of the [[fundamental group]] of a connected [[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.
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| The most common application of the plus construction is in [[algebraic K-theory]]. If <math>R</math> is a [[unital algebra|unital]] [[ring (mathematics)|ring]], we denote by <math>GL_n(R)</math> the group of [[inverse element|invertible]] <math>n</math>-by-<math>n</math> [[matrix (mathematics)|matrices]] with elements in <math>R</math>. <math>GL_n(R)</math> embeds in <math>GL_{n+1}(R)</math> by attaching a <math>1</math> along the diagonal and <math>0</math>s elsewhere. The [[direct limit]] of these groups via these maps is denoted <math>GL(R)</math> and its [[classifying space]] is denoted <math>BGL(R)</math>. The plus construction may then be applied to the perfect normal subgroup <math>E(R)</math> of <math>GL(R) = \pi_1(BGL(R))</math>, generated by matrices which only differ from the [[identity matrix]] in one off-diagonal entry. For <math>i>0</math>, the <math>n</math>th [[homotopy group]] of the resulting space, <math>BGL(R)^+</math> is the <math>n</math>th <math>K</math>-group of <math>R</math>, <math>K_n(R)</math>.
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| == See also ==
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| * [[Semi-s-cobordism]]
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| ==References==
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| *{{citation|last=Adams|year=1978|title=Infinite loop spaces|pages=82–95|isbn=0-691-08206-5|publisher=Princeton University Press|location=Princeton, N.J.}}
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| *{{Citation | last1=Kervaire | first1=Michel A. | title=Smooth homology spheres and their fundamental groups | id={{MR|0253347}} | year=1969 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=144 | pages=67–72 | doi=10.2307/1995269}}
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| *{{citation|first=Daniel|last= Quillen|title=The Spectrum of an Equivariant Cohomology Ring: I|journal= [[Annals of Mathematics. Second Series]]|volume= 94|issue=3 |year=1971| pages= 549–572|doi=10.2307/1970770 }}.
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| *{{citation|first=Daniel|last= Quillen|title=The Spectrum of an Equivariant Cohomology Ring: II|journal= [[Annals of Mathematics. Second Series]]|volume= 94|issue=3 |year=1971| pages= 573–602|doi=10.2307/1970771 }}.
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| *{{citation|first=Daniel|last= Quillen|title=On the cohomology and K-theory of the general linear groups over a finite field|journal= [[Annals of Mathematics. Second Series]]|volume= 96|issue=3 |year=1972| pages= 552–586|doi=10.2307/1970825}}.
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| ==External links==
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| *{{eom|id=Plus-construction}}
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| {{topology-stub}}
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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Latest revision as of 15:23, 26 November 2014
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